Geotech Geol Eng (2010) 28:61–77 DOI 10.1007/s10706-009-9279-7
ORIGINAL PAPER
Effect of Soil Disturbance on Consolidation By Prefabricated Vertical Drains Installed in a Rectangular Pattern D. Basu • M. Prezzi • M. R. Madhav
Received: 24 March 2008 / Accepted: 20 October 2009 / Published online: 5 November 2009 Ó Springer Science+Business Media B.V. 2009
Abstract In this paper, the effect of soil disturbance caused by installation of prefabricated vertical drains (PVDs) on the rate of consolidation is studied by twodimensional finite element analysis. A transition zone lying between the highly disturbed smear and the undisturbed zones is considered in the analysis. The hydraulic conductivity in the transition zone is assumed to vary linearly from a low value in the smear zone to the original in situ value in the undisturbed zone. The analysis uses the actual band shape of the PVD and the rectangular shape of the unit cell. A parametric study is carried out to investigate the effects of the degree of soil disturbance, the size of the smear and the transition zones, the PVD spacing and the mandrel size and shape. Design guidelines, where the smear and transition zones are replaced by an expanded smear zone producing the same effect, are provided so that
D. Basu Department of Civil and Environmental Engineering, University of Connecticut, 261 Glenbrook Road, Unit 2037, Storrs, CT 06269, USA e-mail:
[email protected];
[email protected] M. Prezzi (&) School of Civil Engineering, Purdue University, 550 Stadium Mall Drive, West Lafayette, IN 47907, USA e-mail:
[email protected] M. R. Madhav J.N. Technical University, Hyderabad 500 034, India e-mail:
[email protected]
existing analytical solutions considering only the smear zone can be used for analysis and design. A comparison with experimental results shows that consideration of the transition zone is important for correct estimation of the degree of consolidation. Keywords Prefabricated vertical drain Consolidation Smear Soil disturbance Ground improvement Finite element analysis
1 Introduction Ground improvement of thick deposits of soft, saturated clays using preloading in combination with vertical drains is a common practice in Geotechnical Engineering (Johnson 1970; Jamiolkowski et al. 1983; Holtz 1987; Bergado et al. 1993a). Vertical drains are installed at regular intervals in a square, rectangular or triangular pattern (Bergado et al. 1996), with centerto-center distances varying from about 1.0 to 3.5 m or more (Holtz 1987). The excess pore pressure generated due to preloading is quickly dissipated due to the reduced drainage path and predominant horizontal flow within the clay deposits (except near the top surface or close to a highly permeable silt/sand seam) because the hydraulic conductivity is generally greater in the horizontal direction due to depositional anisotropy. This results in rapid increase in shear strength and stiffness of the soil.
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Band-shaped prefabricated vertical drains (PVDs) have been used at various sites throughout the world since the early 70’s (Holtz 1987; McDonald 1985; Crawford et al. 1992; Bergado et al. 1993a, 1997, 2002; Lo and Mesri 1994; Grondin et al. 1991; Stark et al. 1999; Chu et al. 2004). The most commonly used PVDs consist of a plastic core surrounded by a filter sleeve with a typical cross-section dimension of 100 mm 9 4 mm (Holtz 1987). The filter sleeve prevents fine soil particles from entering the drain while allowing easy entry of pore water into the plastic core, which acts as a vertical drainage channel (Akagi 1994). Prefabricated vertical drains are generally installed by closed-ended mandrels, which displace and drag the soil down in order to make room for the PVD (Hird and Moseley 2000). Consequently, the soil surrounding the PVD is disturbed. As a result of soil disturbance, the hydraulic conductivity k decreases in the vicinity of the PVDs and the consolidation process is slowed down. Extensive research has been performed to investigate the extent of the disturbed zones surrounding the PVDs and to quantify the reduction in hydraulic conductivity in the disturbed zones (Casagrande and Poulos 1969; Akagi 1977; McDonald 1985; Bergado et al. 1991, 1993a; Indraratna and Redana 1998; Chai and Miura 1999; Hird and Moseley 2000; Hird and Sangtian 2002; Chu et al. 2004). Most of these studies assume (1) a single disturbed zone (also