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Behaviour of a PVD unit cell under vacuum pressure and a new method for consolidation analysis
Computers and Geotechnics 120 (2020) 103415
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Research Paper
Behaviour of a PVD unit cell under vacuum pressure and a new method for consolidation analysis
T
Jin-chun Chaia,⁎, Hong-tao Fua,⁎, Jun Wangb, Shui-Long Shenc a
Department of Civil Engineering and Architecture, Saga University, Japan College of Architecture and Civil Engineering, Wenzhou University, Wenzhou, Zhejiang 325035, China c Department of Civil and Environmental Engineering, College of Engineering, Shantou University, Shantou, Guangdong 515063, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Vacuum consolidation Laboratory tests Finite element analysis PVD
The behaviour of a prefabricated vertical drain (PVD) unit cell with clays of high initial water content has been investigated by laboratory model tests and finite element analysis (FEA). The results indicate that there is considerable horizontal soil movement towards the PVD. The soils in the zone near the PVD are in horizontal compression, and the other zones are in horizontal tension. The surface settlements are not uniform, and the soil at the periphery of the unit cell settles more than the soil near the PVD. The model test results indicate that the phenomenon of soil particle separation does not occur, and it implies that the main mechanism of apparent “clogging” is due to non-uniform consolidation. The results of the FEA indicate that at the end of vacuum consolidation, the soil adjacent to the PVD has a higher effective vertical stress as well as undrained shear strength, but the stress state is very close to the critical state line. Then, a new explicit consolidation analysis method has been established/verified for considering the variations of consolidation properties, including apparent clogging during the consolidation process.
1. Introduction There are a number of land reclamation projects using dredged clays as fill materials [1]. In addition, to maintain port facilities, large amounts of dredged clayey soils are generated every year in many countries, such as China and Japan. To accelerate the consolidation process of the dredged clays, vacuum consolidation with installation of prefabricated vertical drains (PVDs) is a commonly used method (e.g. [2,3]). While using vacuum consolidation for dredged clays with high initial water contents, there are reported successful cases, but there are also reported cases in which vacuum consolidation could not result in the designed improvement effect [4]. The reported main reason for unsuccessful improvement is “clogging” around the PVD. The fundamental mechanisms of the clogging can be the migration of finer soil particles to around the PVD (particle separation) [5] and/or a nonuniform consolidation-induced apparent “smear” effect around the PVDs [6]. However, there is limited evidence to clarify which of the mechanisms is the dominant one. The theories normally used for calculating the rate of PVD-induced consolidation are called PVD unit cell consolidation theories (e.g., [7,8]). A PVD unit cell is a cylinder consisting of a PVD and its improved soil volume. The theories were derived assuming that the strain
⁎
in the unit cell is mainly in the vertical direction. However, field evidence shows that for a PVD unit cell with clays of high initial water content, vacuum consolidation can induce considerable horizontal soil movement towards the PVD and form a hard soil column around the PVD (e.g., [9–11]). For such cases, the vacuum consolidation does not yield the expected results. In the past few decades, there have been a large number of publications in the literature about PVD-induced consolidation. There are solutions considering both the radial and vertical drainages (e.g., [12,13]); solutions considering large strain [14]; a design chart for considering the variation of the coefficient of consolidation during the consolidation process [15]; and a method for considering the effect of non-uniform consolidation during the consolidation process [6]. However, there is no report about the magnitude of horizontal strain in a PVD unit cell as well as its possible effect on the average degree of consolidation of the unit cell and the method(s) to consider it. In this study, the fundamental mechanism of clogging and the magnitude of horizontal strain within a PVD unit cell were investigated by laboratory model tests and finite element analysis (FEA). The model test results are presented first. Then, the results of the FEA using a prototype PVD unit cell are presented with discussions in terms of the distribution of horizontal strain and stress states within the unit cell.
Corresponding authors. E-mail addresses: [email protected] (J.-c. Chai), [email protected] (H.-t. Fu), [email protected] (J. Wang), [email protected] (S.-L. Shen).
https://doi.org/10.1016/j.compgeo.2019.103415 Received 2 August 2019; Received in revised form 20 November 2019; Accepted 22 December 2019 Available online 22 January 2020 0266-352X/ © 2019 Elsevier Ltd. All rights reserved.
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Fig. 1. Sketch of the model and the top sealing method.
pressure to the mini-PVD through an air/water separation tank (the right upper corner in Fig. 1(b)), and start the test. (5) Measure the following items: (a) Vacuum pressure at the back-end of the model (Fig. 1(a)) by the data-logging device. (b) Amount of water drained out (water in the air/water separation tank). (c) Surface settlements along the A-A′ line (Fig. 1(a)) by the Vernier callipers manually. (d) After vacuum consolidation, the water content of the soil, the dry soil mass per unit area, and the grain size distribution curves of the clays at different radial distances from the miniPVD. The soil samples were obtained by a small soil sampler with an inner diameter of 13.5 mm and wall thickness of 1.0 mm.
Finally, a new explicit method for considering the effect of apparent clogging as well as the variation of the coefficient of consolidation during the consolidation process on the degree of consolidation is proposed. The method has been validated using the results of FEAs and a model test reported in the literature. 2. Laboratory model tests 2.1. Model and test procedure A small-scale model was used. The model box is a trapezoidal shape made by PVC to simulate approximately 1/8 of a PVD unit cell. The inside dimensions of the box are illustrated in Fig. 1(a). The straight edges (back-end) are a simple approximation of an arc. It is considered that this simplification may not have considerable effect on the test results. With this small model, the vacuum pressure can be easily applied to the mini-PVD without penetrating through the sealing sheet. A piezometer is installed at the back end of the box (corresponding to the periphery side of a PVD unit cell) for measuring the vacuum pressures (Fig. 1(a)). The main purpose of the small-scale model test is to clarify the issues: (1) is there soil particle separations during a vacuum consolidation, and (2) is there horizontal soil movement within the model ground. The test procedures are as follows:
2.2. Material used Remoulded Ariake clay in Saga, Japan, was used. Ariake clay is a marine clay, and its natural water content is generally more than 100%. It has a high compressibility and low strength (e.g., [16,17]). The soil was sampled approximately 2.0 m below the ground surface. Some physical properties and odometer test results of the soil are listed in Table 1. The soil was thoroughly mixed at a designed initial water content in a container and then covered by a plastic sheet and cured for at least 24 h before used for the model test. A mini-PVD was cut from a commercial PVD with dimensions of 5.5 mm in thickness, 35 mm in width and 120 mm in length. The filter was made of polyester and the core of polyolefin.
(1) Smear grease on the inside walls of the model box for reducing the friction between the soil and the model box. Install the mini-PVD in the position as illustrated in Fig. 1(a) (corresponding to the centre of a unit cell). (2) Place well-mixed clay in the box up to 130 mm in thickness layer by layer, and carefully level the surface. (3) Apply a thin layer of sealant on the top edges of the model box and then seal the top surface using a flexible rubber sheet (approximately 0.3 mm in thickness). To consider possible settlement of the model ground, set the sheet in a loose condition with several folds. Then, apply a confining pressure on the sheet and the top edges of the model box using two wooden plates, one at the bottom and the other at the top of the model, and tighten those using threaded rods and nuts (Fig. 1(b)). Make an opening in the top plate to allow air pressure to be applied on the top surface of the sheet as well as to observe the top surface of the model ground and to measure the settlement during the consolidation process. (4) Connect the piezometer to a data-logging device. Apply vacuum
Table 1 Selected physical and mechanical properties of the soil. Grain size (%) > 5 μm
< 5 μm
28
72
Specific gravity, G
Liquid limit wl (%)
Plastic limit wp (%)
λ
kv (10−9 m/s)
cv (10−8 m2/s)
2.63
115.8
60.8
0.32
1.98
7.87
Note: λ = slope of the virgin compression lines in e − ln p′ plot (p′ is the consolidation stress); kv = permeability; and cv = coefficient of consolidation. The values of kv and cv are for vertical consolidation stress from 20 to 40 kPa. 2
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2.3. Cases tested This study targets the consolidation of dredged clay slurry, which normally has a water content of 1.0–2.0 times its liquid limit [18]. Two cases (Case 1 and Case 2) were tested with initial water contents (w0) of 150% and 185%, and corresponding unit weights of 13.1 kN/m3, and 12.6 kN/m3, respectively. The initial water contents approximately correspond to 1.3 and 1.6 times the liquid limit of the clay. For a soil with an initial water content less than 2.0 times its liquid limit, there will be self-weight induced consolidation but no particle sedimentation process will occur (e.g. [19,20]). For both cases, the applied vacuum pressure at the mini-PVD location was −60 kPa. A small air-shooting device was used to generate a vacuum pressure from compressed air. With this device a steady vacuum pressure of about −60 kPa can be easily maintained and this value was adopted for the tests. The tests lasted until there was no measurable water draining from the model ground.
