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Article

Experimental and Theoretical Explanations for the Initial Difference in the Hydraulic Head in Aquitards

1
Beijing Key Laboratory of Geotechnical Engineering for Deep Foundation Pit of Urban Rail Transit, Beijing Urban Construction Exploration & Surveying Design Research Institute Co., Ltd., Beijing 100101, China
2
Key Laboratory of Water Cycle and Related Land Surface Processes, Institute of Geographic Sciences and Natural Resources Research, Chinese Academy of Sciences, Beijing 100101, China
*
Author to whom correspondence should be addressed.
Water 2022, 14(19), 3042; https://doi.org/10.3390/w14193042
Submission received: 27 July 2022 / Revised: 13 September 2022 / Accepted: 21 September 2022 / Published: 27 September 2022
(This article belongs to the Section Hydrogeology)

Abstract

:
Accurate estimation of the buoyancy forces exerted on underground structures is a problem in geotechnical engineering that directly impacts the construction safety and cost of these structures. Therefore, studying the buoyancy resistance of underground structures has great scientific and practical value. In this study, an initial difference in the hydraulic head, Δh0, was discovered to be present in aquitards through analysis of water-level data collected from the observation of real-world structures and in laboratory control tests. That is, seepage occurs beyond a threshold Δh0. Analysis of test data reveals that a deviation from Darcy’s law is the theoretical basis for Δh0 and that Δh0 equals the initial hydraulic gradient multiplied by the length of the seepage path. The general consistency between the experimentally measured and theoretically calculated values of Δh0 validates the theoretical explanation for Δh0. The results of this study provide a basis for scientifically calculating the buoyancy resistance required for the construction of underground structures.

1. Introduction

Urban development has led to the construction of large numbers of underground structures that, together with groundwater and soil masses, form complex systems in which these structures interact. Due to the presence of groundwater, these systems can cause a multitude of problems to real-world underground structures. Buoyancy resistance is prominent among these problems [1]. Apart from being a major problem in geotechnical engineering, buoyancy resistance is also a challenging issue encountered with real-world underground structures that directly impacts their safety and cost. Therefore, studying the buoyancy resistance of underground structures has great scientific and practical value [2,3].
In current engineering practice, groundwater in formation layers is generally regarded as static water, and the buoyancy forces on underground structures from groundwater are calculated using Archimedes’ principle. The interparticle pores in sand and pebble layers are large. As a result, free water that fills these pores is able to move freely between the particles. Therefore, Archimedes’ principle is suitable for highly permeable beds [4,5]. Extensive experimental results show that this approach is applicable to such highly permeable layers composed of sand and pebbles [6,7].
Aquitards composed of clayey soil have low connectivity due to the presence of particle-bound water. Therefore, calculating the buoyancy forces on underground structures in aquitards is a complex task. Relevant research results differ considerably [8,9,10,11]. Some studies have shown a high level of consistency between the buoyancy forces measured in experiments and those calculated using Archimedes’ principle [12,13,14,15]. Cui et al. (1999) determined the buoyancy force on an underground structure in two media (sandy and clayey soils) in a model test and did not find a significant reduction in the pore-water pressure (PWP) [16]. Through extensive laboratory tests, Zhang and Chen (2008) found that the PWP reduction coefficient of bound water in clayey soils is very small, so as to be negligible in real-world engineering scenarios [12]. Xiang et al. (2010) experimentally demonstrated that saturated clayey soils in a long-term stable state can fully transfer PWP and that the effects of bound water do not need to be considered [17]. In contrast, other studies have reported an appreciable difference between the buoyancy forces measured in experiments and those calculated using Archimedes’ principle, while noting that a reduction in the theoretical value needs to be considered in the calculation of the buoyancy force from groundwater in clayey soils and introducing methods to determine the reduction coefficient [18,19,20]. Zhou et al. (2019) experimentally determined the variation in the buoyancy force on an underground structure in an aquitard with depth and found that the reduction coefficient increased from 0.25 to 0.52 as the depth increased [3]. Song et al. (2017) conducted an experiment in which the half-interval search method was used to determine the buoyancy forces exerted by groundwater on foundations embedded in clay soil [19]. They found that the measured buoyancy forces on foundations in clay soil were lower than the corresponding theoretical values but were consistent with large-scale field measurements. Zhang et al. (2019) experimentally evaluated the buoyancy effect on underground grain silos in sandy and clayey soils and determined the values (0.95 and 0.79, respectively) of the buoyancy reduction coefficient for these two types of soils [21]. Zhang et al. (2018) analyzed the patterns of variation in the PWP with depth in sandy and clayey soils through centrifugal model tests and found the following. (1) The measured values of the PWP were consistent with its theoretical values in sandy soils, while the measured values of the PWP were lower than its theoretical values in clayey soils. (2) The PWP reduction coefficient varied with depth and stabilized at approximately 0.68 at depths greater than 10 m [22]. By designing and conducting model tests at multiple scales, Zhou (2006) found that groundwater buoyancy in clay layers is only 75% of the conventional theoretical value [18]. Despite the substantial experimental efforts in the abovementioned studies, a consensus has yet to be established on the reduction in water buoyancy in clay layers. Research on the theoretical basis of the reduction phenomenon lacks depth.
Currently, the reduction coefficient (i.e., the ratio of the measured value to the theoretical value) is still used to depict the reduction in water buoyancy in clay layers. This method neglects the lack of a direct proportional relationship between the measured and theoretical values of buoyancy. The line of best fit between the experimentally measured and theoretical values of groundwater buoyancy in clay layers has an intercept that is defined in this study as the “initial difference in the hydraulic head h, Δh0.” This study attempts to demonstrate the presence of Δh0 in aquitards through model-based continuous buoyancy monitoring tests and provide a theoretical explanation for Δh0, which is subsequently validated based on measurements.