called smear zone) with a constant hydraulic conductivity khs and (2) axisymmetric flow (thus, the cross sections of the drain and its area of influence, often called the unit cell, are converted to equivalent circles). Following the assumption of axisymmetric flow, the smear zone cross section is also converted to an equivalent circle with the center coinciding with the center of the circular drain (thus, the smear zone cross section is actually annular in shape with the inner boundary coinciding with the drain boundary). The radial extent (i.e., the diameter of the outer boundary) of the smear zone is mostly expressed in terms of the equivalent mandrel diameter dm,eq (obtained by equating the actual cross sectional area of the mandrel to an equivalent circular area). Different authors proposed different values for the smear-zone diameter which are given in Table 1. In Table 2, the degree of disturbance in the smear zone (described in terms of the ratio khs/kho where khs is the hydraulic
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Geotech Geol Eng (2010) 28:61–77 Table 1 Extent of smear zone Authors
Extent of smear zone
Casagrande and Poulos (1969) Smear zone area = mandrel area Jamiolkowski et al. (1983)
Smear zone diametera = (2.5–3)dbm,eq
Holtz and Holm (1973);
Smear zone diameter = 2dm,eq
Hansbo (1986, 1987, 1973); Bergado et al. (1991, 1993b, 1973) Hansbo (1997); Chai and Miura (1999);
Smear zone diameter = (2–3)dm,eq
Hird and Moseley (2000) Mesri et al. (1994)
Smear zone diameter = (2–4)dm,eq
a Smear zone diameter = Diameter of the outer boundary of the smear zone b
dm,eq = Equivalent mandrel diameter measured from the center of the drain
Table 2 Degree of disturbance in the smear zone Authors
Degree of disturbance
Casagrande and Poulos (1969)
khs/kho = 0.001
Hansbo (1986); Hird and Moseley (2000)
khs/kho = 0.33
Bergado et al. (1993a, b)
khs/kho = 0.1 = kvo/kho
Hansbo (1997)
khs/kho = 0.25–0.3
Hansbo (1987); Bergado et al. (1996)
khs/kho = kvo/kho
Bergado et al. (1991)a
khs/kho = 0.5–0.66
khs = Hydraulic conductivity in the smear zone for horizontal flow kho = In situ hydraulic conductivity for horizontal flow kvo = In situ hydraulic conductivity for vertical flow a
Based on laboratory experiments
conductivity in the smear zone for horizontal flow and kho is the in situ hydraulic conductivity for horizontal flow) proposed by various authors are reported. Experimental investigations by Onoue et al. (1991), Madhav et al. (1993), Indraratna and Redana (1998) and Sharma and Xiao (2000), however, suggest that the hydraulic conductivity is not constant
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63 1
0.8
0.8
0.6
0.6
khs /k ho
khs /k ho
1
0.4
0.4
σ' = 118 kPa
0.2
0.2
σ' = 235 kPa 0
0
4
8
12
16
20
Normalized Distance, r/rm,eq Fig. 1 Normalized hydraulic conductivity profiles in the disturbed zone from field samples collected by Madhav et al. (1993) (rm,eq is the equivalent mandrel radius and r is the radial distance from the center of drain)
0
0
4
8
12
16
20
Normalized Distance, r/rm,eq Onoue et al. (1991) Onoue et al. (1991) Indraratna and Redana (1998) Indraratna and Redana (1998)
within the disturbed zone. Madhav et al. (1993) performed laboratory tests on samples collected from a site where PVDs had been installed with a square mandrel (Fig. 1). Based on their results, Madhav et al. (1993) proposed that there are two distinct zones within the disturbed zone: the smear zone immediately surrounding the PVD in which the hydraulic conductivity remains constant at khs, and the transition zone surrounding the smear zone in which the hydraulic conductivity gradually increases from khs to the in situ value kho as the horizontal distance from the drain increases (Fig. 1). Thus, the outer boundary of the smear zone coincides with the inner boundary of the transition zone, and the outer boundary of the transition zone coincides with the inner boundary of the undisturbed zone (the undisturbed zone is the outer part of the unit cell in which the soil is not disturbed). The horizontal extent of the outer boundary of the smear zone was found to be about the same as the width of the square mandrel, while that of the outer boundary of the transition zone was about 12 times the width of the square mandrel. In the smear zone, khs/kho was found to be approximately equal to 0.2. A similar two-zone model for the disturbed zone was also proposed by Onoue et al. (1991) (Fig. 2).
