Model-scale tests to examine water pressures acting on potentially buoyant underground structures in clay strata

Zhisheng Ren, Qixian Lu, Kaiwen Liu, Pengpeng Ni, Guoxiong Mei,**

a School of Civil Engineering, Southern Marine Science and Engineering Guangdong Laboratory(Zhuhai),Guangdong Key Laboratory of Oceanic Civil Engineering,Guangdong Research Center for Underground Space Exploitation Technology, Sun Yat-sen University, Guangzhou, 510275, China

b Key Laboratory of Disaster Prevention and Structural Safety of Ministry of Education,College of Civil Engineering and Architecture,Guangxi University,Nanning,530004, China

c State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan, 430072, China

d Key Laboratory of High-speed Railway Engineering (MOE), School of Civil Engineering, Southwest Jiaotong University, Chengdu, 610031, China

Keywords:Buoyant force Reservoir impoundment Pore pressure generation Reduction factor Uplift pressure Time lag effect

A B S T R A C T

1. Introduction

Underground space is increasingly utilized worldwide, especially the construction of subway systems, utility tunnels, buried pipeline networks, underground shopping malls, and parking garages. For underground structures near a reservoir, fluctuations of reservoir water level can result in the generation of pore pressure in the soil (Zhang et al., 2020), eventually influencing the development of uplift pressure. During design of underground infrastructure, the flotation problem must be taken into consideration carefully;otherwise,both transient and sustained hydrostatic uplift forces can cause instable upheaval issues (Wang et al., 2020). The action of sustained hydrostatic uplift is due to the seepage effect of groundwater (Zhang et al., 2018; Ren et al., 2020); whereas the initiation of transient uplift force results from the pore pressure generation in instantaneously deformed soils during earthquakes(Koseki et al.,1997;Chian and Madabhushi,2012;Kang et al.,2013).To sum up, the change of pore pressure in the surrounding soil is the primary reason that affects the mobilized uplift force acting on underground structures.However,the pore pressure generation in clays due to an increase in reservoir level,and its associated effect on uplift pressure are not fully understood.

Conventionally, the buoyant force of an underground structure is estimated from the displaced volume of groundwater based on Archimedes’ principle. When the surrounding medium is highly permeable,such as sands,gravels and fractured rocks,the seepage flow is fast,and the calculated buoyant force can be used in design directly. When the surrounding medium has a low permeability,such as clays, the generation of pore pressure often shows a time lag effect, leading to reduced uplift pressure (Song et al., 2017; Ni et al., 2019). Hence, uplift design of underground structures in clays is often problematic, and the usage of a reduction factor for buoyant force needs to be discussed.Some researchers claimed that the Archimedes’buoyant force should also be used for structures in clays conservatively (Cui et al., 1999; Zhang, 2007; Xiang et al.,2010). However, conservative flotation design for large-scale underground structures can result in overly high costs. From modelscale laboratory tests, the measured uplift pressure often lies within a range of 0.7-0.8 times the Archimedes’ buoyant force(Song et al., 2017; Ni et al., 2019; Zhang et al., 2019). Ren et al.(2020) reported a case study for a large-scale underground parking garage under a shopping mall in Shenzhen, China, and found that the construction cost can be reduced by 50%-60% when a reduction factor for buoyant force is adopted.

The upheaval motion of underground structure is resisted by the combined effect of the self-weight, the overburden pressure, and the skin friction at the soil-structure interface (Ni et al., 2019).Mitigation techniques have been developed to change the contribution from different components to increase the overall uplift resistance for an underground structure. For example, Damoun et al. (2016) compared the capacity-based techniques of ground improvement, chemical grouting, horn-type structures, and plate connections above pipelines with the demand-based approach of drainage pipes, and suggested to choose plate connections due to the economic and operational considerations.Increasing the uplift capacity is the most straightforward method. If the design allows,the soil resistance above underground structures can be increased using a greater burial depth or a larger base area (Palmer et al.,2003; Cheuk et al., 2008; White et al., 2008; Deshmukh et al.,2010; Saeedzadeh and Hataf, 2011; Liu et al., 2012; Jung et al.,2013; Robert and Thusyanthan, 2015). For hollow type structures,the wall thickness can be increased to enhance the contribution of uplift resistance from the self-weight, such as the case of deep tunnel shafts in soft clays in Mexico City as reported by Auvinet-Guichard et al. (2010). Skin frictions at the soil-structure interface can also be increased using long friction piles or anchors(Sakr et al.,2005;Khatri and Kumar,2010;Tafreshi et al.,2014;Ni et al.,2017b).Alternatively, reducing the demand of uplift force can result in more economic flotation design for underground structures. For example, skirted foundations (Gourvenec et al., 2009; Acosta-Martinez et al., 2012; Mana et al., 2013) and suction caissons(Luke et al., 2005) are widely used to provide uplift resistance of foundation for offshore structures, due to the contribution of negative pore pressure underneath the footing. Different drainage channel structures are introduced for underground structures,such as drainage holes (Obead and AL-Baghdadi, 2014; Ni et al., 2017a,2018a), drainage galleries (El-Razek and Elela, 2002), dewatering schemes (Hong and Ng, 2013; Mohamed, 2014; Hong et al., 2015),and cutoff walls (Liu and Song, 2006; Wang et al., 2019). All water pressure relief systems(Wong,2001;Ni et al.,2018b)can facilitate the dissipation of pore pressure in the surrounding soil,minimizing the development of uplift force.