2. Materials and Methods

2.1. Tests

It is exceedingly difficult to measure the buoyancy force on a real-world underground structure and impossible to artificially control its boundary conditions in the field. In comparison, the boundary conditions (e.g., the surrounding soil layer, water level H, and lateral resistance) of an underground structure module (USM) can be controlled in a laboratory test setting. Hence, laboratory tests were conducted in this study to experimentally and theoretically explain the Δh0 in aquitards.
(1)
Test design
External environmental elements (i.e., a soil layer and groundwater) are required to determine the variation in the buoyancy force on a USM in a silty soil layer with known water-supply conditions. To achieve this, a model test box (MTB) consisting of a USM, an enclosure structure module (ESM), a test soil layer, a water-supply system, and monitoring systems was employed to conduct continuous buoyancy monitoring tests. The design of the MTB allowed the test soil layer to be replaced, the delivery head hd to be adjusted, and the H in the test soil layer (Hs) and the buoyancy force exerted by the water on the USM (F2) to be monitored in real time.
The USM consisted of a rigid circular steel plate and a tetrahedron. The rigid circular steel plate was located on the bottom of the ESM and overlain by the tetrahedron, which served as a fulcrum for a tension–compression sensor located above the USM.
The ESM, composed of a rigid cylinder and a waterproof rubber bottom, ensured that the enclosure structure was devoid of water during the test. Moreover, the internal USM was not placed in direct contact with the wall of the enclosure to prevent lateral friction.
Silty clay and fine sand were employed as the two types of soil layers in the test. The silty clay layer, representing aquitards, was the focus of this study. The fine sand layer, representing highly permeable layers, was used to examine the reliability of the test model. The F2 monitoring test results obtained using the two layers were compared.
The water-supply system consisted of an external water tank, an inlet pipe, and an inverted filter layer. The external water tank was used to adjust hd. The inlet pipe, made of a plastic flexible tube, connected the external water tank and MTB. The inverted filter layer, composed of medium sand, was primarily used to ensure a uniform water supply to the test soil layer above it.
The MTB was equipped with three systems to monitor hd, Hs, and F2, respectively. A PWP sensor was used to automatically monitor hd. Similarly, PWP sensors were also used to automatically monitor the Hs at seven observation holes located at different depths. A tension–compression sensor was employed to automatically monitor F2.
(2)
Composition of the test setup
The test setup consisted mainly of an MTB, a test soil layer, a tetrahedron, a rigid circular steel plate, an enclosure structure, waterproof rubber, a reaction beam, Hs observation holes, a water-supply tank, and an inverted filter layer. Figure 1 and Figure 2 show the test setup in detail.
The MTB was 1000 mm long, 1000 mm wide, and 1100 mm tall. The enclosure structure consisted of a bottomless steel cylinder with a height of 400 mm and an inner diameter of 350 mm. The USM was composed of a tetrahedron and a rigid circular steel plate with an outer diameter of 335 mm. Throughout the test, water was supplied laterally to the bottom of the MTB. Seven Hs observation holes were prepared in the test soil layer at depths of 200, 300, 400, 500, 600, 700, and 800 mm. These holes are denoted by G1, G2, G3, G4, G5, G6, and G7, respectively. A 2 cm-long filter was placed at the bottom of each hole to observe the Hs at the corresponding depth. A 10 cm-thick inverted filter layer composed of medium sand was placed at the bottom of the MTB and was overlain by the test soil layer. The micro-PWP and tension–compression sensors used in the MTB were custom-made products (Figure 3 and Figure 4). Table 1 summarizes their basic parameters.
(3)
Test procedure
The test procedure is detailed in Figure 5 below.
(1)
Soil-sample preparation and filling. The fine sand and silty clay used in the tests were both retrieved from construction sites in Changchun. In this study, the effects of structured soils on test results were not considered. Therefore, remolded soils were used instead of undisturbed soils. The retrieved soil samples were manually crushed in the laboratory. Each soil sample was sprayed sparingly with water while being placed in the MTB layer by layer. Each layer was no thicker than 20 cm and was manually compacted before the subsequent layer was placed on top of it. The same procedure was followed to fill the MTB with each of the two test soil samples (i.e., fine sand and silty clay). Table 2 and Table 3 summarize the physical property metrics of the test soil samples placed in the MTB. The fine sand layer was first tested to examine test reliability. Subsequently, a continuous F2 monitoring test was performed with the silty clay layer;
(2)
Delivery of water from the external water tank. The external water tank was used to provide a continuous water supply to the MTB. hd was kept at the same level as the bottom of the rigid circular steel plate in the MTB. The Hs in the MTB was monitored continuously. After Hs stabilized, the MTB was allowed to stand still for 24 h to allow the test soil layer beneath the USM to be completely saturated;
(3)
F2 monitoring test. The height of the water-supply tank was adjusted to increase hd in a stagewise manner. The extent to which hd was increased at each stage depended on the Hs and F2 response speed inside the MTB. At the start of each stage, water was added to the water tank to maintain a stable hd for a certain period of time, after which the addition of water was terminated. After Hs stabilized, hd was increased again to commence the subsequent stage of the test. Throughout the test, Hs and F2 were monitored automatically and in real time.
To facilitate subsequent discussion and analysis, the bottom of the rigid circular steel plate was used as a reference surface (i.e., the initial h) in each test.