Fig. 3. Measured vacuum pressures.
was drained out, the higher hydraulic gradient around the mini-PVD was no longer required, and the vacuum propagation rate was increased and the measured vacuum pressure at the time of termination the test was almost the same as that of w0 = 150% case.
2.4. Test results (1) Amount of water drained out (Q)
(3) Final distributions of the settlement, water content and mass per unit area
Fig. 2 shows the variation of the amount of water that drained out from the model ground. After approximately 35 days, the drainage rate was close to zero. The higher the initial water content, the larger the amount of the water that drained out. The model ground had a volume of approximately 4.04 l (4.04 × 10−3 m3), and the corresponding average volumetric strains are calculated and indicated on the vertical axis on the right side of Fig. 2.
Fig. 4(a)–(c) show the final distributions of the surface settlement along the A-A′ line (Fig. 1(a)), water content and normalized mass per unit area. The normalized mass per unit area is the value of the dry mass per unit area at a given location divided by the average mass per unit area in the model. The settlements along the A-A′ line are the largest at a given radial distance. The settlements near the walls of the model box were smaller due to friction between the soil and the wall and/or adhesion of the clays to the wall (even it was greased). An interface friction angle of about 3° was reported for a greased PVC plate and Ariake clay by Chai and Nguyen [21]. It can be seen that the settlements were not uniform and that the nearer to the mini-PVD, the less
(2) Measured vacuum pressures Fig. 3 shows the measured vacuum pressures at the back-end of the model. Before the termination of the tests, the measured vacuum pressure was approximately −35 kPa, and the absolute value was less than the applied value of −60 kPa at the mini-PVD location. The possible reason is that the soil horizontally moved towards the miniPVD, and formed a harder zone around the mini-PVD, causing apparent clogging of the mini-PVD which reduced the radial propagation of the vacuum pressure. For the case of w0 = 185%, at the earlier stage the measured vacuum pressures were lower than that of the case of w0 = 150%. The reason considered is the higher the initial water content, the stronger the non-uniform consolidation, and there is more water needs to be drained out, which requires a higher hydraulic gradient in the zone around the mini-PVD. As a result, larger amount of the applied vacuum pressure was consumed around the mini-PVD and reduced the propagation rate to the back-end of the model. Then when the most water
Fig. 4. Final distributions of the surface settlement, water content and mass per unit area.
Fig. 2. Variations of the amount of water that drained out. 3
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the settlement was (Fig. 4(a)). Due to the adhesion of the clay to the wall of the model box and/or the effect of interface friction, at the backend (corresponding to the periphery of the unit cell), the settlement was not the largest. The nearer to the mini-PVD, the lower the water content of the clays was (Fig. 4(b)). For the test with an initial water content of 185%, it resulted in more settlement (Fig. 4(a)) but still higher final water content. Further, there were horizontal soil movements towards the mini-PVD within the model ground (Fig. 4(c)), and adjacent to the mini-PVD, the dry soil mass per unit area was approximately 1.5 times the average value. Although the effect is not very obvious, Fig. 4(c) shows that the higher the initial water content, the stronger was the tendency of the horizontal soil movement towards the mini-PVD. The horizontal soil movement can form a harder zone around the mini-PVD, which can cause apparent clogging of the mini-PVD and reduce the radial propagation rate of the vacuum pressure. (4) Grain size distributions For the test with w0 = 185%, after the vacuum consolidation test, the soils were sampled at approximate radial distances of r = 10 mm, 110 mm and 210 mm from the mini-PVD, and grain size distribution tests were conducted. The results are shown in Fig. 5. The three curves are almost identical, and there is no sign of soil particle separation. Although the model test has its limitations, such as the effect of interface friction, the scale is very small (not represents the field situation), etc., the results of the model tests provided direct physical evidences that: (1) there was no separation of soil particles, and (2) there was horizontal soil movement within the unit cell during the vacuum consolidation process. Especially, the possible particle separation phenomenon cannot be investigated by a numerical analysis using continuous mechanics.
Fig. 6. FEA model with mesh.
undrained condition to form an under-consolidated model ground, and the force equilibrium was achieved by several iterations. The stress-strain relationship of the clay was modelled by the modified Cam clay (MCC) model [22]. The adopted model parameters are listed in Table 2. The simulated soil properties are the same as the soil tested except the initial water content. The adopted initial water content of 165% (about 1.43 times the liquid limit of the soil) is approximately a middle value of the two model tests. The corresponding initial void ration e0 was 4.37. Hong et al. [19] reported that even for slurry with an initial water content (w0) of approximately 1.5wl, there is an apparent yield stress, which was named the “remoulded yield stress”. Chai et al. [20] reported that for Ariake clay with an initial water content of approximately 1.5wl, the remoulded yield stress is approximately 2.0 kPa. This value was adopted for setting up the isotropic initial effective stresses in the model ground. Further, for Holocene Ariake clay in Saga, Japan, its permeability (k) varies with void ratio (e) follows Taylor’s (1948) equation [23].
3. Numerical investigation To provide some further insight understandings about a PVD unit cell with clays of high initial water content under the field condition, vacuum consolidation of a prototype PVD unit cell (not the model test) was simulated by finite element analysis (FEA). 3.1. Analysis model
k = k 0·10−(e0 − e)/ Ck
The adopted finite element analysis model with the mesh is shown in Fig. 6. The model simulates a PVD unit cell with a diameter (De) of 1.0 m and a height (depth) of 5.0 m. For easy analysis/interpretation of the results, only horizontal radial drainage was allowed. The initial condition of the model simulates a newly reclaimed clay deposit, and its self-weight-induced consolidation is not completed. To do so, the model was initially established without turning on the gravity force of the model ground. Then the gravity force was turned on under an
(1)
where e = void ratio; e0 = initial void ratio; k0 = initial permeability corresponding to e0; and Ck = a constant, the slope in the ln(k) − e plot. The value of Ck is approximately 1.2 for Holocene Ariake clay [20]. With the soil properties listed in Table 1 and the initial conditions of the model ground described in the above, an initial horizontal permeability (kh0) of approximately 1.0 × 10−8 m/s was evaluated by Eq. (1). In addition, during the consolidation process, permeability was varied with void ratio according Eq. (1). The PVD has an equivalent diameter (dw) of 50 mm. The whole model-ground was remoulded, and no smear effect was considered. Further, it was assumed that the PVD had a larger discharge capacity (qw) and that well resistance could be ignored. Eight-node quadrilateral Table 2 Model parameters. ν
λ
κ
M
Ck
De (m)
dw (mm)
qw
0.15
0.32
0.032
1.2
1.2
1.0
50
∞
Note: ν = Poisson’s ratio; λ and κ = slopes of the unloading–reloading compression line in e − ln p′ plot (p′ is the consolidation stress), respectively; Μ = slope of the critical state line in the p′-q plot (q is the deviator stress); Ck = the constant in Taylor’s (1948) equation (Eq. (1)); De = diameter of the unit cell; dw = equivalent diameter of the PVD; and qw = discharge capacity of the PVD.
Fig. 5. Grain size distribution curves. 4
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elements with excess pore water degree of freedom only at the vertex nodes were used for the model ground. The applied vacuum pressure was −80 kPa. Coupled analyses were carried out using the CRISP-AIT program [24], which is based on the original CRISP program [25]. The phenomenon of large deformation was approximately simulated by updating nodal coordinates at the end of each incremental step. To compare the degree of consolidation (DOC) from the FEA and that from Hansbo’s [8] solution, a representative value of the coefficient of consolidation in the horizontal direction (chrep) was calculated by a representative effective stress determined by the following equation [26]:
′ = σvrep
σvi′ σvf′
(2)
where σ′vi and σ′vf = the initial and final vertical effective stresses, respectively. With the parameters/conditions adopted, σ′vrep = 13.4 kPa and chrep = 6.26 × 10−8 m2/s were evaluated.