Researchers often conduct numerical simulations to understand the uplift performance of dam structures, such as the works of Dewey et al.(1994),Liu et al.(2011),and Hu and Ma(2016),where the effect of reservoir impoundment associated time-dependent uplift can be characterized. Numerical results need to be assessed by experimental data to provide design implications. Despite the transient or sustained nature of hydrostatic uplift forces, modelscale laboratory tests are often carried out to simulate the behavior of underground structures.The use of elevated gravity can help to reproduce the correct stress level in the soil (Koseki et al.,1997; Gourvenec et al., 2009; Acosta-Martinez et al., 2012; Kang et al., 2013; Mana et al., 2013), but the analysis is often limited by the scale effect, where the soil particles are also scaled up. At 1gconditions (wheregis the acceleration of gravity), the self-weight of foundation model varies to find the critical floating state (Song et al., 2017; Ni et al., 2019). In all cases, the uplift force is measured or calculated without considering the time lag effect of pore pressure generation in the soil.In other words,the influence of reservoir water level on buoyant force reduction of nearby structures in clays cannot be analyzed. Further experimental evidence on the correlation between reservoir impoundment and buoyant force reduction for underground structures in clays is needed.

This analysis presents a unique design of a U-shaped test chamber, where seven pore pressure sensors are assigned in the horizontal branch,and the water level in one vertical branch varies to cause the sustained hydrodynamic uplift movement for a foundation model in another vertical branch. Two series of tests are conducted:one series is focused on the evaluation of pore pressure generation in clays in the horizontal direction due to the nearby reservoir impoundment, and another series is performed to calculate the reduction in uplift force acting on the foundation model.A mathematical model for pore pressure generation in clays is established, and the calculated results are compared against experimental measurements. The variation of pore pressure underneath the foundation model is also measured to explore the correlation with the time lag effect of uplift force.

2. Model-scale laboratory tests

2.1. Test chamber

A U-shaped test chamber is designed and illustrated in Fig. 1.Perspex (plexiglass) plates are used to manufacture the U-shaped test chamber, and holes are left in the front Perspex window to enable the instrumentation using pore pressure sensors. Seven pore pressure transducers(PPTs)are installed horizontally with an even spacing of 26.7 cm in the horizontal branch of the test chamber, as illustrated with numbers in Fig.1. In the left vertical branch,the water level(i.e.H1is measured from the water surface to the bottom of the test chamber in the left vertical branch)can be controlled, and the resulted hydraulic gradient between the left and the right vertical branch (i.e.H2is measured from the water surface to the bottom of the test chamber in the right vertical branch) results in the occurrence of seepage flow. One series of tests is conducted to measure the pore pressure generation only without a foundation model, and another series of tests is carried out to evaluate the floating performance of foundation model in the right vertical branch. The foundation model has dimensions of 17 cm in diameter and 30 cm in height. It should be emphasized that all dimensions are selected carefully to fulfill all the similitude requirements. The mass of the foundation model is 643 g. The water level inside the foundation model can be controlled to simulate scenarios with different surcharge pressures.

Fig. 1. Schematic diagram of U-shaped test chamber: (a) Measurement of the pore pressure generation, and (b) Floating test of foundation model (unit: cm).

The law of similitude has been well established for model-scale tests.Physical parameters in the prototype-scale can be reproduced from those in the model-scale using similitude ratios (Ni et al.,2016). Usually, the quantities of density, gravity, friction angle,Poisson’s ratio and strain are scaled by 1, the quantities of dimension, displacement, cohesion, Young’s modulus, stress and lateral pressure are scaled by a factor ofn(i.e.nis the similitude ratio),and the quantities of permeability, velocity, rainfall intensity and time are scaled by a factor of －－－n√. It is reasonable to consider that pore pressure generation can also be scaled byn. The present modelscale test is carried out under a 1genvironment, which can hardly result in representative results in the prototype-scale properly. Caution must be taken during interpretation of experimental measurements using similitude ratios, and all following results are presented without applying similitude ratios to avoid misinterpretation (Ni et al., 2018a).

The right vertical branch has a width of 40 cm, which is about 2.4 times the diameter of foundation model.It should be noted that the size of the test chamber is often selected to be 5 times the dimension of the buried structure to minimize the boundary effect(Ni et al., 2017b). However, the use of large model dimension is because the soil-structure interaction often involves large deformations in the surrounding soil.In the investigation,the seepage effect is primarily studied. The foundation model initiates to float once the uplift demand exceeds its resistance, and the test is terminated immediately after floatation of foundation model.Two linear variable displacement transducers (LVDTs) are mounted above the foundation model to monitor the floating state. The influence of flotation of structure can hardly spread to lateral walls,and the boundary effect is, hence, considered to be minimal.