2.2. Force Analysis

Because the USM was not subject to lateral friction in the test, only the vertical forces are analyzed. Figure 6 shows the forces acting on the USM, as determined according to the principle of static equilibrium. The equilibrium equation is given below.
F1 + F2 = G0 + Ps
where G0 is the dead weight of the USM (N), Ps is the pressure detected by the tension–compression sensor (N), F1 is the vertical reaction from the soil particles (N), and F2 is the buoyancy force exerted by the groundwater (N).
Water was initially delivered into the MTB to allow Hs to be flush with the bottom plate of the USM. Under this condition, F2 = 0 and Ps was a fixed value (Ps0; 338 and 250 N in the fine sand and silty clay layers, respectively). Based on Equation (1), F1 = Ps0 + G0 (G0 = 39.9 N). The test was started by increasing Hs. The variation in the reading of the tension–compression sensor was in fact the variation in the total pressure of the water and soil due to the comprehensive action of F2. Therefore, F1 is assumed to have remained constant throughout the test, while the variation in the total pressure is regarded as F2. The following equation holds in a stable state:
F2 = PsP0
The measured F2 (mm) is converted to h (hb, the buoyancy head), i.e.,
hb = F2/Ag
where A is the area of the bottom of the rigid circular steel plate with a diameter of 335 mm (88,141.31 mm2) and g is the gravitational acceleration (10 N/kg).
Considering that limited test data were obtained, the measured data were fitted to reflect the variation trend in hb for all hd values.

3. Results and Discussion

3.1. Presence of Δh0

3.1.1. Presence of Δh0 in the Tests

The values of F2 in the fine sand and silty clay layers were monitored. The dynamic variation in F2 under instantaneous water-supply conditions was determined. The relationship between the stable hd and F2 was analyzed through comparison.
(1)
Dynamic variation in F2 under instantaneous water-supply conditions.
Figure 7 and Figure 8 show the patterns of dynamic variation in F2 in the fine sand and silty clay layers, respectively, as determined based on the values of hd and hb obtained from the tests.
Figure 7 and Figure 8 show that after hd was increased instantaneously at each stage, the H in the water-supply tank decreased with time, while F2 gradually increased. The F2 acting on the USM in the fine sand layer responded rapidly to hd. The test conducted in the fine sand layer involved four stages and lasted for 10.5 h in total. During the first 0.5 h of each stage, water was continuously delivered into the MTB, resulting in a rapid increase in hb. Two hours later, hb tended to stabilize. The measured value of hb in the fine sand layer was almost the same as the value of hd. The F2 acting on the USM in the silty clay layer responded slowly to hd. The test conducted in the silty clay layer involved seven stages and lasted for approximately 572 h in total. On average, it took 60 h for hb to stabilize during each stage. The stable value of hb was appreciably lower than the value of hd.
A comparison of Figure 7 and Figure 8 shows the following. The measured value of F2 acting on the USM in the simulated highly permeably layer was basically consistent with its theoretical value, suggesting no reduction in h. In contrast, the measured value of F2 acting on the USM in the simulated aquitard was considerably lower than its theoretical value, suggesting a reduction in h.
(2)
Comparison of the stable hd and hb.
After water was instantaneously delivered into the MTB during each stage, hd and hb eventually became stable and remained unchanged. Under this condition, the groundwater throughout the MTB was in a static state. The stable values of hd and hb were linearly fitted for analysis, as shown in Figure 9a,b.
Figure 9a shows a slope of nearly 1 for the line of best fit between the stable hd and hb measured during the test conducted in the fine sand layer, suggesting no reduction in h in the simulated highly permeable layer. The transverse intercept of the line of best fit shown in Figure 9a is 2.6 mm, which can be ascribed to experimental error and is negligible.
In Figure 9b, the stable hd is not directly proportional to the stable hb determined during the test conducted in the silty clay layer. The line of best fit between the stable hd and hb does not pass through the origin. A transverse intercept of 19.2 mm can be observed for Hs at the bottom of the USM. Analysis of the linear fitting equation reveals that there is no hb for the USM when hd is smaller than this transverse intercept and that F2 only occurs when hd exceeds this transverse intercept. This transverse intercept is defined in this study as the initial difference in h, Δh0. Observation of the lines of best fit shows a prominent Δh0 in the silty clay layer that plays a significant role in the calculation of F2.