Fig. 8. Final surface settlement profile.
3.2. FEA results (1) Horizontal strains (εh) The distributions of the horizontal strains in the model ground are shown in Fig. 7(a) and (b) for an average DOC of approximately 18% (earlier stage) and 100% (final), respectively. Both figures show that in the zone near the PVD, εh is in compression, and in the zones away from the PVD, εh is in tension. A comparison of Fig. 7(a) and (b) indicates that the non-uniform distributions of εh are more significant at the earlier stage of consolidation. Further, at the final stage (Fig. 7(b)), the shallower the depth, the more non-uniform the horizontal strains are. (2) Settlements Fig. 9. Comparison of the time-settlement curves.
Fig. 8 shows the final surface settlement profile. The settlement at the periphery of the PVD unit cell is larger than that at the centre (PVD location). The difference is approximately 54 mm. The non-uniform settlements occur only within shallow depths, and below approximately 0.5 m depth, the settlements are almost uniform, as shown in Fig. 9. (3) Final distributions of stresses and void ratios Fig. 10 shows the final distribution of the effective vertical (σ′z), radial (σ′r) and tangential (σ′θ) stresses at a depth of 2.5 m (initial value) of the model ground. The zone near the PVD consolidated much faster than the zone near the periphery of the unit cell and caused horizontal movement of the soils (Fig. 7). Then, during the consolidation of the soil near the periphery in the later stage, shear deformation was induced within the unit cell, which increased σ′z and σ′θ in the zone near the PVD. The value of σ′z in the soil adjacent to the PVD was approximately 3.0 times that in the soil near the periphery of the unit cell. For
Fig. 10. Distributions of the stresses with radial distance.
Fig. 7. Distributions of horizontal strains. 5
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Fig. 11. Distribution of the final void ratios. Fig. 13. Distribution of the undrained shear strength.
σ′r, the consolidation of the soils near the periphery caused a reduction of the horizontal compressive strains (comparing Fig. 7(a) and (b)) near the PVD, which caused a certain reduction of σ′r in the zone adjacent to the PVD at r < 0.05 m. For r > 0.05 m, σr slightly reduced with increasing r. Fig. 11 shows the final distribution of void ratios in the radial direction at 2.5 m depth. The non-uniform deformation caused non-uniform effective stress distributions and ultimately a non-uniform void ratio distribution. The nearer to the PVD, the smaller the void ratio is.
unit cell, the stress and density states are highly non-uniform even at the end of vacuum consolidation. The existing theories for analysing the consolidation/deformation of a unit cell do not consider this kind of non-uniformity. It is worth checking the applicability of the existing consolidation theories/methods for a PVD unit cell with clays of high initial water content.
4. a new method for analysing PVD-induced consolidation (4) Stress paths and undrained shear strengths 4.1. Variation of the coefficient of consolidation with the effective stress Fig. 12 shows the stress paths at r = 0.03 m, 0.255 m, and 0.475 m and at 0.5 m and 2.5 m depths. At r = 0.03 m (adjacent to the PVD), during the process of consolidation, the effective stress paths almost reached the critical state line (CSL) in the p′ − q plot (p′ is effective mean stress and q is deviator stress). For the points away from the PVD, the effective stress paths moved along approximately constant q/p′ ratio lines and far from the CSL. This tendency did not change with depth (comparing Fig. 12(a) and (b)). The simulated final distribution of the undrained shear strength (su) [27] is shown in Fig. 13. The values of su in the zone adjacent to the PVD were much higher than (approximately four times) those in other zones. During the process of vacuum consolidation, su in the zone adjacent to the PVD was almost fully mobilized (Fig. 12).
The coefficient of consolidation in the horizontal direction (ch) is a function of the permeability (kh) and the coefficient of volume compressibility (m) of a soil, i.e.,
ch =
kh mγw
(3)
where γw = the unit weight of water. Adopting a linear e – ln(p′) relationship, and in a normally consolidated state, m is:
m=
λ (1 + e ) p′
(4)
where λ = the compression index in the e − ln(p′) plot. Using the linear e − log(kh) relationship proposed by Taylor [23], the variation of kh with e can be expressed by Eq. (1). Substituting Eqs. (1) and (4) into Eq. (3), the following expression for ch can be obtained.
(5) Discussions The results of FEA presented above clearly indicate that for a PVD
Fig 12. Effective stress paths. 6
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consolidation of a PVD unit cell. To consider the variations of ch and the apparent smear effect into the PVD-induced consolidation analysis, a new pragmatic method has been proposed in this study. 4.3. Proposed new method for calculating the average degree of consolidation Hansbo’s [8] solution is widely used for analysing PVD-induced consolidation, and the average DOC (Uh) is:
Uk = 1 − exp(−8Tk / μ)
(6)
where μ = the effect of spacing between the PVDs, the smear effect and well resistance on DOC, and Th = the time factor. These parameters can be expressed as:
ch =
Fig. 14. The zones for ch increasing/decreasing with increasing p′.
μ = ln(n/ s ) + (Kh/ Ks ) ln(s ) −
kh0 ((1 + e0)/ λ − ln(p′ / p0′ ))·p′ γw (p′ / p0′ )2.3λ / Ck
Th = (5)
3 2π·l 2·Kh + 4 3qw
ch t De2
(7) (8)
where De = the diameter of the unit cell, n = De/dw, s = ds/dw (ds is the diameter of the smear zone), l = the drainage length of a PVD, kh = the horizontal permeability of the soil, and ks = the horizontal permeability of the smear zone. Eq. (6) was derived under the condition of constant ch and μ. To consider the effect of varying ch and μ during the consolidation process, we propose the following incremental calculation method. The key points of the method are as follows:
where p′0 = the initial effective mean stress. There are soils in which ch increases with increasing p′ and there are soils in which ch decreases with increasing p′. As a general qualitative tendency, a soil is less compressible (smaller λ value) and the rate of reduction of permeability with void ratio is smaller (larger Ck value), ch will increase with the increase of p′. While for a soil with a larger λ and a smaller Ck values, ch will reduce with the increase of p′ Therefore whether ch will increase or decrease with increasing p′is a function of the combination of λ and Ck. In this study, it has been found that in λ–λ/Ck plot, the zones ch increase or decrease with increase of p′ can be identified as in Fig. 14. Within the zone between two dashed lines, the variation of ch with p′ is small, and for some cases, ch is approximately a constant with the variation of p′. Above the dashed zone, ch decreases with increasing p′, and below the dashed zone, ch increases with increasing p′. For the cases assumed in the FEA presented above (λ = 0.32 and Ck = 1.2), ch increases with increasing p′. The initial kh = 10−8 m/s and m = 0.03 m2/kN (initial vertical effective stress of 2 kPa), which results in a ch = 3.4 × 10−8 m2/s. Under a vertical effective stress of 82 kPa (80 kPa vacuum pressure plus 2 kPa initial stress), kh = 1.0 × 10−9 m/s and m = 9.6 × 10−4 m2/kN and then ch = 1.1 × 10−7 m2/s can be calculated, which is about 3.2 times of the initial value. Zhou and Chai [6] reported that the non-uniform consolidation-induced apparent smear effect also varies during the consolidation process. All existing explicit solutions for PVD-induced consolidation assume a constant ch and a constant smear effect.