2.2. Test setup

Clays are collected at the site and used as backfill materials in this study. Laboratory element tests are conducted to measure the physical properties of clay.It should be emphasized that few trials are conducted in the test chamber,from which the state conditions of soil in the model-scale test can be determined. The dry density and bulk density are 1.35 g/cm3and 1.97 g/cm3, respectively. The liquid limit and plastic limit are 37.8% and 19.2%, and the water content is 44.8%, exceeding the liquid limit by about 120%. The specific gravity is 2.46, and the void ratio is 0.82. From the perspective of consolidation, the anisotropy coefficient is often defined to quantify the ratio of horizontal to vertical consolidation coefficient(Chen et al.,2020).Therefore,the hydraulic conductivity in the horizontal (kh) and vertical directions (kv) should also be different. With the increase of anisotropy coefficient (i.e.kh), the obstruction of seepage flow in the horizontal direction becomes smaller, resulting in an increased consolidation rate (Yao et al.,2019; Chen et al., 2020). Hence, the mechanism of pore pressure generation could be affected by the anisotropy coefficient,showing less time lag effect in pore pressure response. Given that the consolidation pressure is not significant in the model-scale test,the soil response is assumed to be isotropic,especially for the hydraulic conductivity with a constant value of 2.54 ×10-7m/s.

The horizontal branch of the U-shaped test chamber needs to be backfilled first. In order to simulate the seepage flow in the soil,seepage channels at the soil-sidewall interface must be eliminated completely. Silicone grease is applied at the inner boundaries of sidewalls before backfilling. The purpose of silicone grease at all boundaries is two-fold:(1)seepage channels are not allowed at the soil-sidewall interface, and (2) the friction between soil and sidewall is reduced,which can further minimize the boundary effect(Ni et al., 2019; Ren et al., 2020). Clays are mixed with water to reach the height ofh2, after which surcharge loading is applied, as demonstrated in Fig. 2. Chu et al. (2004) claimed that the consolidation time could be reduced significantly when a surcharge load is imposed. In the model-scale test of Ni et al. (2018a), the full saturation condition can be reached after 2 weeks, based on the interpretation of variations of soil thickness. In this study, surcharge pressure is sustained for one month to consolidate the soil,during which surplus water and a composite system of geotextiles and double polyethylene sheets are used to reduce trapped air bubbles and eliminate water evaporation(Hird and Moseley,2000;Ni et al.,2018a). After one month, the ground surface settles by ～1 cm,further clays are backfilled to experience a second stage of surcharge preloading,until the final height of the soil in the horizontal branch is stabilized ath2. The preloading duration is selected to consider the trade-off between the sufficient degree of consolidation and the feasibility of the test. It should be noted that the surface settlement is not significant after one month, and the pore pressure readings do not change too much. Perspex plate is then used to cover the horizontal branch,and all plates are sealed using cyanoacrylate adhesive and covered with silicone sealant.Clays are continued to be backfilled in the right vertical branch to the height ofh4=40 cm.In order to achieve a uniform ground,the surface of the previous consolidated layer is scratched,and the addition of the new layer can then have a good contact with the previous layer.Similarly, surcharge preloading is implemented to consolidate the soil for one month, as shown in Fig. 2.

Fig. 2. Experimental setup: (a) Surcharge preloading consolidation of clay in the horizontal branch, (b) Surcharge preloading consolidation of clay in the right vertical branch, and (c) Instrumentation.

The tests are conducted in two stages: (1) measurement of the pore pressure generation,and(2)floating test of foundation model.At the second stage,a square pit with a plan area of 20 cm×20 cm and a depth of 16 cm is excavated in the soil in the right vertical branch to place the foundation model. It is acknowledged that disturbance cannot be avoided during the excavation and placement of the foundation model, although the excavation stage is implemented as slow as possible. All boundaries of foundation model are also implemented with a thin layer of silicone grease to avoid the formation of seepage channels and minimize the adhesion at the soil-structure interface.It should be emphasized that the interface friction cannot be ignored, but the use of silicone grease can facilitate the uplift mobility of the model. In the calculation of buoyant force,the friction component is also considered,but can be eliminated by some mathematical derivations. The bottom of foundation model is fixed at the height ofh3= 24 cm, and the contact between the foundation base and the soil is tight.The tight contact at all interfaces is achieved by a short preloading consolidation stage near the perimeter of the model,once layered soils are compacted around the foundation again. Below the foundation model,a pore pressure sensor is installed to assess the uplift force as the product of pore pressure and base area. Saturated clays are added in layers in the over-excavated pit, until the final height ofh4= 40 cm is reached.The whole system is settled for 7 d prior to commencement of the test. In the trial test, the mechanical and hydraulic properties of the later placed saturated clay are measured,which are found to have negligible difference compared to the original placed saturated clay. The whole test setup procedure needs to be repeated for each floating test.The burial depth of foundation model is measured carefully to guarantee the repeatability of the testing program. Considering the low selfweight of foundation model, water is filled in the model to make sure that uplift movement does not occur initially.Plastics are used to wrap and cover the model to prevent evaporation.