3.1.2. Presence of Δh0 in Real-World Engineering Settings

Δh0 can also be found during routine drilling operations. A confined aquifer is overlain by a thick silty clay layer. A dry soil or unsaturated wet soil layer is encountered during the initial stage of drilling in the silty clay layer. This layer is referred to as the vadose zone. Further drilling in the silty clay layer exposes free water. An H that remains unchanged for a prescribed period of time after the drilling operation is paused is often taken as the stable initial H (H0) (Figure 10a). A saturated zone is encountered as the drilling depth increases. Here, the stable H observed in the borehole continues to increase (Figure 10b,c). The stable H in the borehole is exactly the same as the confined H (Hc) in the aquifer when the borehole just reaches the confined aquifer (Figure 10d). At this location, the Hc is significantly higher than the stable H0.
Zhang (1980) referred to the saturated layer between the location of the stable H0 and the roof of an aquifer as the water-bearing zone (WBZ) of the aquitard above the aquifer [23]. An underground structure in a WBZ is subject to a positive buoyancy force from water. However, because the H in a WBZ is invariably lower than the Hc, the measured value of the buoyancy force from the water in the WBZ is invariably lower than the value calculated based on the Hc. The Δh0 at a specific location in a WBZ equals the Hc minus the H at the location. The Δh0 in a WBZ equals the h of the confined water minus the stable H0. When the Hc remains stable and unchanged, the stable H0 also remains unchanged. As the Hc increases, the difference in hh) surpasses the Δh0 in the original stable state. As a result, groundwater begins to seep upward in the silty clay layer, accompanied by an increase in the stable H0.

3.2. Theoretical Explanation for Δh0

Under normal circumstances, the seepage velocity V of groundwater in a formation layer is directly proportional to the hydraulic gradient I according to the well-known Darcy’s law. However, some researchers obtained results from laboratory seepage tests conducted in saturated silty clay layers that deviate from Darcy’s law [20,24,25,26]. These test results show that the V–I curve for a clay layer does not pass through the origin, that no seepage occurs when I is lower than a certain value I0, and that the V–I curve is a straight line when I > I0 (Figure 11).
The seepage curve deviates from Darcy’s law, as shown in Figure 11, and can be described using the approximate expression proposed by Poza (1950) shown below [27]:
V = K (II0)
the intercept I0 of the above equation is the initial I often mentioned in the context of groundwater seepage flow.
In the critical seepage state when I = I0, I0 equals the ratio of the Δh in the critical seepage state (i.e., Δh0) to the length of the corresponding seepage path, L0, as expressed below.
I0 = Δh0/L0
The Δh in the critical seepage state is the initial Δh (i.e., Δh0). In other words, Δh0 exists in a test soil layer with a certain thickness. When the Δh between the two sides of the seepage path is smaller than Δh0, the groundwater is in a static state. The groundwater is only able to seep through the soil layer when the Δh between the two sides of the seepage path is greater than Δh0. Therefore, in the tests conducted in this study, when hd < Δh0, the groundwater did not seep through the test soil layer and rise above the bottom of the USM. As a result, the USM was not subject to F2. In contrast, when hd > Δh0, seepage occurred in the test soil layer, resulting in an increase in Hs above the bottom of the USM. Therefore, in the calculation of the buoyancy force from the groundwater in an aquitard, hd minus Δh0, should be used as H, while Δh0 = I0 × L0.