(1) At the ith time increment, use values of ch and μ corresponding to the stress state at the end of the i − 1th time increment. (2) To ensure that the continuity of the calculated DOC at the beginning of the ith time increment is the same as that at the end of the i − 1th time increment, use a theoretically determined imaginary time at the beginning of the ith time increment. The procedure of the calculation is as follows: (a) For the first time increment Δt1, calculate U1 = ΔU1 using Eq. (6) (b) Starting at the second time increment: (i) For the ith time increment, calculate the total effective mean stress increment at the end of the i − 1th time increment: Δp′i−1 = Ui−1∙p (p is the total applied load up to i − 1th time increment). Then, the effective mean stress at the end of Δti−1 is p′i−1 = p′0 + Δp′i−1 (p′0 is the initial effective mean stress) (ii) Calculate the void ratio corresponding to p′i−1 as:
ei − 1 = e0 − λ ln(pi′− 1 / p0′ )
4.2. Considering the effect of horizontal strain
(9)
(iii) Calculate mi−1 and khi−1 using Eqs. (4) and (1), respectively, and chi−1 by Eq. (3). (iv) Calculate the change of the apparent smear effect using the method proposed by Zhou and Chai [6]. For Uh < 50%, the apparent smear effect does not change; for 50 < Uh < 95% (0.087 < Th/ μ < 0.375), kh/ks linearly reduces from a value larger than 1.0 to 1.0 (Fig. 13 in Zhou and Chai [6]). Then, calculate μi−1 by Eq. (7). (v) Calculate an imaginary time for the start of ith time increment, t0i, as:
Zhou and Chai [6] proposed a method for considering the effect of non-uniform consolidation on the average degree of consolidation of a PVD unit cell. The method was based on the results of the FEA, which inexplicitly included the effect of the horizontal movement of soils within a PVD unit cell. The FEAs conducted by Zhou and Chai [6] for PVD unit cells were under the surcharge loads. In this study, only considering the horizontal radial drainage, consolidations of a PVD unit cell under both a vacuum pressure and a distributed load of the same magnitude as the vacuum pressure under the free vertical strain conditions were conducted. The results indicate that the deformations of the PVD unit cell for the both cases are identical, i.e. the vacuum pressure and the surcharge load induced the same soil deformation of the PVD unit cell. Therefore, it is considered that the apparent smear (clogging) effect estimated by Zhou and Chai’s method includes both the effects of non-uniform consolidation and horizontal strain in a PVD unit cell. However, there was no rational way established for considering its effect in the calculation of the average degree of
t0i =
μi − 1 ·chi − 2 (t0i − 1 + Δti − 1) μi − 2 ·chi − 1
(10)
where t01 = 0 and ch0 and μ0 are the initial values. (vi) Calculate ΔUhi and Uhi as:
7
ΔUhi = exp(−8chi − 1 t0i/ μi − 1) − exp(−8chi − 1 (t0i + Δti )/ μi − 1)
(11)
Uhi = Uhi − 1 + ΔUhi
(12)
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Fig. 15. Comparison of DOCs from different methods.
Fig. 16. Comparison of measured and calculated DOCs.
All the calculations can be carried out using an Excel spreadsheet. However, since the DOC is a function of μ, to find a time corresponding to 95% DOC, some iterations are required. First, the time for Uh = 95% with a constant μ value can be used as a starting value for the iterations.
comparison of the measured and calculated DOCs is given in Fig. 16, which clearly shows that the results from the proposed method are very close to the measured data. The results in Figs. 15 and 16 indicate that the proposed method can consider the effect of non-uniform consolidation and the horizontal movement of soil within a PVD unit cell and the variations of the coefficient of consolidation and the apparent smear effect in the calculation of the average DOC of a PVD unit cell.
4.4. Comparison with the results of the FEA With the conditions adopted for the FEA presented above, from Zhou and Chai [6], with ds/dw = 4, and De = 1.0 m, a value of (kh/ks)e of approximately 3.3 can be estimated. Fig. 15 shows the comparison of average DOCs calculated by the excess pore water pressures from the FEA, the proposed method, and using Eq. (6) with chrep = 6.26 × 10−8 m2/s. The figure clearly shows that the DOCs from the proposed method are very close to those of the FEA. The theory (Eq. (6)), which does not consider the apparent smear effect, over-estimates the DOCs significantly.
5. Conclusions The behaviour of a prefabricated vertical drain (PVD) unit cell with clays of high initial water content has been investigated by laboratory model tests and finite element analysis (FEA). Then, a new explicit method for consolidation analysis of a PVD unit cell has been established. Based on the test and analysis results, the following conclusions can be drawn.
4.5. Comparison with a model test results
(1) Horizontal soil movement within a PVD unit cell. The results from both the model test and the FEA indicate that there is considerable horizontal soil movement towards the PVD. The soils in the zone near the PVD are in horizontal compression, and the other zones are in horizontal tension. The compressive strain can be more than 10%, and the extension strain can be 2–3%. (2) Non-uniform surface settlements. The surface settlements of a PVD unit cell are not uniform, and at the periphery of a unit cell settles more than the region near the PVD. The results of the FEA indicate that at deeper locations (> 0.5 m), the settlements are almost uniform. (3) Non-uniform vertical stress and strength distribution. The results of the FEA indicate that under the assumed conditions and at the end of a vacuum consolidation, the effective vertical stress in the soil adjacent to the PVD is approximately 3.0 times that in the soil near the periphery of the unit cell. The soil adjacent to the PVD also has a higher undrained shear strength, but the stress state is very close to the critical state line. (4) Mechanism of apparent clogging. The model test results indicate that the phenomenon of soil particle separation did not occur, which implies that the main mechanism of apparent clogging is due to non-uniform consolidation and horizontal soil movement. (5) A new method for consolidation analysis. A new incremental consolidation calculation method has been established for considering the variations of consolidation properties, including apparent clogging during the consolidation process. The calculated average degree of consolidation agreed well with the results of the FEA and with the results of a model test reported in the literature.
Chai et al. [28] reported the results of a laboratory PVD unit cell model test. The model ground had a diameter of 0.45 m and height of 0.8 m. A mini-PVD was installed at the centre of the model ground. Ariake clay with an initial water content of approximately 180% was used to form the model ground. The reported initial void ratio was 4.75, and the compression index (Cc) of the clay was 0.99. As discussed previously, for Ariake clay, the slope in the ln(kh) − e plot is approximately 1.2 (corresponding to the value of Ck in Taylor’s equation). Then, an initial permeability of 1.5 × 10−8 m/s was extrapolated from the reported average permeability for the consolidation pressure from 40 to 80 kPa. The loads applied were a 40 kPa vacuum pressure and a 10 kPa surcharge load. The reason for adopting a combined load was to prevent possible gaps between the top surface of the model ground and the loading piston above it. Three piezometers were installed in the middle height of the model ground at radial distances of 50, 100 and 200 mm from the centre for measuring the variations of pore water pressures (u) during the consolidation process. Dividing the model ground into three zones with two division lines at the middle of each two adjacent piezometers and assuming that the measured values of u by each piezometer represented the average values of each zone, the average DOC of the model ground was calculated, which is regarded as the measured value. Using Zhou and Chai’s [6] method, an initial apparent smear effect of (kh/ks)e = 4.0 was evaluated. Assuming a remoulded yield stress of 2.0 kPa, the average DOCs of the model ground by the proposed method was calculated. Then, using Eq. (2), a representative effective stress of 10 kPa and a corresponding representative horizontal coefficient of consolidation chrep = 4.77 × 10−8 m2/s were evaluated. Then the average DOC by Hansbo’s solution [8] was calculated also. The 8
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CRediT authorship contribution statement [11]
Jin-chun Chai: Conceptualization, Methodology, Formal analysis, Writing - original draft. Hong-tao Fu: Data curation, Investigation. Jun Wang: Resources. Shui-Long Shen: Writing - review editing.
[12]
Declaration of Competing Interest [13]
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.
[14]
[15]
Acknowledgements
[16]
This work was supported by the National Natural Science Foundation of China with a grant number 51578514.