Indraratna and Redana (1998) Indraratna and Redana (1998) Indraratna and Redana (1998) Indraratna and Redana (1998) Sharma and Xiao (2000) Sharma and Xiao (2000) Sharma and Xiao (2000)
Fig. 2 Normalized hydraulic conductivity profiles from laboratory model studies (rm,eq is the equivalent mandrel radius and r is the radial distance from the center of drain)
They found, based on laboratory experiments with circular steel drains, that the outer diameter of the smear zone is about 1.5 times the drain diameter; the outer diameter of the transition zone, however, is equal to about 6–7 times the drain diameter. They also found that, in the smear zone, khs/kho varies between 0.2 and 0.6. The results of experiments (Fig. 2) on PVDs (installed using circular mandrels) by Indraratna and Redana (1998) suggest that the outer diameter of the transition zone is more than seven times the mandrel diameter (the outer
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Table 3 Extent of transition zone Authors
Extent of transition zone
Onoue et al. (1991)d
Transition zone diametera = (6–7)dbm,eq
Madhav et al. (1993)c
Transition zone diameter = 12dm,eq
Indraratna and Redana (1998)d
Transition zone diameter [ 7dm,eq
Sharma and Xiao (2000)d
Transition zone diameter = (6–10)dm,eq
a Transition zone diameter = Diameter of the outer boundary of transition zone b dm,eq = Equivalent mandrel diameter measured from the center of the drain pffiffiffiffiffiffiffiffi c Square mandrel was used: dm,eq = 4=p mandrel width d
Circular mandrels were used: dm,eq = Actual mandrel diameter
In this paper, the effect of the transition zone, in addition to the smear zone, on the rate of consolidation by PVDs is studied through a twodimensional finite element (FE) analysis. The PVDs are assumed to be installed in a rectangular pattern, and the actual band shape of the PVD and the rectangular shape of the zone of influence (unit cell) are considered in the analysis without converting them into equivalent circles. The error caused by the assumption of equivalent circular drain and unit cell is also quantified. A method of replacing the transition zone by an expanded smear zone having the same effect is proposed so that standard solutions considering a single smear zone can be used in design. Moreover, a method of converting rectangular domains to equivalent circular domains is outlined.
2 Scope of Present Study boundaries of the transition zone could not be determined from the results), while those by Sharma and Xiao (2000), who also used circular mandrels, suggest that the transition-zone outer diameter is about 6–10 times the mandrel diameter (see Table 3). The presence of a transition zone has also been confirmed indirectly in many case studies (e.g., Holtz and Holm 1973) in which the void ratio or water content has been found to be a function of the radial distance from the drain (void ratio and water content have been explicitly correlated to the hydraulic conductivity in the literature by Gabr et al. (1996) and Sathananthan and Indraratna (2006)). However, the transition zone has not yet been explicitly accounted for in design, mainly because measurement of hydraulic conductivity as a function of the radial distance from PVDs is generally not done in the field. Consequently, most case studies have so far been analyzed by back calculating the hydraulic conductivity of the smear zone with the assumption that the hydraulic conductivity is spatially constant in the smear zone (Bergado et al. 1993a, b). In fact, only a few research studies have considered the transition zone; these include the numerical studies of Madhav et al. (1993) and Hawlader et al. (2002) and analytical solutions developed by Chai et al. (1997) and Basu et al. (2006). In practice, analysis and design are mostly done based on the analytical solution of Hansbo (1981), which considers only the smear zone with a constant hydraulic conductivity.
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This paper considers PVDs with cross-sectional dimensions of 100 mm 9 4 mm installed in a rectangular pattern, with spacings of sx and sy along the x and y directions, respectively. The unit cell of each PVD (except those at the boundaries) is a rectangle in cross section with dimensions sx 9 sy (Fig. 3), and the PVD is located centrally within it. The water within the unit cell flows only to the PVD located at
y sx tx lx sy
ty
undisturbed zone transition zone smear zone
ly
x unit cell boundary
PVD
kh k ho k hs x
Fig. 3 Unit cell with smear and transition zones
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its center; there is no flow across the boundaries between adjacent cells. For a homogeneous deposit, all the unit cells (except those at the boundaries) can be assumed to behave identically with respect to soil consolidation. One unit cell is considered in the analysis, and the average degree of consolidation U of the soil within the unit cell is studied as a function of time for different degrees of soil disturbance (by assuming different values for khs/kho), for different dimensions for the smear and transition zones and for different PVD spacings and mandrel sizes. The hydraulic conductivity kh for horizontal flow is assumed to increase linearly in the transition zone from khs to kho as the distance from the drain increases (Fig. 3). The Terzaghi-Rendulic theory of consolidation (Terzaghi 1925; Rendulic 1935, 1936) is used in the present analysis. Since two-dimensional horizontal flow is assumed in this analysis, the results are valid strictly for thick clay deposits in which vertical flow has little effect on the overall rate of consolidation. However, the applicability of the results is likely to be less restrictive because Leo (2004) showed that the vertical flow, in general, has negligible contribution to the overall degree of consolidation.