3.3. Validation of Δh0

The scenario simulated in the MTB used in this study closely resembles that in the WBZ of a real-world aquitard in the city of Changchun. A test soil sample was taken from the construction site of the Underground Rail Transit Project of Changchun. The aquitard mainly consists of silty clay, for which the measured physical property metrics are presented in Table 3. To validate the hypothesis that Δh0 = I0 × L0, the variation in the Hs at different depths (equivalent to the H in a WBZ) and the stable H0 in the MTB with hd (equivalent to Hc) was monitored continuously along with F2. First, the variation in Hs and Δh (i.e., hd minus Hs) was analyzed to determine the values of Δh0 at different depths, which were subsequently compared with the intercept of the fitting equation in Figure 9b. Then, the relationship between hd and Hs was analyzed. Finally, the I0 in the silty clay layer was determined through a laboratory test. On this basis, Δh0 was calculated using Equation (5). Moreover, the measured and calculated values of Δh0 were compared.

3.3.1. Variation in Hs and Δh

Figure 12 shows the variation in hd and the Hs at each depth and their difference under instantaneous water-supply conditions. As seen in Figure 12, after hd was increased instantaneously during each stage, the groundwater in the soil layer in the MTB began to seep. Consequently, Hs increased and eventually tended to stabilize. The variation in Hs was generally consistent with that in hb. Moreover, after hd was increased instantaneously during each stage, Δh peaked instantaneously and then gradually decreased until the seepage flow stopped. In this process, the Δh at each depth tended to be a nonzero stable value. Thus, as the groundwater transitioned from a seepage state to a static state, hd remained higher than Hs. In addition, the Δh formed in a static state remained stable.
Table 4 summarizes the stable values of Δh at different depths at stages 1–7. At a depth of 400 mm, Hs observation hole G3 was located just below the bottom plate of the USM. The average value of Δh at observation hole G3 measured during the seven stages was 19.0 mm, which is almost identical to the intercept (19.2 mm) of the fitting equation in Figure 9b. This finding demonstrates that this transverse intercept is not a result of system error, but is instead the objectively existing Δh0. Therefore, the stable Δh measured at the observation hole at each depth was the Δh0 at this depth.
Figure 13 shows the variation in Δh0 at different depths during the multistage test process. Horizontally, the Δh0 at any depth did not change appreciably as hd increased. The corresponding broken line is almost a horizontal straight line. Vertically, Δh0 and the corresponding L0 both decreased as the depth increased. Therefore, Δh0 is independent of hd, but is closely linked with L0.

3.3.2. Variation in the Stable hd and H0

After the groundwater entered a static state during each test stage, the stable H0 was measured by drilling. The boreholes drilled at stages 1–7 are denoted by Z1–7. Figure 14 shows the stable hd and H0 at each stage. Analysis of Figure 14 shows the following. As the stable hd increased, the stable H0 and the thickness of the WBZ both increased. However, the stable H0 was lower than the stable hd at each stage. The Δh0 in the WBZ of the test soil layer equaled the stable hd minus the stable H0. Table 5 summarizes the measured values of Δh0 and the corresponding L0. As seen in Table 5, an increase in L0 led to an increase in Δh0.

3.3.3. Comparison of the Theoretically Calculated and Measured Values of Δh0

Six samples (denoted by 1–6) of the test silty clay layer used were analyzed in the laboratory to determine their I0. Table 6 summarizes the test results. As shown in Table 6, the average I0 in the test silty clay layer was 0.032. The theoretical value of Δh0 corresponding to each value of L0 in Table 4 and Table 5 was calculated by multiplying the average I0 by the value of L0. Table 7 summarizes the calculation results. Figure 15 compares the theoretical and measured values of Δh0. Analysis of Figure 15 reveals that the theoretical and measured values of Δh0 basically fall near the 1:1 line and the correlation coefficient R2 between them is 0.971, suggesting that the two values are nearly consistent. This finding demonstrates that Δh0 = I0 × L0.
In summary, the above test results show that the groundwater level at any location in the WBZ of an aquitard is lower than the supply water level upstream of the seepage flow due to the action of I0, that Δh0 exists perennially, regardless of whether the groundwater is in a seepage or static state, and that Δh0 = I0 × L0. Therefore, the reduction in h due to Δh0 should be considered in the calculation of the buoyancy force from the groundwater in an aquitard.

4. Conclusions

To address the difficulty in accurately estimating buoyancy resistance required for the construction of underground structures, model tests were conducted in this study to observe the buoyancy force on a USM in an aquitard. A phenomenon in geotechnical engineering, the occurrence of a Δh0, was discovered and demonstrated based on observation data collected from real-world engineering settings and laboratory buoyancy tests. It was inferred from the laboratory buoyancy observation test data that a deviation from Darcy’s law is the theoretical basis for Δh0; that is, Δh0 = I0 × L0. A comparison shows a 1:1 linear correlation between the experimentally measured and theoretically calculated values of Δh0, thus experimentally validating the theoretical explanation for Δh0. The results of this study provide a basis for scientifically calculating the buoyancy resistance required for the construction of underground structures.