Before each test, two LVDTs with a capacity of 25 mm and an accuracy of 0.001 mm are mounted above the foundation model to measure the uplift movement.Displacement measurements at two discrete locations are helpful to evaluate the uniformness of the soil;the measured data could differ if the soil is not uniform to alter the distribution of pore pressure, leading to different uplift force.The pore pressure sensor has a capacity of 50 kPa and an accuracy of 0.01 kPa. Careful calibration of pore pressure sensor has been carried out prior to the instrumentation of the foundation model, to establish the correlation between readings and measurements.Essentially, the full saturation condition should be guaranteed for these pore pressure sensors all the time. Seven sensors in the horizontal branch and one sensor underneath the foundation model are connected to a HCSC-32 data acquisition system,as well as two LVDTs. This enables a real-time monitoring of the pore pressure generation in the soil and the floating state of foundation model. It should be noted that waterproof cover and silicone sealant are used around holes in the test chamber, where wires of pore pressure sensors are placed. Details of instrumentation and the test setup are given in Fig. 2.

2.3. Test program

One test series is to evaluate the seepage flow in the horizontal branch by increasing the water level in the left vertical branch (H1). At each water level, the stability criterion in the soil is established when the pore pressure data at all sensors become basically stable, i.e. the rate of pore pressure variation is less than 0.1 kPa/h. Once the equilibrium state of all pore pressure sensors is reached, the water level,H1, is raised by 40 cm, 50 cm and 60 cm in the left vertical branch, respectively, after which the corresponding patterns of pore pressures at all points are recorded.

The second test series is to evaluate the floating state of foundation model. The water level in the left vertical branch (H1) is increased, until the water level in the right vertical branch is stabilized at a predefined height (H2). After bothH1andH2become stable,the floating test of foundation model starts.The self-weight of foundation model is then gradually reduced by draining the water out from the model using a syringe. At the early stage of testing, the discharged mass of water is 50 g. After 50% of the Archimedes’ buoyant force is reached, each discharged mass of water is reduced to 10 g. Once the foundation model initiates to float by the indicator of displacement measurement, the foundation model is excavated to measure the weight using an electronic scale. Excavation is necessary because the foundation model has already floated, and it needs to be buried again for the next test.Different initial combinations ofH1andH2are tested,and the test procedure is repeated with a fixed burial depth of 16 cm for the foundation model. In this study, floating tests with six initial heights ofH2=44 cm,46 cm,48 cm,50 cm,52 cm and 54 cm are conducted.

2.4. Force analysis

In floating test, the foundation model is partially buried in the soil, and its force equilibrium before and after the initiation of floatation can be established, as illustrated in Fig. 3. The static condition is then written as follows:

Fig. 3. Force equilibrium of foundation model (a) before and (b) after the initiation of floatation.

whereFrepresents the uplift force,Pcorresponds to the reaction force acting on the foundation base,andGdenotes the self-weight of foundation model.It should be emphasized that the influence of adhesion at the soil-structure interface on the force equilibrium is neglected in the analysis.

The uplift movement of foundation model can mobilize the friction at the soil-structure interface,f. When the foundation model initiates to float, it corresponds to a critical state that the buoyant force, the self-weight and the soil-structure interface friction establish a balanced condition. Although the foundation model still has contact with the soil, the reaction force at the foundation base becomes zero. Now the force equilibrium is expressed by

Previous studies (Song et al., 2017; Ni et al., 2019; Zhang et al.,2019; Ren et al., 2020) indicated that the buoyant force for underground structures in clays should be reduced due to the time lag effect for deriving the uplift force.Therefore,a reduction factor for buoyant force is assumed as μ, and the force equilibrium is then rewritten as

whereFwis the Archimedes’ buoyant force. For underground structures in highly permeable materials, the reduction factor for buoyant force can be simplified as μ = 1.

Under a fixed water level condition (H2is constant), the force equilibrium of foundation model upon the initiation of floatation can be expressed by

When the water level(H2)changes,the friction mobilized at the soil-structure interface upon the initiation of floatation is invariant.Essentially, the friction at the soil-structure interface is influenced by the contact area,the surface roughness and the stress state.The burial depth in all floating tests is the same, such that the contact area and the stress state do not change. The flexural rigidity of sidewalls of foundation model is high, and as such the increase in water level cannot cause deformations, leading to a fixed surface roughness.The force equilibrium for the second floating test can be derived as

Combining Eqs. (4) and (5), the reduction factor for buoyant force can be calculated as follows:

3. Mathematical model for pore pressure generation

3.1. Problem description

A mathematical model for pore pressure generation in the horizontal branch is established following three main assumptions:

(1) The clayey soil is completely saturated,and seepage flow can occur under hydraulic gradient freely (i.e. adsorbed water between pores is neglected);

(2) Seepage flow can be described by Darcy’s law, where the hydraulic gradient is applied at the left boundary instantaneously, and the right boundary is perfectly pervious;

(3) The pore pressure generation only occurs within the soil medium, and there is no energy loss at the soil-sidewall interface.