[17]
References
[18]
[1] Tang C, Ji L, Jia YS. Study of implement scheme for using Yangtze estuary waterways dredged soil to Hengsha east shoal reclamation. Eng Sci 2013;6:91–8. https:// doi.org/10.3969/j.issn.1009-1742.2013.06.016. [2] Chai JC, Miura N, Bergado DT. Preloading clayey deposit by vacuum pressure with cap-drain: analyses versus performance. Geotext Geomembr 2008;26(3):220–30. https://doi.org/10.1016/j.geotexmem.2007.10.004. [3] Shinsha H, Kumagai T, Miyamoto K, Hamaya T. Execution for the volume reduction of dredged soil using the vacuum consolidation method with horizontal pre-fabricated drains. Geotech Eng J Jpn Geotech Soc 2013;8(1):97–108. https://doi.org/10. 3208/jgs.8.97. [4] Bao SF, Lou Y, Dong ZL, Mo HH, Chen PS, Zhou RB. Causes and countermeasures for vacuum consolidation failure of newly-dredged mud foundation. Chin J Geotech Eng 2014;36(7):1350–9. https://doi.org/10.11779/CJGE201407020. [5] Xu K, Lin SF, Geng ZZ, Han RR. Experimental study on effects of vacuum loading modes on clogging of drainage board filtration membranes. Chin J Geotech Eng 2016;38(z2):123–9. https://doi.org/10.11779/CJGE2016S2020. [6] Zhou Y, Chai JC. Equivalent ‘smear’ effect due to non-uniform consolidation surrounding a PVD. Geotechnique 2017;67(5):410–9. https://doi.org/10.1680/jgeot. 16.P.087. [7] Barron RA. Consolidation of fine-grained soils by drain wells. Transactions of the American Society of Civil Engineers; 1948. p. 718–42. [8] Hansbo S. Consolidation of fine-grained soils by prefabricated drains. In: Proc. 10th Int Conf Soil Mech Found Eng; 1981; Stockholm 3, p. 677–82. [9] Wang J, Cai YQ, Ma JJ, Chu J, Fu HT, Wang P, Jin YW. Improved vacuum preloading method for consolidation of dredged clay-slurry fill. J Geotech Geoenviron Eng 2016;142(11):06016012. https://doi.org/10.1061/(ASCE)GT.1943-5606. 0001516. [10] Wang J, Ni JF, Cai YQ, Fu HT, Wang P. Combination of vacuum preloading and lime
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Research Paper
Behaviour of a PVD unit cell under vacuum pressure and a new method for consolidation analysis
T
Jin-chun Chaia,⁎, Hong-tao Fua,⁎, Jun Wangb, Shui-Long Shenc a
Department of Civil Engineering and Architecture, Saga University, Japan College of Architecture and Civil Engineering, Wenzhou University, Wenzhou, Zhejiang 325035, China c Department of Civil and Environmental Engineering, College of Engineering, Shantou University, Shantou, Guangdong 515063, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Vacuum consolidation Laboratory tests Finite element analysis PVD
The behaviour of a prefabricated vertical drain (PVD) unit cell with clays of high initial water content has been investigated by laboratory model tests and finite element analysis (FEA). The results indicate that there is considerable horizontal soil movement towards the PVD. The soils in the zone near the PVD are in horizontal compression, and the other zones are in horizontal tension. The surface settlements are not uniform, and the soil at the periphery of the unit cell settles more than the soil near the PVD. The model test results indicate that the phenomenon of soil particle separation does not occur, and it implies that the main mechanism of apparent “clogging” is due to non-uniform consolidation. The results of the FEA indicate that at the end of vacuum consolidation, the soil adjacent to the PVD has a higher effective vertical stress as well as undrained shear strength, but the stress state is very close to the critical state line. Then, a new explicit consolidation analysis method has been established/verified for considering the variations of consolidation properties, including apparent clogging during the consolidation process.
1. Introduction There are a number of land reclamation projects using dredged clays as fill materials [1]. In addition, to maintain port facilities, large amounts of dredged clayey soils are generated every year in many countries, such as China and Japan. To accelerate the consolidation process of the dredged clays, vacuum consolidation with installation of prefabricated vertical drains (PVDs) is a commonly used method (e.g. [2,3]). While using vacuum consolidation for dredged clays with high initial water contents, there are reported successful cases, but there are also reported cases in which vacuum consolidation could not result in the designed improvement effect [4]. The reported main reason for unsuccessful improvement is “clogging” around the PVD. The fundamental mechanisms of the clogging can be the migration of finer soil particles to around the PVD (particle separation) [5] and/or a nonuniform consolidation-induced apparent “smear” effect around the PVDs [6]. However, there is limited evidence to clarify which of the mechanisms is the dominant one. The theories normally used for calculating the rate of PVD-induced consolidation are called PVD unit cell consolidation theories (e.g., [7,8]). A PVD unit cell is a cylinder consisting of a PVD and its improved soil volume. The theories were derived assuming that the strain
⁎
in the unit cell is mainly in the vertical direction. However, field evidence shows that for a PVD unit cell with clays of high initial water content, vacuum consolidation can induce considerable horizontal soil movement towards the PVD and form a hard soil column around the PVD (e.g., [9–11]). For such cases, the vacuum consolidation does not yield the expected results. In the past few decades, there have been a large number of publications in the literature about PVD-induced consolidation. There are solutions considering both the radial and vertical drainages (e.g., [12,13]); solutions considering large strain [14]; a design chart for considering the variation of the coefficient of consolidation during the consolidation process [15]; and a method for considering the effect of non-uniform consolidation during the consolidation process [6]. However, there is no report about the magnitude of horizontal strain in a PVD unit cell as well as its possible effect on the average degree of consolidation of the unit cell and the method(s) to consider it. In this study, the fundamental mechanism of clogging and the magnitude of horizontal strain within a PVD unit cell were investigated by laboratory model tests and finite element analysis (FEA). The model test results are presented first. Then, the results of the FEA using a prototype PVD unit cell are presented with discussions in terms of the distribution of horizontal strain and stress states within the unit cell.
Corresponding authors. E-mail addresses: [email protected] (J.-c. Chai), [email protected] (H.-t. Fu), [email protected] (J. Wang), [email protected] (S.-L. Shen).
https://doi.org/10.1016/j.compgeo.2019.103415 Received 2 August 2019; Received in revised form 20 November 2019; Accepted 22 December 2019 Available online 22 January 2020 0266-352X/ © 2019 Elsevier Ltd. All rights reserved.
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Fig. 1. Sketch of the model and the top sealing method.
pressure to the mini-PVD through an air/water separation tank (the right upper corner in Fig. 1(b)), and start the test. (5) Measure the following items: (a) Vacuum pressure at the back-end of the model (Fig. 1(a)) by the data-logging device. (b) Amount of water drained out (water in the air/water separation tank). (c) Surface settlements along the A-A′ line (Fig. 1(a)) by the Vernier callipers manually. (d) After vacuum consolidation, the water content of the soil, the dry soil mass per unit area, and the grain size distribution curves of the clays at different radial distances from the miniPVD. The soil samples were obtained by a small soil sampler with an inner diameter of 13.5 mm and wall thickness of 1.0 mm.
Finally, a new explicit method for considering the effect of apparent clogging as well as the variation of the coefficient of consolidation during the consolidation process on the degree of consolidation is proposed. The method has been validated using the results of FEAs and a model test reported in the literature. 2. Laboratory model tests 2.1. Model and test procedure A small-scale model was used. The model box is a trapezoidal shape made by PVC to simulate approximately 1/8 of a PVD unit cell. The inside dimensions of the box are illustrated in Fig. 1(a). The straight edges (back-end) are a simple approximation of an arc. It is considered that this simplification may not have considerable effect on the test results. With this small model, the vacuum pressure can be easily applied to the mini-PVD without penetrating through the sealing sheet. A piezometer is installed at the back end of the box (corresponding to the periphery side of a PVD unit cell) for measuring the vacuum pressures (Fig. 1(a)). The main purpose of the small-scale model test is to clarify the issues: (1) is there soil particle separations during a vacuum consolidation, and (2) is there horizontal soil movement within the model ground. The test procedures are as follows:
2.2. Material used Remoulded Ariake clay in Saga, Japan, was used. Ariake clay is a marine clay, and its natural water content is generally more than 100%. It has a high compressibility and low strength (e.g., [16,17]). The soil was sampled approximately 2.0 m below the ground surface. Some physical properties and odometer test results of the soil are listed in Table 1. The soil was thoroughly mixed at a designed initial water content in a container and then covered by a plastic sheet and cured for at least 24 h before used for the model test. A mini-PVD was cut from a commercial PVD with dimensions of 5.5 mm in thickness, 35 mm in width and 120 mm in length. The filter was made of polyester and the core of polyolefin.