Author Contributions

Conceptualization, C.D. and Y.X.; methodology, Y.X.; validation, W.P. and S.H.; formal analysis, Y.X. and S.L.; investigation, C.D.; resources, H.M. and Y.X.; data curation, H.M.; writing—original draft preparation, Y.X.; writing—review and editing, C.D.; visualization, S.H. and S.L.; supervision, C.D. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the Open Research Fund Program the Beijing Key Laboratory of Geotechnical Engineering for Deep Foundation Pit of Urban Rail Transit (202103).

Institutional Review Board Statement

This study does not require ethical approval.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data presented in this study are available in this article.

Acknowledgments

The authors would like to thank Jianquan Zhang and Wenxin Gao for the suggestions and support in this research.

Conflicts of Interest

The authors declare no conflict of interest.

Abbreviation

PWPPore-water pressure
USMUnderground structure module
MTBModel test box
ESMEnclosure structure module
Symbols
ΔhHydraulic head difference
Δh0Initial hydraulic head difference
IHydraulic gradient
I0Initial I
L0Length of the corresponding seepage path
HsHead of the test soil layer
H0Stable initial head
HcConfined head in the aquifer
G0Dead weight of the USM
F1Vertical reaction of soil particles
F2Buoyancy force exerted by groundwater
PsPressure detected by the tension–compression sensor
Ps0The force on the soil sample the when the buoyancy from the groundwater is zero
AArea of the bottom of a rigid circular steel plate
GGravitational acceleration
hbBuoyancy head from groundwater
hdDelivery head