Fig.4 depicts the schematics of the mathematical model for pore pressure generation.In the calculation,the influence of self-weight on seepage flow is ignored.It is assumed that pore pressure within the entire soil medium is zero before the application of hydraulic gradient ofu0at the left boundary.The right boundary is perfectly pervious, and hence the pore pressure is alwaysu= 0. The horizontal branch has a height ofAand a length ofL;whereas the right vertical branch has a width of 2Aand a height ofH.At the interface between the two branches, equilibrium can be established that both the pore pressure and the volume of flow are continuous.

Fig. 4. Schematic diagram of mathematical model for pore pressure generation.

Fig. 5. Variations of pore pressure with time: (a) Water level raised by 40 cm, (b)Water level raised by 50 cm, and (c) Water level raised by 60 cm.

3.2. Model establishment

The percentage of the soil mass that can transfer pore pressure over the mass of solid soil particles can be written as

where ? is the percentage;m?represents the soil mass that can transfer pore pressure;msrepresents the mass of solid soil particles,which is calculated by subtractingm?from the total soil mass;V?andVare the volumes of the soil mass that can transfer pore pressure and the total soil mass, respectively; and γ?and γsatare the unit weights of water and saturated soil, respectively.

Hence, the volume of the soil mass that can transfer pore pressure is derived by

Using a finite element approximation, the volume, dV, of an element with a width of dxat a distance ofxmeasured from the left boundary can be calculated by

Therefore,in this element,the volume of the soil mass that can transfer pore pressure can be simplified as

At a specific time, dt, the differential volume, ΔQ, of flow through the element can be computed as follows:

According to Darcy’s law, the volume of flow,q, is written as

wherekis the hydraulic conductivity; andirepresents the hydraulic gradient, i.e.

wherehdenotes the hydraulic head.

Substituting Eqs. (12) and (13) into Eq. (11), it gives

Due to seepage flow, pore water and soil particles experience compression. The compressive modulus of water is defined asE?,and the compressive modulus of soil particles is denoted asEs.The volume change of pore water and soil particles (i.e. ΔV?and ΔVs)within the soil element is then derived by

The volume of flow equals the summation of ΔV?and ΔVsas follows:Substituting Eqs. (14)-(16) into Eq. (17), it gives

whereCrepresents the transfer coefficient for pore pressure,which is defined asC=kEsEw(1 + ?)/{Es?γsat+Ew[γ?(1 + ?) -?γsat]}.

3.3. Derivation of solution

The established mathematical model for pore pressure generation can be expressed by

wherea2=C.

The initial condition is written as

The boundary conditions are expressed by

At the interface between two branches,the continuity condition is written as

The technique of separation of variables (also known as the Fourier method) is employed in this investigation to solve the partial differential equation of Eq. (19). Basically, the variables of spatial location and time occur on a different side of the equation.Hence, the magnitude of pore pressure at an arbitrary location within the lengthLcan be derived:

where

From Eq.(24),one can infer that the magnitude of pore pressure at differentspatiallocationsisafunctionofk，Es，Ew， γsat， γ?， ? andt.For a specific soil, all parameters ofk，Es，Ew， γsat， γ?and ? are deterministic,and the variation of pore pressure is simply a function of timet.This partially demonstrates that the pore pressure generation in the soil shows a time lag effect(Ni et al.,2019;Ren et al.,2020).

4. Results and analysis

4.1. Influence of hydraulic gradient on the pore pressure generation

The influence of hydraulic gradient on the pore pressure generation is systematically investigated through the first series of tests,

where the water level is increased by three levels of 40 cm, 50 cm and 60 cm in the left vertical branch. At different water levels, the variations of pore pressure with time obtained from seven pore pressure sensors are plotted in Fig. 5. It can be seen that each pore pressure sensor is triggered at different times,since they are aligned at different spatial locations. The triggering time of pore pressure sensor at different water levels is summarized in Table 1. The pore pressure sensor 1(PPT1)is placed at a distance that is the closest to the left boundary where the hydraulic head is applied,whereas the pore pressure sensor 7(PPT7)is located at a distance that is near the right boundary. With an increase in distance from the hydraulic head,the length of the seepage path increases,leading to a delayed triggering time of pore pressure sensor. This demonstrates that the time required to trigger changes in pore pressure at a far distance is longer. Over the same timeframe, the pore pressure increment at a greater distance is smaller,and as such the stabilized pore pressure is smaller. Although clay has a low permeability, it can transfer pore pressure, but the time lag effect is apparent, which is an increasing function of the length of the seepage path.The larger the hydraulic gradient, the shorter the triggering time required at a given spatial location. It should be emphasized that although the triggering time of pore pressure sensor reduces with the increase of hydraulic gradient, the time lag effect can still be observed for pore pressure sensors at a greater distance.

Table 1 Triggering time of pore pressure sensor.