(1) Smear grease on the inside walls of the model box for reducing the friction between the soil and the model box. Install the mini-PVD in the position as illustrated in Fig. 1(a) (corresponding to the centre of a unit cell). (2) Place well-mixed clay in the box up to 130 mm in thickness layer by layer, and carefully level the surface. (3) Apply a thin layer of sealant on the top edges of the model box and then seal the top surface using a flexible rubber sheet (approximately 0.3 mm in thickness). To consider possible settlement of the model ground, set the sheet in a loose condition with several folds. Then, apply a confining pressure on the sheet and the top edges of the model box using two wooden plates, one at the bottom and the other at the top of the model, and tighten those using threaded rods and nuts (Fig. 1(b)). Make an opening in the top plate to allow air pressure to be applied on the top surface of the sheet as well as to observe the top surface of the model ground and to measure the settlement during the consolidation process. (4) Connect the piezometer to a data-logging device. Apply vacuum
Table 1 Selected physical and mechanical properties of the soil. Grain size (%) > 5 μm
< 5 μm
28
72
Specific gravity, G
Liquid limit wl (%)
Plastic limit wp (%)
λ
kv (10−9 m/s)
cv (10−8 m2/s)
2.63
115.8
60.8
0.32
1.98
7.87
Note: λ = slope of the virgin compression lines in e − ln p′ plot (p′ is the consolidation stress); kv = permeability; and cv = coefficient of consolidation. The values of kv and cv are for vertical consolidation stress from 20 to 40 kPa. 2
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2.3. Cases tested This study targets the consolidation of dredged clay slurry, which normally has a water content of 1.0–2.0 times its liquid limit [18]. Two cases (Case 1 and Case 2) were tested with initial water contents (w0) of 150% and 185%, and corresponding unit weights of 13.1 kN/m3, and 12.6 kN/m3, respectively. The initial water contents approximately correspond to 1.3 and 1.6 times the liquid limit of the clay. For a soil with an initial water content less than 2.0 times its liquid limit, there will be self-weight induced consolidation but no particle sedimentation process will occur (e.g. [19,20]). For both cases, the applied vacuum pressure at the mini-PVD location was −60 kPa. A small air-shooting device was used to generate a vacuum pressure from compressed air. With this device a steady vacuum pressure of about −60 kPa can be easily maintained and this value was adopted for the tests. The tests lasted until there was no measurable water draining from the model ground.
Fig. 3. Measured vacuum pressures.
was drained out, the higher hydraulic gradient around the mini-PVD was no longer required, and the vacuum propagation rate was increased and the measured vacuum pressure at the time of termination the test was almost the same as that of w0 = 150% case.
2.4. Test results (1) Amount of water drained out (Q)
(3) Final distributions of the settlement, water content and mass per unit area
Fig. 2 shows the variation of the amount of water that drained out from the model ground. After approximately 35 days, the drainage rate was close to zero. The higher the initial water content, the larger the amount of the water that drained out. The model ground had a volume of approximately 4.04 l (4.04 × 10−3 m3), and the corresponding average volumetric strains are calculated and indicated on the vertical axis on the right side of Fig. 2.
Fig. 4(a)–(c) show the final distributions of the surface settlement along the A-A′ line (Fig. 1(a)), water content and normalized mass per unit area. The normalized mass per unit area is the value of the dry mass per unit area at a given location divided by the average mass per unit area in the model. The settlements along the A-A′ line are the largest at a given radial distance. The settlements near the walls of the model box were smaller due to friction between the soil and the wall and/or adhesion of the clays to the wall (even it was greased). An interface friction angle of about 3° was reported for a greased PVC plate and Ariake clay by Chai and Nguyen [21]. It can be seen that the settlements were not uniform and that the nearer to the mini-PVD, the less
(2) Measured vacuum pressures Fig. 3 shows the measured vacuum pressures at the back-end of the model. Before the termination of the tests, the measured vacuum pressure was approximately −35 kPa, and the absolute value was less than the applied value of −60 kPa at the mini-PVD location. The possible reason is that the soil horizontally moved towards the miniPVD, and formed a harder zone around the mini-PVD, causing apparent clogging of the mini-PVD which reduced the radial propagation of the vacuum pressure. For the case of w0 = 185%, at the earlier stage the measured vacuum pressures were lower than that of the case of w0 = 150%. The reason considered is the higher the initial water content, the stronger the non-uniform consolidation, and there is more water needs to be drained out, which requires a higher hydraulic gradient in the zone around the mini-PVD. As a result, larger amount of the applied vacuum pressure was consumed around the mini-PVD and reduced the propagation rate to the back-end of the model. Then when the most water
Fig. 4. Final distributions of the surface settlement, water content and mass per unit area.
Fig. 2. Variations of the amount of water that drained out. 3
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the settlement was (Fig. 4(a)). Due to the adhesion of the clay to the wall of the model box and/or the effect of interface friction, at the backend (corresponding to the periphery of the unit cell), the settlement was not the largest. The nearer to the mini-PVD, the lower the water content of the clays was (Fig. 4(b)). For the test with an initial water content of 185%, it resulted in more settlement (Fig. 4(a)) but still higher final water content. Further, there were horizontal soil movements towards the mini-PVD within the model ground (Fig. 4(c)), and adjacent to the mini-PVD, the dry soil mass per unit area was approximately 1.5 times the average value. Although the effect is not very obvious, Fig. 4(c) shows that the higher the initial water content, the stronger was the tendency of the horizontal soil movement towards the mini-PVD. The horizontal soil movement can form a harder zone around the mini-PVD, which can cause apparent clogging of the mini-PVD and reduce the radial propagation rate of the vacuum pressure. (4) Grain size distributions For the test with w0 = 185%, after the vacuum consolidation test, the soils were sampled at approximate radial distances of r = 10 mm, 110 mm and 210 mm from the mini-PVD, and grain size distribution tests were conducted. The results are shown in Fig. 5. The three curves are almost identical, and there is no sign of soil particle separation. Although the model test has its limitations, such as the effect of interface friction, the scale is very small (not represents the field situation), etc., the results of the model tests provided direct physical evidences that: (1) there was no separation of soil particles, and (2) there was horizontal soil movement within the unit cell during the vacuum consolidation process. Especially, the possible particle separation phenomenon cannot be investigated by a numerical analysis using continuous mechanics.
Fig. 6. FEA model with mesh.
undrained condition to form an under-consolidated model ground, and the force equilibrium was achieved by several iterations. The stress-strain relationship of the clay was modelled by the modified Cam clay (MCC) model [22]. The adopted model parameters are listed in Table 2. The simulated soil properties are the same as the soil tested except the initial water content. The adopted initial water content of 165% (about 1.43 times the liquid limit of the soil) is approximately a middle value of the two model tests. The corresponding initial void ration e0 was 4.37. Hong et al. [19] reported that even for slurry with an initial water content (w0) of approximately 1.5wl, there is an apparent yield stress, which was named the “remoulded yield stress”. Chai et al. [20] reported that for Ariake clay with an initial water content of approximately 1.5wl, the remoulded yield stress is approximately 2.0 kPa. This value was adopted for setting up the isotropic initial effective stresses in the model ground. Further, for Holocene Ariake clay in Saga, Japan, its permeability (k) varies with void ratio (e) follows Taylor’s (1948) equation [23].
3. Numerical investigation To provide some further insight understandings about a PVD unit cell with clays of high initial water content under the field condition, vacuum consolidation of a prototype PVD unit cell (not the model test) was simulated by finite element analysis (FEA). 3.1. Analysis model
k = k 0·10−(e0 − e)/ Ck
The adopted finite element analysis model with the mesh is shown in Fig. 6. The model simulates a PVD unit cell with a diameter (De) of 1.0 m and a height (depth) of 5.0 m. For easy analysis/interpretation of the results, only horizontal radial drainage was allowed. The initial condition of the model simulates a newly reclaimed clay deposit, and its self-weight-induced consolidation is not completed. To do so, the model was initially established without turning on the gravity force of the model ground. Then the gravity force was turned on under an
(1)
where e = void ratio; e0 = initial void ratio; k0 = initial permeability corresponding to e0; and Ck = a constant, the slope in the ln(k) − e plot. The value of Ck is approximately 1.2 for Holocene Ariake clay [20]. With the soil properties listed in Table 1 and the initial conditions of the model ground described in the above, an initial horizontal permeability (kh0) of approximately 1.0 × 10−8 m/s was evaluated by Eq. (1). In addition, during the consolidation process, permeability was varied with void ratio according Eq. (1). The PVD has an equivalent diameter (dw) of 50 mm. The whole model-ground was remoulded, and no smear effect was considered. Further, it was assumed that the PVD had a larger discharge capacity (qw) and that well resistance could be ignored. Eight-node quadrilateral Table 2 Model parameters. ν
λ
κ
M
Ck
De (m)
dw (mm)
qw
0.15
0.32
0.032
1.2
1.2
1.0
50
∞
Note: ν = Poisson’s ratio; λ and κ = slopes of the unloading–reloading compression line in e − ln p′ plot (p′ is the consolidation stress), respectively; Μ = slope of the critical state line in the p′-q plot (q is the deviator stress); Ck = the constant in Taylor’s (1948) equation (Eq. (1)); De = diameter of the unit cell; dw = equivalent diameter of the PVD; and qw = discharge capacity of the PVD.