References

  1. Bikçe, M.; Örnek, M.; Cansız, Ö.F. The effect of buoyancy force on structural damage: A case study. Eng. Fail. Anal. 2018, 92, 553–565. [Google Scholar] [CrossRef]
  2. Tang, H.Y.; Yang, D.; Wang, C.J.; Guo, H.X.; Dong, Q.P. Anti-floating problem and retrofit measures for a damaged basement. Mater. Res. Innov. 2015, 19, S8-219–S8-222. [Google Scholar] [CrossRef]
  3. Zhou, J.K.; Lin, C.H.; Chen, C.; Zhao, X.Y. Reduction of groundwater buoyancy on the basement in weak-permeable/impervious foundations. Adv. Civil. Eng. 2019, 2019, 7826513. [Google Scholar] [CrossRef]
  4. Huerta, D.A.; Sosa, V.; Vargas, M.C.; Ruiz-Suárez, J.C. Archimedes’ principle in fluidized granular systems. Phys. Rev. E 2005, 72, 031307. [Google Scholar] [CrossRef] [PubMed]
  5. dos Santos, F.C.; Santos, W.M.S.; Berbat, S.D. An analysis of floating bodies and the principle of archimedes. Rev. Bras. Ensino Fis. 2007, 29, 295–298. [Google Scholar]
  6. Barbot, A.; Lopes, J.B.; Soares, A.A. Relating water and energy by experimenting with urban water cycle, desalination of sea water and archimedes’ law. In Proceedings of the 6th International Technology, Education and Development Conference, Valencia, Spain, 5–7 March 2012; pp. 2534–2544. [Google Scholar]
  7. Kang, W.T.; Feng, Y.J.; Liu, C.S.; Blumenfeld, R. Archimedes’ law explains penetration of solids into granular media. Nat. Commun. 2018, 9, 1101. [Google Scholar] [CrossRef]
  8. Hisatake, M. A Proposed methodology for analysis of ground settlements caused by tunneling, with particular reference to the “buoyancy” effect. Tunn. Undergr. Space Technol. 2011, 26, 130–138. [Google Scholar] [CrossRef]
  9. Liu, X.Y.; Yuan, D.J. Mechanical analysis of anti-buoyancy safety for a shield tunnel under water in sands. Tunn. Undergr. Space Technol. 2015, 47, 153–161. [Google Scholar] [CrossRef]
  10. Xu, Z.J.; Yu, H.H. Non-contact experiment investigation of the interaction between the soil and underground granary subjected to water buoyancy. Appl. Sci. 2021, 11, 8988. [Google Scholar] [CrossRef]
  11. Zhang, J.W.; Cao, J.; Mu, L.; Wang, L.; Li, J. Buoyancy force acting on underground structures considering seepage of confined water. Complexity 2019, 2019, 7672930. [Google Scholar] [CrossRef]
  12. Zhang, D.; Chen, L. Experiment on the computing method of anti-floating in underground structures. Sichuan Build. Sci. 2008, 34, 105–108. (In Chinese) [Google Scholar]
  13. Li, X.L.; Wang, B.L.; Li, X.; Ji, C. Analysis of anti-floating measures and construction parameters of shield tunnel through river under ultra-shallow covering. In Proceedings of the International Conference on Civil Engineering, Architecture and Building Materials, Jinan, China, 24–26 May 2013; pp. 628–634. [Google Scholar]
  14. Zhang, S.Q.; Yuan, W.J.; Zhang, H.H.; Huang, L.; Zhong, Y.F. Analytical and numerical anti-floating study concerning the shield tunnel across the river. In Proceedings of the International Conference on Structures and Building Materials, Guiyang, China, 9–10 March 2013; pp. 1087–1092. [Google Scholar]
  15. Tang, M.X.; Hu, H.S.; Zhang, C.L.; Hou, M.X.; Chen, H. Determination of anti-floating water level in guangzhou based on the distribution of the underground aquifers. In Proceedings of the GeoShanghai International Conference on Geoenvironment and Geohazard, Shanghai, China, 27–30 May 2018; pp. 468–480. [Google Scholar]
  16. Cui, Y.; Cui, J.; Wu, S. A model test of shallow underground structure buoyancy. Spec. Struct. 1999, 16, 32–36. (In Chinese) [Google Scholar]
  17. Xiang, K.; Zhou, S.; Zhan, C. Model test study of buoyance on shallow underground structure. J. Tongji Univ. (Nat. Sci.) 2010, 38, 346–352. (In Chinese) [Google Scholar]
  18. Zhou, P. Groundwater Uplift Mechanism Study under Complex Urban Environment. Master’s Thesis, China University of Geosciences (Beijing), Beijing, China, 2006. [Google Scholar]
  19. Song, L.; Kang, X.; Mei, G. Buoyancy force on shallow foundations in clayey soil: An experimental investigation based on the “Half Interval Search”. Ocean. Eng. 2017, 129, 637–641. [Google Scholar] [CrossRef]
  20. Song, L.; Huang, Q.; Yan, D.; Mei, G. Experimental study on effect of hydraulic gradient on permeability of clay. Chin. J. Geotech. Eng. 2018, 40, 1635–1641. (In Chinese) [Google Scholar]
  21. Zhang, Q.; Ouyang, L.; Wang, Z.; Liu, H.; Zhang, Y. Buoyancy reduction coefficients for underground silos in sand and clay. Indian Geotech. J. 2019, 49, 216–223. [Google Scholar] [CrossRef]
  22. Zhang, Z.; Cao, P.; Jia, M. Centrifugal model test of pore water pressure in soft clay. In Proceedings of the GeoShanghai 2018 International Conference: Fundamentals of Soil Behaviours, Shanghai, China, 28–30 May 2018. [Google Scholar]
  23. Zhang, Z. The Bond Water Dynamics Theory; Geology Press: Beijing, China, 1980; pp. 80–83. [Google Scholar]
  24. Kutiĺek, M. Non-darcian flow of water in soils—Laminar region: A review. Dev. Soil Sci. 1972, 2, 327–340. [Google Scholar]
  25. Mesri, G.; Rokhsar, A. Theory of consolidation for clays. J. Geotech. Geoenviron. Eng. 1974, 100, 889–904. [Google Scholar]
  26. Deng, Y.; Xie, H.; Huang, R.; Liu, C. Law of nonlinear flow in saturated clays and radial consolidation. Appl. Math. Mech. 2007, 28, 1427–1436. [Google Scholar] [CrossRef]
  27. Law, K.T.; Lee, C.F. Initial gradient in a dense glacial till. In Proceedings of the 10th International Conference on Soil Mechanics and Foundation Engineering, Stockholm, Sweden, 15–19 June 1981; Volume 1, pp. 441–446. [Google Scholar]
Figure 1. Cross-sectional schematic of the MTB (units: mm).
Figure 1. Cross-sectional schematic of the MTB (units: mm).
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Figure 2. Top view of the MTB (units: mm).
Figure 2. Top view of the MTB (units: mm).
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Figure 3. Micro-PWP.