4.2. Comparison between calculated and measured pore pressures

The established mathematical model for pore pressure generation is programmed, from which the variations of pore pressure with time at different spatial locations can be solved to compare against experimental measurements. Fig. 6 shows the comparison of measured and calculated pore pressure changes at two increased water levels of 50 cm and 60 cm in the left vertical branch.It should be noted from Fig.5 that the magnitude of pore pressure increment for the case with a water level increase of 40 cm is not significantly different from the case with a water level increase of 50 cm.Hence,the mathematical model calculates similar results for the two cases.For the sake of simplicity,the comparison for the case with a water level increase of 40 cm is not included.All input parameters for the calculations are determined based on the results from laboratory element tests and the local engineering experience. At a specific time oft= 2000 s, the measured pore pressure changes are also compared with those calculated using the developed analytical solution as presented in Fig. 7. The use oft= 2000 s for demonstration is due to the consideration that the time lag effect is more or less eliminated in response of pore pressure sensors, at which the true uplift behavior can be interpreted. In general, the proposed analytical solution can capture the pattern of pore pressure changes at different spatial locations,especially at a later stage of testing.Att=2000 s,all data points fall essentially along the 45°line, which demonstrates that the mathematical model can reasonably predict the stabilized behavior of pore pressure within the soil. It should be noted that a great difference between calculated and measured pore pressure changes is observed for pore pressure sensor 1 (PPT1) at the initial stage of testing (i.e. beforet=400 s),where the measured values are apparently smaller than the calculations.Att=2000 s,most data points are slightly above the 45°line,showing that the calculated pore pressures exceed the measured values. However, there are still many differences between the results derived using the proposed mathematical model and the experiment. When the experimentally measured pore water pressure tends to be stable, the mathematical model still shows the increasing result,but with a reduced increasing rate.The convergence is not reached due to the nature of the model,which is the limitation of the present work. Further work should be implemented to include a plateau for the solution.

Fig.6. Comparison of measured and calculated pore pressure changes:(a)Water level raised by 50 cm, and (b) Water level raised by 60 cm.

Fig. 7. Comparison of measured and calculated pore pressure changes at t = 2000 s:(a) Water level raised by 50 cm, and (b) Water level raised by 60 cm.

Fig. 8. Variations of H1 with H2 at the end of the test.

These differences can be attributed to three reasons: (1) the time lag effect can be resulted from the working principle of pore pressure sensor.A pore pressure sensor contains two parts:a high air-entry filter part and a sensing part.Porous stone and steel casing are used to form a filter,through which pore water can pass to reach the water reservoir. Silicon diaphragm in the water reservoir then deforms to generate electronical signals, reflecting digital changes of pore pressure. Hence, the reaction time of pore pressure sensor can cause the time lag effect of measurement; (2) in the mathematical model, the hydraulic gradient is applied at the left boundary instantaneously, which can be hardly reproduced by increasing the water level in the left vertical branch in the test;(3)at the initial seepage stage, the viscous effect of clay results in turbulent flow, especially near the boundary of hydraulic gradient(e.g.PPT1).As time elapses,steady state seepage can be established in the soil.In the test,it takes time for seepage flow(from turbulent flow to steady state seepage) to reach the position of PPT1, with evidence of delayed triggering time as reported in Table 1. In the mathematical model, steady state seepage with a constant hydraulic conductivity is assumed,which is not the case for the initial seepage stage in the test.

Table 2 Measurements of self-weight of foundation model upon the initiation of flotation.

In practice, the hydraulic head should be determined as the difference between the highest water level in history and the groundwater level at the site to conduct a conservative uplift design. The saturated hydraulic conductivity can be measured easily by retrieving soils from the site and performing laboratory element tests (falling head tests) on remoulded specimens. The hydraulic conductivity for unsaturated soils can be determined by the soil-water characteristic curve(Pei et al.,2020;Xie et al.,2020).

4.3. Reduction in uplift force

In floating tests,six initial heights ofH2=44 cm,46 cm,48 cm,50 cm, 52 cm and 54 cm in the right vertical branch are adopted.Upon the initiation of floatation, the test is terminated, and the foundation model is weighted. Measurements of self-weight of foundation model from all six floating tests are tabulated in Table 2.Two tests with different self-weights can be considered as one scenario to calculate the reduction factor for buoyant force using Eq. (6). It is clear that all measured reduction factors for buoyant force are less than 1(falling within the range of 0.85-0.87).Again,this demonstrates that the uplift force of underground structure in clays shows the time lag effect.From scenario 1 to scenario 5,with the increase of the water level ofH2, the hydraulic head becomes higher, and the reduction factor for buoyant force becomes larger.

Table 3 Comparison of reduction factor for buoyant force reported by different researchers.

In each floating test,the water level in both the left(H1)and the right vertical branch (H2) is stabilized at a certain height. Fig. 8 illustrates the variation ofH1withH2at the end of the test. It is interesting that the water level in the left vertical branch is always higher than that in the right vertical branch. The correlation betweenH1andH2can be roughly described by a linear fit. This is because clays can transfer pore pressure under certain hydraulic gradients, but the low permeability reduces the seepage flow and hinders the pore pressure generation at a far distance. Therefore,the magnitude of pore pressure within the soil medium cannot be simply estimated based on the theoretical calculation of hydrostatic pressure; instead, there is a certain degree of reduction for Archimedes’ buoyant force.The water level in the left and right vertical branches is, hence, different. Zhang (2004) measured the pore pressures in a deep foundation pit, and found that the measured values were always less than the hydrostatic pressures when the groundwater level varied. The observations of this study are consistent with the findings of Zhang (2004).