3 Smear and Transition Zones The shape and size of the smear and transition zones depends mostly on the mandrel cross section (Chai and Miura 1999). Four different rectangular mandrel cross sections (125 9 50, 150 9 50, 120 9 120 and 150 9 150, all in mm) that have been used in practice are considered in the analysis (Bergado et al. 1993b; Madhav et al. 1993; Saye 2003; American Wick Drain Corporation 2005). These rectangular mandrels are likely to create smear and transition zones that are rectangular or nearly rectangular (e.g., oval) in plan, although the exact shape is not known with certainty. In this analysis, the smear and transition zones are assumed to be rectangular with dimensions lx 9 ly and tx 9 ty, respectively (Figs. 3, 4). Although the size of the smear zone is available in the literature in terms of the equivalent mandrel diameter (Table 1), in this analysis, it is assumed to depend on the smaller dimension of the mandrel cross section. The basis for this assumption is that, as the mandrel is pushed into the ground, the primary
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transition zone boundary mandrel ty = pd
1 2
ly = pd
smear zone boundary
(p−1)d a
d 1 2
(p−1)d
lx = a + (p−1)d tx = a + (p−1)d Smear Zone: 2 ≤ p ≤ 3 Transition Zone: 6 ≤ p ≤ 12 Fig. 4 Dimensions of the smear and transition zones in terms of mandrel size
motion of the soil is in the direction perpendicular to the larger dimension of the mandrel. The thickness of the smear zone surrounding the mandrel is assumed to be constant along the entire mandrel perimeter (Fig. 4). Therefore, for a mandrel of cross section a 9 d with a [ d, the dimensions ly and lx (Fig. 3) of the smear zone are given by (Fig. 4): ly ¼ pd
ð1Þ
lx ¼ a þ ðp 1Þd
ð2Þ
where p is a parameter determining the extent of the smear zone. Since the literature indicates that the smear-zone outer diameter is about 1–4 times the equivalent mandrel diameter dm,eq with most researchers agreeing on the range (2–3)dm,eq (Table 1), the width of the smear zone ly is assumed in this paper to be 2–3 times the mandrel thickness d (i.e., 2 B p B 3). Since the actual mechanism of soil movement during mandrel penetration is not well understood, simulations were done with different values of lx [lx was increased by up to 50% from the value calculated using Eq. (2)] for fixed values of ly to ensure that the assumption regarding the shape of the smear zone is acceptable. It was observed that, for a given ly, the rate of consolidation remains practically the same irrespective of the values of lx. Thus, it is primarily the thickness of the smear zone, and not the overall shape of the rectangle, that determines the
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consolidation rate. This suggests that as long as the thickness ly remains more or less constant, the exact shape of the smear zone (whether it is rectangular in shape or slightly elliptical) will not affect the consolidation rate to a large extent. This observation is consistent with the FE analysis of Chai and Miura (1999), according to which, the difference in the consolidation rate due to the assumption of a circular smear zone and a rectangular smear zone with the same area is negligible. The dimensions of the transition zone are determined exactly the same way as those of the smear zone. They are obtained from Eqs. (1) and (2) by replacing ly and lx by ty and tx. The experimental studies (Onoue et al. 1991; Madhav et al. 1993; Indraratna and Redana 1998; Sharma and Xiao 2000) conducted to determine the extent of the transition zone suggest that the outer diameter of the transition zone is about (6–12)dm,eq. Accordingly, the range 6 B p B 12 is chosen for ty and tx. 4 Hydraulic Conductivity The hydraulic conductivity khs in the smear zone is varied in the analyses from 0.05 to 0.5 times the hydraulic conductivity kho of the undisturbed zone, although most of the analyses are performed with khs/kho = 0.2, which is approximately the average of the range of values reported in the literature (Table 2). In the transition zone, the hydraulic conductivity is assumed to increase linearly from khs to kho as the distance from the PVD surface increases (Fig. 3). This is based on the experimental observations of Onoue et al. (1991), Madhav et al. (1993), Indraratna and Redana (1998) and Sharma and Xiao (2000) shown in Figs. 1 and 2; a linear variation fits well the experimental data. At any distance within the transition zone, the variation of the hydraulic conductivity in the x direction is given by: kht ðxÞ ¼ khs þ