Fig. 5. Grain size distribution curves. 4
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elements with excess pore water degree of freedom only at the vertex nodes were used for the model ground. The applied vacuum pressure was −80 kPa. Coupled analyses were carried out using the CRISP-AIT program [24], which is based on the original CRISP program [25]. The phenomenon of large deformation was approximately simulated by updating nodal coordinates at the end of each incremental step. To compare the degree of consolidation (DOC) from the FEA and that from Hansbo’s [8] solution, a representative value of the coefficient of consolidation in the horizontal direction (chrep) was calculated by a representative effective stress determined by the following equation [26]:
′ = σvrep
σvi′ σvf′
(2)
where σ′vi and σ′vf = the initial and final vertical effective stresses, respectively. With the parameters/conditions adopted, σ′vrep = 13.4 kPa and chrep = 6.26 × 10−8 m2/s were evaluated.
Fig. 8. Final surface settlement profile.
3.2. FEA results (1) Horizontal strains (εh) The distributions of the horizontal strains in the model ground are shown in Fig. 7(a) and (b) for an average DOC of approximately 18% (earlier stage) and 100% (final), respectively. Both figures show that in the zone near the PVD, εh is in compression, and in the zones away from the PVD, εh is in tension. A comparison of Fig. 7(a) and (b) indicates that the non-uniform distributions of εh are more significant at the earlier stage of consolidation. Further, at the final stage (Fig. 7(b)), the shallower the depth, the more non-uniform the horizontal strains are. (2) Settlements Fig. 9. Comparison of the time-settlement curves.
Fig. 8 shows the final surface settlement profile. The settlement at the periphery of the PVD unit cell is larger than that at the centre (PVD location). The difference is approximately 54 mm. The non-uniform settlements occur only within shallow depths, and below approximately 0.5 m depth, the settlements are almost uniform, as shown in Fig. 9. (3) Final distributions of stresses and void ratios Fig. 10 shows the final distribution of the effective vertical (σ′z), radial (σ′r) and tangential (σ′θ) stresses at a depth of 2.5 m (initial value) of the model ground. The zone near the PVD consolidated much faster than the zone near the periphery of the unit cell and caused horizontal movement of the soils (Fig. 7). Then, during the consolidation of the soil near the periphery in the later stage, shear deformation was induced within the unit cell, which increased σ′z and σ′θ in the zone near the PVD. The value of σ′z in the soil adjacent to the PVD was approximately 3.0 times that in the soil near the periphery of the unit cell. For
Fig. 10. Distributions of the stresses with radial distance.
Fig. 7. Distributions of horizontal strains. 5
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Fig. 11. Distribution of the final void ratios. Fig. 13. Distribution of the undrained shear strength.
σ′r, the consolidation of the soils near the periphery caused a reduction of the horizontal compressive strains (comparing Fig. 7(a) and (b)) near the PVD, which caused a certain reduction of σ′r in the zone adjacent to the PVD at r < 0.05 m. For r > 0.05 m, σr slightly reduced with increasing r. Fig. 11 shows the final distribution of void ratios in the radial direction at 2.5 m depth. The non-uniform deformation caused non-uniform effective stress distributions and ultimately a non-uniform void ratio distribution. The nearer to the PVD, the smaller the void ratio is.
unit cell, the stress and density states are highly non-uniform even at the end of vacuum consolidation. The existing theories for analysing the consolidation/deformation of a unit cell do not consider this kind of non-uniformity. It is worth checking the applicability of the existing consolidation theories/methods for a PVD unit cell with clays of high initial water content.
4. a new method for analysing PVD-induced consolidation (4) Stress paths and undrained shear strengths 4.1. Variation of the coefficient of consolidation with the effective stress Fig. 12 shows the stress paths at r = 0.03 m, 0.255 m, and 0.475 m and at 0.5 m and 2.5 m depths. At r = 0.03 m (adjacent to the PVD), during the process of consolidation, the effective stress paths almost reached the critical state line (CSL) in the p′ − q plot (p′ is effective mean stress and q is deviator stress). For the points away from the PVD, the effective stress paths moved along approximately constant q/p′ ratio lines and far from the CSL. This tendency did not change with depth (comparing Fig. 12(a) and (b)). The simulated final distribution of the undrained shear strength (su) [27] is shown in Fig. 13. The values of su in the zone adjacent to the PVD were much higher than (approximately four times) those in other zones. During the process of vacuum consolidation, su in the zone adjacent to the PVD was almost fully mobilized (Fig. 12).
The coefficient of consolidation in the horizontal direction (ch) is a function of the permeability (kh) and the coefficient of volume compressibility (m) of a soil, i.e.,
ch =
kh mγw
(3)
where γw = the unit weight of water. Adopting a linear e – ln(p′) relationship, and in a normally consolidated state, m is:
m=
λ (1 + e ) p′
(4)
where λ = the compression index in the e − ln(p′) plot. Using the linear e − log(kh) relationship proposed by Taylor [23], the variation of kh with e can be expressed by Eq. (1). Substituting Eqs. (1) and (4) into Eq. (3), the following expression for ch can be obtained.
(5) Discussions The results of FEA presented above clearly indicate that for a PVD
Fig 12. Effective stress paths. 6
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consolidation of a PVD unit cell. To consider the variations of ch and the apparent smear effect into the PVD-induced consolidation analysis, a new pragmatic method has been proposed in this study. 4.3. Proposed new method for calculating the average degree of consolidation Hansbo’s [8] solution is widely used for analysing PVD-induced consolidation, and the average DOC (Uh) is:
Uk = 1 − exp(−8Tk / μ)
(6)
where μ = the effect of spacing between the PVDs, the smear effect and well resistance on DOC, and Th = the time factor. These parameters can be expressed as:
ch =
Fig. 14. The zones for ch increasing/decreasing with increasing p′.
μ = ln(n/ s ) + (Kh/ Ks ) ln(s ) −
kh0 ((1 + e0)/ λ − ln(p′ / p0′ ))·p′ γw (p′ / p0′ )2.3λ / Ck
Th = (5)
3 2π·l 2·Kh + 4 3qw
ch t De2
(7) (8)
where De = the diameter of the unit cell, n = De/dw, s = ds/dw (ds is the diameter of the smear zone), l = the drainage length of a PVD, kh = the horizontal permeability of the soil, and ks = the horizontal permeability of the smear zone. Eq. (6) was derived under the condition of constant ch and μ. To consider the effect of varying ch and μ during the consolidation process, we propose the following incremental calculation method. The key points of the method are as follows:
where p′0 = the initial effective mean stress. There are soils in which ch increases with increasing p′ and there are soils in which ch decreases with increasing p′. As a general qualitative tendency, a soil is less compressible (smaller λ value) and the rate of reduction of permeability with void ratio is smaller (larger Ck value), ch will increase with the increase of p′. While for a soil with a larger λ and a smaller Ck values, ch will reduce with the increase of p′ Therefore whether ch will increase or decrease with increasing p′is a function of the combination of λ and Ck. In this study, it has been found that in λ–λ/Ck plot, the zones ch increase or decrease with increase of p′ can be identified as in Fig. 14. Within the zone between two dashed lines, the variation of ch with p′ is small, and for some cases, ch is approximately a constant with the variation of p′. Above the dashed zone, ch decreases with increasing p′, and below the dashed zone, ch increases with increasing p′. For the cases assumed in the FEA presented above (λ = 0.32 and Ck = 1.2), ch increases with increasing p′. The initial kh = 10−8 m/s and m = 0.03 m2/kN (initial vertical effective stress of 2 kPa), which results in a ch = 3.4 × 10−8 m2/s. Under a vertical effective stress of 82 kPa (80 kPa vacuum pressure plus 2 kPa initial stress), kh = 1.0 × 10−9 m/s and m = 9.6 × 10−4 m2/kN and then ch = 1.1 × 10−7 m2/s can be calculated, which is about 3.2 times of the initial value. Zhou and Chai [6] reported that the non-uniform consolidation-induced apparent smear effect also varies during the consolidation process. All existing explicit solutions for PVD-induced consolidation assume a constant ch and a constant smear effect.