Figure 3. Micro-PWP.
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Figure 4. Tension–compression sensor.
Figure 4. Tension–compression sensor.
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Figure 5. Flowchart of the test procedure.
Figure 5. Flowchart of the test procedure.
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Figure 6. Force analysis of the USM.
Figure 6. Force analysis of the USM.
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Figure 7. Variation in hd and hb during the test conducted in fine sand.
Figure 7. Variation in hd and hb during the test conducted in fine sand.
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Figure 8. Variation in hd and hb during the test conducted in the silty clay layer.
Figure 8. Variation in hd and hb during the test conducted in the silty clay layer.
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Figure 9. Relationship between the stable hd and hb during a test conducted in the fine sand and silty clay layers.
Figure 9. Relationship between the stable hd and hb during a test conducted in the fine sand and silty clay layers.
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Figure 10. Schematic showing the variation in H observed during drilling of a confined aquifer [23] ((a): Stable initial water table; (b,c): Stable water table observed at different depths after entering the unsaturated zone; (d): Stable water table in confined aquifer).
Figure 10. Schematic showing the variation in H observed during drilling of a confined aquifer [23] ((a): Stable initial water table; (b,c): Stable water table observed at different depths after entering the unsaturated zone; (d): Stable water table in confined aquifer).
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Figure 11. VI curves from seepage tests.
Figure 11. VI curves from seepage tests.
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Figure 12. Curves showing the dynamic variation in Hs and Δh at different depths.
Figure 12. Curves showing the dynamic variation in Hs and Δh at different depths.
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Figure 13. Variation in Δh0 of the observation hole at different depths during the multistage test process.
Figure 13. Variation in Δh0 of the observation hole at different depths during the multistage test process.
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Figure 14. Schematic showing the stable hd and H0 (units: mm).
Figure 14. Schematic showing the stable hd and H0 (units: mm).
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Figure 15. Comparison of the theoretically calculated and measured values ofΔh0.
Figure 15. Comparison of the theoretically calculated and measured values ofΔh0.
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Table 1. Sensor parameters.
Table 1. Sensor parameters.
NameTypeMeasuring RangeComposite Error (%F·S)Diameter × Thickness (mm)MaterialImpedance (Ω)
Micro-PWP sensorYSV32010–30 kPa≤0.05Φ15 × 20Stainless steel350
Tension–compression sensorH-20–100 kg≤0.05Φ40 × 30Stainless steel400 ± 10
Table 2. Physical property metrics of the fine sand sample used in the test.
Table 2. Physical property metrics of the fine sand sample used in the test.
Test Soil SampleMoisture Content
W/%
Wet Density
ρ/(g·cm–3)
Porosity
e
Saturation
Sr/%
Dry Density
ρd/(g·cm–3)
Permeability Coefficient
k (cm·s–1)
Sand26.71.960.6496.11.635.59 × 10–3
Table 3. Physical property metrics of the silty clay sample used in the test.
Table 3. Physical property metrics of the silty clay sample used in the test.
Test Soil SampleW/%ρ/(g·cm–3)Porosity n/%Sr/%ρd/(g·cm–3)Plasticity Index IpLiquidity Index ILPermeability Coefficient
k (cm·s–1)
Silty clay21.11.7144.695.91.3513.40.461.45 × 10–5
Table 4. Δh0 and L0 for observation holes at different depths and stages.
Table 4. Δh0 and L0 for observation holes at different depths and stages.
Observation HoleDepth (mm)L0 (mm)Δh0 (mm)
Stage 1Stage 2Stage 3Stage 4Stage 5Stage 6Stage 7Average
G1200800 2726272526
G2300700 22232320232223
G34006001819201918201919
G45005001516171615171516
G56004001213121313141313
G670030081091010111010
G780020065677756
Table 5. Comparison of the stable hd and H0 in the MTB.
Table 5. Comparison of the stable hd and H0 in the MTB.
Test StageBoreholeStable hd (mm)Stable H0 (mm)Δh0 (mm)L0 (mm)
1Z11008119681
2Z218015723757
3Z321018624786
4Z427024426844
5Z533030129901
6Z636033030930
7Z741037733977
Table 6. Test results for I0 in the silty clay layer.
Table 6. Test results for I0 in the silty clay layer.
Test Sample123456Average
I0 (dimensionless)0.0310.0330.0350.0330.0290.0320.032
Table 7. Theoretically calculated values of Δh0 in the silty clay layer.
Table 7. Theoretically calculated values of Δh0 in the silty clay layer.
Observation Hole/BoreholeDepth (mm)L0 (mm)Theoretically Calculated Value of Δh0 (mm)
G120080026
G230070022
G340060019
G450050016
G560040013
G670030010
G78002006
Z131968122
Z224375724
Z321478625
Z415684427
Z59990129
Z67093030
Z72397731
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Xu, Y.; Du, C.; Ma, H.; Pang, W.; Huang, S.; Li, S. Experimental and Theoretical Explanations for the Initial Difference in the Hydraulic Head in Aquitards. Water 2022, 14, 3042. https://doi.org/10.3390/w14193042

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Xu Y, Du C, Ma H, Pang W, Huang S, Li S. Experimental and Theoretical Explanations for the Initial Difference in the Hydraulic Head in Aquitards. Water. 2022; 14(19):3042. https://doi.org/10.3390/w14193042

Chicago/Turabian Style

Xu, Yongliang, Chaoyang Du, Haizhi Ma, Wei Pang, Suhang Huang, and Shimin Li. 2022. "Experimental and Theoretical Explanations for the Initial Difference in the Hydraulic Head in Aquitards" Water 14, no. 19: 3042. https://doi.org/10.3390/w14193042

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