Fig. 9. Variations of pore pressure with time at the bottom of foundation model.

In practice, engineers often use the highest reservoir level in uplift design for underground structures, and calculate the Archimedes’buoyant force based on hydrostatic pressures.In this study,the water level in the left vertical branch (H1) corresponds to the occurrence of reservoir impoundment, which represents the highest water level. The foundation model is buried in clays at a certain distance from the reservoir. Due to the time lag effect of seepage flow in clays, the water level in the right vertical branch(H2)is apparently smaller than that in the left vertical branch (H1)caused by the increase in reservoir level. Therefore, one can infer that it is appropriate to adopt a reduction factor for water level in uplift design, leading to a reduced hydraulic head. This also demonstrates that the Archimedes’buoyant force should be reduced for use to estimate the uplift force. The reduction factor for buoyant force is then defined using the variation of water level as follows:

When the hydraulic heads(H1-H2)are 8.19 cm,8.37 cm,8.6 cm,8.48 cm,8.39 cm and 8.79 cm,the corresponding reduction factors for buoyant force are calculated as 0.843,0.846,0.848,0.855,0.861 and 0.86, respectively. One can see that the hydraulic head in different tests does not change much, and hence the calculated reduction factors are also limited within a small range of 0.84-0.86.Regardless of the values ofH1andH2, uplift of foundation model always occurs, once the hydraulic head exceeds ～8 cm. It is interesting that the range of reduction force estimated from the variation of water level using Eq.(25)is close to the range of 0.84-0.87 calculated from the weight measurements in the two tests using Eq. (6), as shown in Table 2. Overall, all calculated reduction factors for buoyant force fall within the range of 0.84-0.87.

Table 3 summarizes the reduction factors for buoyant force measured from other laboratory tests. Some researchers claimed that it is not necessary to introduce a reduction factor for buoyant force in uplift design (Cui et al., 1999; Zhang 2007; Xiang et al.,2010). However, Ren et al. (2020) compared different design schemes for a large-scale underground parking garage, and found that the use of reduced uplift force is still safe,leading to a reduced construction cost by 50%-60%. The observation of this work is generally consistent with other similar investigations,suggesting a reduction factor for buoyant force within the range of 0.5-0.82(Luke et al., 2005; Song et al., 2017; Ni et al., 2019; Zhang et al.,2019; Ren et al., 2020). The difference of reduction factor measured from these studies could be induced by the different types of clayey soils.

4.4. Pore pressure at the foundation base

Fig. 9 records the variations of pore pressure with time at the bottom of foundation model. Before the initiation of flotation, the pore pressure is generally stable. Once the foundation model initiates to float, the pore pressure drops significantly, and increases afterwards to a stabilized level that is higher than the value prior to flotation.This is because clay has a low hydraulic conductivity,and negative pore pressure can be generated before the initiation of uplift. Due to the time lag effect of pore pressure generation in clays, pore water cannot act on the foundation base immediately,leading to a sudden reduction of pore pressure.Upon the initiation of flotation,a gap is formed between the base of foundation model and the soil. The self-weight of foundation model is continued to reduce, such that the gap underneath the foundation model becomes larger. Pore water can then seep into the gap, causing the increase of pore pressure. The water level in the right vertical branch is raised due to the seepage flow to the gap,which results in a higher stabilized pore pressure at the end of the test.Overall,one can infer that the buoyant force before flotation could account for about 0.8 times the theoretical value,and the reduction in buoyant force can be caused by the low permeability of the soil.

5. Discussions

Marsland and Randolph (1978) derived an analytical solution to evaluate the variations of water pressure in pervious strata underlying marsh. In their study, the River Thames was assumed to be a straight channel, and the sandy gravel layer was horizontal with a specific thickness and uniform physical properties,extending to an infinite boundary. The alluvium and upper chalk layers were considered to be impervious. Due to the fluctuation of water level, water can seep through the sandy gravel layer freely, following a sine function with time. It should be noted that the solution of Marsland and Randolph (1978) was developed at a much larger scale for time-dependent pore pressure transmission through sandy gravels in the horizontal plane, whilst the current analytical solution was proposed based on the model-scale laboratory test results to analyze the buildup of water pressure in clays. Essentially, the form of the solution of Marsland and Randolph (1978) was a complementary error function provided earlier for a heat flow type problem by Carslaw and Jaeger (1959).Despite the scale difference, the “step input” function in the derivation of Marsland and Randolph (1978) can describe the phenomenon of water flow in a horizontal soil layer for the calibration purpose.