2x lx ðkho khs Þ tx lx
ðlx =2 x tx =2Þ
5 Analysis 5.1 Differential Equation and Boundary and Initial Conditions The differential equation governing the TerzaghiRendulic two-dimensional consolidation for a homogeneous and isotropic soil deposit is given by: 2 ou o u o2 u ¼ ch þ ð4Þ ot ox2 oy2 where u = u(x, y, t) is the excess pore pressure; t is the time; x and y are the spatial coordinates; and ch is the coefficient of consolidation for flow in the horizontal direction. In the case of soil disturbance, Eq. (4) can be modified as: ou khd o2 u o2 u ¼ ch þ ð5Þ ot kho ox2 oy2 where khd is the hydraulic conductivity in the disturbed zone. If the element lies within the smear zone, then khd is equal to khs. On the other hand, if the element lies in the transition zone, then khd is equal to kht. It is assumed that soil disturbance only affects the hydraulic conductivity and not the compressibility of the soil (Hansbo 1981; Madhav et al. 1993). Since the PVD is located centrally within the unit cell, there is a symmetric flow pattern with respect to the coordinate axes (Fig. 3). For this reason, it is sufficient to analyze one of the four quadrants. The first quadrant (within which x C 0 and y C 0) is chosen for analysis. The PVD thickness of 4 mm is neglected. This makes the domain of analysis a rectangular area with dimension sx/2 9 sy/2 (Fig. 5).
y Neumann boundary,
Neumann boundary, ∂u =0 ∂y
sy/2
ð3Þ where kht is the hydraulic conductivity in the transition zone for flow in the horizontal direction. A similar expression for the variation of kht in the y direction, kht (y), can be obtained by replacing y for x, ly for lx, and ty for tx.
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∂u =0 ∂x
x Dirichlet (drain) boundary, u = 0
sx/2
Fig. 5 Domain of analysis and boundary conditions
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Free flow occurs across the boundary that represents the interface between the PVD and soil (referred to as drain boundary). Thus, a Dirichlet boundary condition (in which a value for the primary variable u is specified) is prescribed along the drain boundary. For the remaining boundaries, no flow is possible across them. Consequently, the Neumann boundary condition qu/qn = 0 is prescribed at those boundaries (n denotes the direction of the unit normal vector at the boundary surface; Fig. 5). The initial condition imposed is uini = u(x, y, 0) = 100 within the domain and at all the boundaries, except at the drain boundary (0 B x B 50 mm, y = 0), where the pore pressure dissipates instantaneously. Hence, the initial condition prescribed at the drain boundary is uini = 0. This is also the boundary condition for the drain boundary at all times (t [ 0) during the consolidation process (Fig. 5). 5.2 Degree of Consolidation and Time Factor
Fig. 6 Typical finite element mesh
The degree of consolidation U at any particular time for two-dimensional consolidation can be expressed in terms of integrals over the unit cell domain as:
5.3 Finite Element Analysis and Time Integration
sx sy
R2 R2 U ¼1
uðx; y; tÞ dydx
0 0
ð6Þ
sx sy
R2 R2
uini dydx
0 0
Cu_ þ Ku ¼ 0
where u(x, y, t) is the excess pore pressure at any point with coordinates (x, y) at time t, and uini is the initial excess pore water pressure (taken equal to 100). The integrals in Eq. (6) are evaluated by a Gauss-Quadrature integration scheme after obtaining the values of pore pressure u at the Gauss points within each element of the finite element mesh (Cook et al. 2002). The time t is normalized in the calculations by defining the time factor T as: T¼
ch t 2 dc;eq
For the FE analysis, the domain is discretized using four-noded rectangular elements (a typical FE mesh for a 1 m 9 1 m unit cell is shown in Fig. 6). The discretized equation governing the consolidation process, obtained from Eqs. (4) or (5), is given by: ð9Þ
where u is the unknown global pore pressure vector (containing the u values at the nodes of the elements that discretize the domain); the dot () represents a derivative with respect to time; and K and C are global matrices associated with u and its time derivative, respectively. Time integration of Eq. (9) is done following an implicit (Backward Difference) numerical scheme (Lewis et al. 1996).