(1) At the ith time increment, use values of ch and μ corresponding to the stress state at the end of the i − 1th time increment. (2) To ensure that the continuity of the calculated DOC at the beginning of the ith time increment is the same as that at the end of the i − 1th time increment, use a theoretically determined imaginary time at the beginning of the ith time increment. The procedure of the calculation is as follows: (a) For the first time increment Δt1, calculate U1 = ΔU1 using Eq. (6) (b) Starting at the second time increment: (i) For the ith time increment, calculate the total effective mean stress increment at the end of the i − 1th time increment: Δp′i−1 = Ui−1∙p (p is the total applied load up to i − 1th time increment). Then, the effective mean stress at the end of Δti−1 is p′i−1 = p′0 + Δp′i−1 (p′0 is the initial effective mean stress) (ii) Calculate the void ratio corresponding to p′i−1 as:
ei − 1 = e0 − λ ln(pi′− 1 / p0′ )
4.2. Considering the effect of horizontal strain
(9)
(iii) Calculate mi−1 and khi−1 using Eqs. (4) and (1), respectively, and chi−1 by Eq. (3). (iv) Calculate the change of the apparent smear effect using the method proposed by Zhou and Chai [6]. For Uh < 50%, the apparent smear effect does not change; for 50 < Uh < 95% (0.087 < Th/ μ < 0.375), kh/ks linearly reduces from a value larger than 1.0 to 1.0 (Fig. 13 in Zhou and Chai [6]). Then, calculate μi−1 by Eq. (7). (v) Calculate an imaginary time for the start of ith time increment, t0i, as:
Zhou and Chai [6] proposed a method for considering the effect of non-uniform consolidation on the average degree of consolidation of a PVD unit cell. The method was based on the results of the FEA, which inexplicitly included the effect of the horizontal movement of soils within a PVD unit cell. The FEAs conducted by Zhou and Chai [6] for PVD unit cells were under the surcharge loads. In this study, only considering the horizontal radial drainage, consolidations of a PVD unit cell under both a vacuum pressure and a distributed load of the same magnitude as the vacuum pressure under the free vertical strain conditions were conducted. The results indicate that the deformations of the PVD unit cell for the both cases are identical, i.e. the vacuum pressure and the surcharge load induced the same soil deformation of the PVD unit cell. Therefore, it is considered that the apparent smear (clogging) effect estimated by Zhou and Chai’s method includes both the effects of non-uniform consolidation and horizontal strain in a PVD unit cell. However, there was no rational way established for considering its effect in the calculation of the average degree of
t0i =
μi − 1 ·chi − 2 (t0i − 1 + Δti − 1) μi − 2 ·chi − 1
(10)
where t01 = 0 and ch0 and μ0 are the initial values. (vi) Calculate ΔUhi and Uhi as:
7
ΔUhi = exp(−8chi − 1 t0i/ μi − 1) − exp(−8chi − 1 (t0i + Δti )/ μi − 1)
(11)
Uhi = Uhi − 1 + ΔUhi
(12)
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Fig. 15. Comparison of DOCs from different methods.
Fig. 16. Comparison of measured and calculated DOCs.
All the calculations can be carried out using an Excel spreadsheet. However, since the DOC is a function of μ, to find a time corresponding to 95% DOC, some iterations are required. First, the time for Uh = 95% with a constant μ value can be used as a starting value for the iterations.
comparison of the measured and calculated DOCs is given in Fig. 16, which clearly shows that the results from the proposed method are very close to the measured data. The results in Figs. 15 and 16 indicate that the proposed method can consider the effect of non-uniform consolidation and the horizontal movement of soil within a PVD unit cell and the variations of the coefficient of consolidation and the apparent smear effect in the calculation of the average DOC of a PVD unit cell.
4.4. Comparison with the results of the FEA With the conditions adopted for the FEA presented above, from Zhou and Chai [6], with ds/dw = 4, and De = 1.0 m, a value of (kh/ks)e of approximately 3.3 can be estimated. Fig. 15 shows the comparison of average DOCs calculated by the excess pore water pressures from the FEA, the proposed method, and using Eq. (6) with chrep = 6.26 × 10−8 m2/s. The figure clearly shows that the DOCs from the proposed method are very close to those of the FEA. The theory (Eq. (6)), which does not consider the apparent smear effect, over-estimates the DOCs significantly.
5. Conclusions The behaviour of a prefabricated vertical drain (PVD) unit cell with clays of high initial water content has been investigated by laboratory model tests and finite element analysis (FEA). Then, a new explicit method for consolidation analysis of a PVD unit cell has been established. Based on the test and analysis results, the following conclusions can be drawn.
4.5. Comparison with a model test results
(1) Horizontal soil movement within a PVD unit cell. The results from both the model test and the FEA indicate that there is considerable horizontal soil movement towards the PVD. The soils in the zone near the PVD are in horizontal compression, and the other zones are in horizontal tension. The compressive strain can be more than 10%, and the extension strain can be 2–3%. (2) Non-uniform surface settlements. The surface settlements of a PVD unit cell are not uniform, and at the periphery of a unit cell settles more than the region near the PVD. The results of the FEA indicate that at deeper locations (> 0.5 m), the settlements are almost uniform. (3) Non-uniform vertical stress and strength distribution. The results of the FEA indicate that under the assumed conditions and at the end of a vacuum consolidation, the effective vertical stress in the soil adjacent to the PVD is approximately 3.0 times that in the soil near the periphery of the unit cell. The soil adjacent to the PVD also has a higher undrained shear strength, but the stress state is very close to the critical state line. (4) Mechanism of apparent clogging. The model test results indicate that the phenomenon of soil particle separation did not occur, which implies that the main mechanism of apparent clogging is due to non-uniform consolidation and horizontal soil movement. (5) A new method for consolidation analysis. A new incremental consolidation calculation method has been established for considering the variations of consolidation properties, including apparent clogging during the consolidation process. The calculated average degree of consolidation agreed well with the results of the FEA and with the results of a model test reported in the literature.
Chai et al. [28] reported the results of a laboratory PVD unit cell model test. The model ground had a diameter of 0.45 m and height of 0.8 m. A mini-PVD was installed at the centre of the model ground. Ariake clay with an initial water content of approximately 180% was used to form the model ground. The reported initial void ratio was 4.75, and the compression index (Cc) of the clay was 0.99. As discussed previously, for Ariake clay, the slope in the ln(kh) − e plot is approximately 1.2 (corresponding to the value of Ck in Taylor’s equation). Then, an initial permeability of 1.5 × 10−8 m/s was extrapolated from the reported average permeability for the consolidation pressure from 40 to 80 kPa. The loads applied were a 40 kPa vacuum pressure and a 10 kPa surcharge load. The reason for adopting a combined load was to prevent possible gaps between the top surface of the model ground and the loading piston above it. Three piezometers were installed in the middle height of the model ground at radial distances of 50, 100 and 200 mm from the centre for measuring the variations of pore water pressures (u) during the consolidation process. Dividing the model ground into three zones with two division lines at the middle of each two adjacent piezometers and assuming that the measured values of u by each piezometer represented the average values of each zone, the average DOC of the model ground was calculated, which is regarded as the measured value. Using Zhou and Chai’s [6] method, an initial apparent smear effect of (kh/ks)e = 4.0 was evaluated. Assuming a remoulded yield stress of 2.0 kPa, the average DOCs of the model ground by the proposed method was calculated. Then, using Eq. (2), a representative effective stress of 10 kPa and a corresponding representative horizontal coefficient of consolidation chrep = 4.77 × 10−8 m2/s were evaluated. Then the average DOC by Hansbo’s solution [8] was calculated also. The 8
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CRediT authorship contribution statement [11]
Jin-chun Chai: Conceptualization, Methodology, Formal analysis, Writing - original draft. Hong-tao Fu: Data curation, Investigation. Jun Wang: Resources. Shui-Long Shen: Writing - review editing.
[12]
Declaration of Competing Interest [13]
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.
[14]
[15]
Acknowledgements
[16]
This work was supported by the National Natural Science Foundation of China with a grant number 51578514.
[17]
References
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