Both the solution of Marsland and Randolph (1978) and the current method are employed to analyze a specific problem, with dimensions ofL= 3 m andH= 1 m as illustrated in Fig. 4. The compressive modulus of water is defined asE?= 2100 MPa. The parametric study is conducted to illustrate how different combinations of input parameters can affect the calculation, although some combinations could be not so meaningful.Fig.10a shows the variations of pore pressure with time at the position ofx=0.3 m for different values of hydraulic conductivity,k, with a fixed compressive modulus of soil ofEs=0.1 MPa.With the decrease ofk,the increment of pore pressure becomes smaller. Upon the application ofu0=6 kPa on the left,the time that is required to reach the target pore pressure atx=0.3 m becomes longer with the decrease ofk.This demonstrates that clay with a smallerkvalue shows more apparent time lag effect of pore pressure transmission. Hence, a reduction factor for buoyant force should be applied for calculating the pore pressure for clay. Similarly, Fig.10b depicts the variations of pore pressure with time at the position ofx=0.3 m for different values of compressive modulus of soil,with a fixed value ofk=10-7 m/s. With the increase ofEs, the increment of pore pressure becomes greater.For clayey soils with a high compressibility,the void ratio increases, such that water flow in the soil becomes easier,increasing the rate of pore pressure transmission. When theEsvalue is very small, it is hard for pore pressure to transmit.

Fig.10. Variation of pore pressure with time: (a) Influence of hydraulic conductivity,and (b) Influence of compressive modulus of soil.

The reduction factor is then calculated using the method of Marsland and Randolph (1978) and the proposed analytical solution for comparison (see Fig.11). One can see that when the input parameters are the same in the two solutions,the reduction factor fluctuates in the similar manner. The reduction factor becomes stabilized for soils with a high permeability(k≥10-6m/s)or a low permeability (k≤ 10-10m/s). For soils with an intermediate permeability (10-10m/s ≤k≤10-6m/s), the reduction factor reduces with the hydraulic conductivity. At a fixed hydraulic conductivity,the reduction factor decreases faster with the decrease of compressive modulus of soil. This is because for soils with a low compressive modulus, the degree of compaction is higher, leading to difficulty in pore pressure transmission, along with a smaller reduction factor.It is interesting that the solution of Marsland and Randolph(1978)stabilizes at ～1,when the hydraulic conductivity is high(k≥10-6m/s).At a low hydraulic conductivity(k≤10-10m/s), the reduction factor derived by the method of Marsland and Randolph (1978) approaches zero, and the soil becomes impermeable. In the current method, the reduction factor shows a smooth pattern, changing from 0.9 to around 0.3. In reality, even though the soil has a high permeability,pore pressure transmission in the soil takes some time upon the occurrence of seepage flow.At this circumstance, there must be some hydraulic head loss,resulting in a certain reduction factor that is less than 1.When the soil has a low permeability, pore spaces exist in the soil. The discontinuity of these pores can then hinder the pore pressure transmission process,reflecting by the observation of reduced pore pressure (reduction factor much less than 1).

Fig. 11. Comparison of reduction factor obtained using the method of Marsland and Randolph (1978) and the proposed analytical solution.

6. Conclusions

A U-shaped test chamber is designed to simulate the influence of reservoir impoundment on the uplift performance of underground structures. A mathematical model is proposed to interpret the mechanism of pore pressure generation in clays.The following findings from this investigation are drawn:

(1) Due to the low permeability of clay,the pore pressure sensor at a far distance from the applied hydraulic gradient is triggered at a later time, regardless of the magnitude of hydraulic gradient. This demonstrates that the pore pressure generation in clays has a time lag effect.

(2) The influence of hydraulic gradient on the pore pressure generation is apparent. In general, with the increase of hydraulic gradient, the time lag effect of pore pressure generation becomes less significant.

(3) The mathematical model can be used in uplift design of underground structure to describe the pore pressure generation in clays.

(4) The uplift force of underground structure in the specific clay should be reduced by a reduction factor within a range of 0.84-0.87.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant Nos. 51878185, 52078506, and 52178321).

List of symbols

CTransfer coefficient for pore pressure

EsCompressive modulus of soil particles

E?Compressive modulus of water

eVoid ratio

FUplift force

FwArchimedes’ buoyant force

fFriction at the soil-structure interface

GSelf-weight of foundation model

gAcceleration of gravity

H1Distance from the water surface to the bottom of the test chamber in the left vertical branch

H2Distance from the water surface to the bottom of the test chamber in the right vertical branch

hHydraulic head

iHydraulic gradient

kHydraulic conductivity

khHydraulic conductivity in the horizontal direction

kvHydraulic conductivity in the vertical direction

LLLiquid limit

msMass of solid soil particles

m?Soil mass that can transfer pore pressure

nSimilitude ratio

PReaction force acting on the foundation base

PLPlastic limit

qVolume of flow

RwReduction factor for buoyant force defined using the

variation of water level

tTime

V?Volume of the soil mass that can transfer pore pressure

VVolume of the total soil mass

wWater content

xDistance from the left boundary

γ?Unit weight of water

γsatUnit weight of saturated soil

ΔQDifferential volume of flow through the element

ΔVsVolume change of soil particles

ΔV?Volume change of pore water

μ Reduction factor for buoyant force

? Percentage