ð7Þ
where dc,eq is a representative drainage path length; it is taken equal to the diameter of an equivalent circular unit cell having the same area as that of the actual unit cell: rffiffiffiffiffiffiffiffiffiffi 4sx sy dc;eq ¼ ð8Þ p
6 Results 6.1 Test of Convergence A test of convergence was performed to ensure the accuracy of the numerical solution. The size of the elements was varied until two successive meshes produced identical results (difference \ 0.01%). For
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cases with different dimensions of smear and transition zones, different meshes had to be used, and convergence tests were performed for each of these cases. In general, for the final accepted mesh, the length of the elements adjacent to the PVD was equal to 12.5 mm, increasing gradually to up to a maximum of 150 mm at the cell boundaries. In the calculations for time integration, the dimensionless time factor T was used instead of the real time t. The time step DT used was 0.001 for T B 0.1 and 0.01 for T [ 0.1. 6.2 Comparison with Analytical Solutions with Circular Domains The FE solutions are compared with the analytical solutions (Table 4) with axisymmetric flow conditions for cases with smear and without smear zones. The rectangular domain of the unit cell is converted to an equivalent circle of diameter dc,eq, given by Eq. (8); dc,eq is used in the analytical solutions as the diameter dc of the circular unit cell. The rectangular smear zone is also converted to an equivalent circle with a diameter ds,eq (calculated in a similar way as dc,eq with sx and sy replaced by lx and ly in Eq. (8), respectively); ds,eq is used in the analytical solution as the diameter ds of the circular smear zone. The bandshaped PVD is converted to an equivalent circle of diameter dw,eq obtained by equating the perimeter of the PVD to that of a circle (Hansbo 1981):
2 dw;eq ¼ ðdb þ dt Þ p
ð10Þ
where db and dt are the PVD width and thickness, respectively. dw,eq is taken equal to the diameter dw of the circular drain. For a 100 mm 9 4 mm PVD, dw,eq is equal to 66.2 mm. The comparison for 1 m 9 1 m unit cell and 125 mm 9 50 mm (a 9 d) mandrel dimensions (combination 1) with the smear zone dimensions ly = 100 mm and lx = 175 mm [i.e., p = 2 in Eqs. (1) and (2)] is shown in Fig. 7. The results compare well, particularly for large values of T. The maximum difference, occurring at smaller values of T, is partly due to the fact that equal-strain analytical solutions (where the vertical settlement is assumed uniform throughout the unit cell) and free-strain FE solutions (where no restriction on settlement is imposed) yield different results for U \ 50% (Richart 1959) and partly due to the difference in the shapes of the domains. For U [ 50%, the maximum differences in U between the FE and analytical solutions are 9.2% (occurring at T = 0.4) and 3% (occurring at T = 0.5) for cases with and without smear, respectively. Simulations were also done for 3 m 9 3 m unit cell and 150 mm 9 150 mm mandrel dimensions (combination 2) with p = 2. Similar trends were found with the corresponding
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Table 4 Analytical solutions for radial consolidation U ¼1e
8T l 3 4
(A)
l ¼ ln
l ¼ ln n (Barron 1948) n k 3 ho m þ khs ln m 4 (Hansbo 1981)
(C)
n ¼ ddwc
(D)
m ¼ ddws
(E)
(B)
These solutions are based on the assumption of equal strain, according to which the vertical settlement is uniform throughout the unit cell U = Degree of consolidation
Degree of Consolidation, U (%)
spacing = 1m×1m n = 17.04 80
no smear
60
40
with smear khs /k ho = 0.2
mandrel = 125mm×50mm d = 50mm m = 2.25
20
T = Time factor dc = Circular unit cell diameter dw = Circular drain diameter ds = Circular smear zone diameter kho = In situ hydraulic conductivity for horizontal flow khs = Hydraulic conductivity for horizontal flow in the smear zone
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0 0.001
0.01
0.1
1
10
Time Factor, T Analytical Finite Element
Fig. 7 Comparison between rectangular and circular domains
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6.3 Effect of Soil Disturbance on PVD Consolidation Rate The effect of soil disturbance on the consolidation rate is investigated for the combinations 1 and 2 mentioned above. The degree of disturbance, quantified in terms of khs/kho, is maintained constant at a value of 0.2. The smear and transition zone dimensions are calculated with p = 2 and 12, respectively. The U versus T curves obtained for combination 1 are plotted in Fig. 8. Soil disturbance has a substantial detrimental effect on the consolidation process. T required for 90% consolidation is equal to 0.66, 1.73 and 2.32 for the no-smear, only-smear and smear-plus-transition zone cases, respectively. The increase in T with respect to the reference case of no smear or transition zones is equal to 162% when only a smear zone is considered and is equal to 251% when both the smear and transition zones are considered. The increase in T for the case of smearplus-transition over the only-smear case is 34%. A similar trend is found for combination 2 as well (not shown in Fig. 8). For this case, T values corresponding to 90% consolidation for the no-smear, onlysmear and smear-plus-transition cases are equal to 0.96, 2.94 and 3.7, respectively. The increase in T with respect to the reference case with no smear or transition zones is equal to 206% when a smear zone is considered and is equal to 285% when both the
100
Degree of Consolidation, U (%)
differences in U being 3.75% (occurring at T = 0.4) and 2.3% (occurring at T = 0.5) for cases with and without smear, respectively. The difference in PVD response between that obtained by considering the actual shapes of the PVD and the unit cell and that obtained by converting the PVD and the unit cell to equivalent circles has not been quantified in the literature. Atkinson and Eldred (1982) made an estimate of this difference by comparing the shapes of the unit cells and the relative distances of the different sectors of the unit cells from the drain. They concluded that the use of an equivalent circular unit cell overestimates the degree of consolidation U and that the maximum error in the value of U is about 5%. The numerical simulations in this paper confirm that overestimation of U occurs when equivalent circles are assumed, although only for large time factors (the error in the prediction of U can sometimes be [ 5%).
69
80
khs/kho = 0.2 mandrel = 125mm×50mm d = 50mm spacing = 1m ×1m
60
40
20
0 0.001
0.01
0.1
1
10
Time Factor, T No Disturbance Smear = 2 d, No transition Smear = 2 d, Transition = 12 d Fig. 8 Effect of soil disturbance on the rate of consolidation
smear and transition zones are present. The increase in T for the case of smear-plus-transition over the only-smear case is 26%. 6.4 Effect of Dimensions of Smear and Transition Zones In order to understand the extent to which the variability of the reach of the smear zone affects PVD performance, two dimensions of the smear zone corresponding to p = 2 and 3 are considered with a fixed transition zone dimension with p = 12. Combinations 1 and 2 with khs/kho = 0.2 are evaluated (Fig. 9a). For combination 1, T required for 90% consolidation is 2.32 and 2.51 for a smear zone of 2d and 3d, respectively, a difference of 8.2%. For combination 2, T required for 90% consolidation is 3.7 and 3.91 for a smear zone of 2d and 3d, respectively, a difference of 5.7%. Therefore, the dimension of the smear zone has an impact on the consolidation process. The larger the smear zone, the more the consolidation process is delayed. The effect of size of the transition zone is studied by considering two transition zone dimensions with p = 6 and 12, with a fixed smear zone dimension with p = 2. The same two combinations with
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to U = 90% are equal to 3.34 and 3.7 for 6d and 12d transition zones, respectively, with the difference in T equal to 10.8%. The larger the transition zone the greater the delay in the consolidation process.
khs/kho = 0.2 transition = 12 d
80
mandrel = 125mm×50mm d = 50mm spacing = 1m×1m
60
6.5 Effect of Degree of Disturbance
40
mandrel = 150mm×150mm d = 150mm spacing = 3m×3m
20
0 0.001
0.01
0.1
1
10
100
Time Factor, T smear = 2d smear = 3d
Degree of Consolidation, U (%)
(b) 100
The degree of soil disturbance is investigated by varying the ratio khs/kho from 0.005 to 0.5. The smear and the transition zone dimensions are taken as constants with p = 2 and 12, respectively. The U versus T relationships for combination 1 are shown in Fig. 10. The values of T corresponding to U = 90% are 7.19, 4.07, 2.32, 1.69 and 1.1 for khs/ kho equal to 0.05, 0.1, 0.2, 0.3 and 0.5, respectively. The corresponding values of T for combination 2 (not plotted) are 12.22, 6.67, 3.7, 2.62 and 1.7, respectively. The variation of T with khs/kho follows a power law (Fig. 11). This has been found to be true for T corresponding to other values of U as well.
khs/kho = 0.2 smear = 2d 80
60
100
mandrel = 125mm×50mm d = 50mm spacing = 1m×1m
40
mandrel = 150mm×150mm d = 150mm spacing = 3m×3m
20
0 0.01
0.1
1
10
100
Time Factor, T transition = 12 d transition = 6 d
Degree of Consolidation, U (%)
Degree of Consolidation, U (%)
(a) 100
80
spacing = 1m×1m mandrel = 125mm×50mm smear = 2d transition = 12 d
60
40
20
0 0.001
0.01
0.1
1
10
100
Time Factor, T Fig. 9 Effect of disturbed zone dimension on the rate of consolidation: (a) effect of the smear zone dimension; (b) effect of the transition zone dimension
khs/kho = 0.05 khs/kho = 0.1 khs/kho = 0.2
khs/kho = 0.2 are studied (Fig. 9b). For combination 1, T corresponding to 90% consolidation is equal to 2.04 and 2.32 for transition zone dimensions of 6d and 12d, respectively, the difference in T being equal to 13.7%. For combination 2, T values corresponding
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khs/kho = 0.3 khs/kho = 0.5 Fig. 10 Effect of the degree of disturbance on the rate of consolidation
Geotech Geol Eng (2010) 28:61–77
71
12
100
Degree of Consolidation, U (%)
Time Factor, T
U = 90% smear = 2d transition =12d 8
T = 0.94(khs/kho)-0.86 ρ 2 = 0.9999 spacing = 3 m×3 m mandrel = 150 mm ×150 mm 4
T = 0.63 (k hs/k ho)-0.81 ρ 2 = 0.9999 spacing = 1 m×1 m mandrel = 125 mm ×50 mm 0 0 0.1 0.2
0.3
0.4
0.5
k hs/k ho Fig. 11 Dependence of the time factor on the degree of disturbance (q is the correlation coefficient)
80
mandrel = 125mm×50mm d = 50mm k hs/k ho = 0.2
60
no disturbance 40
smear = 2d transition = 12d 20
0 0.001
0.01
0.1
1
10
100
Time Factor, T Spacing = 1m×1m Spacing = 2m×2m Spacing = 3m×3m Fig. 12 Effect of PVD spacing on the rate of consolidation
Therefore, for a constant value of U, T can be expressed in terms of khs/kho as C2 khs T ¼ C1 ð11Þ kho where C1 and C2 are real positive numbers. The curves shown in Fig. 11 clearly indicate that the degree of disturbance has a significant effect on the consolidation rate. The impact of the degree of soil disturbance on the consolidation process is much more pronounced than that of the dimensions of the smear and transition zones. Consequently, the degree of disturbance needs to be predicted with greater accuracy than the dimensions of the zones of disturbance in order to produce a satisfactory design. 6.6 Effect of PVD Spacing The consolidation rate decreases with increasing PVD spacing. The impact of the PVD spacing on the consolidation rate needs to be quantified in design. In order to do this, three different unit cells—1 m 9 1 m, 2 m 9 2 m and 3 m 9 3 m—are considered together with mandrel dimensions of 125 mm 9 50 mm.
U versus T curves for the ideal case where the PVD installation does not disturb the soil are plotted in Fig. 12. For U = 90%, the decrease in T obtained by reducing the spacing from 3 to 2 m is 8.2%, while that for reducing the spacing from 2 to 1 m is 25.8%. For this ideal case, there would be a significant advantage in reducing PVD spacing. Also plotted in Fig. 12 are the results for the case where soil is disturbed due to PVD installation. The smear and the transition zone dimensions are fixed with p = 2 and 12, respectively, and the degree of disturbance is fixed at khs/kho = 0.2. For U = 90%, the values of T corresponding to 1 m 9 1 m, 2 m 9 2 m and 3 m 9 3 m unit cells are 2.32, 2.59 and 2.68, respectively. T decreases by 3.4% when the unit cell size is reduced from 3 m 9 3 m to 2 m 9 2 m, and by 10.4%, when the unit cell size is reduced from 2 m 9 2 m to 1 m 9 1 m. The incremental benefit in reducing the spacing from 3 to 2 m is negligible, although there is some benefit in reducing the spacing from 2 to 1 m. The benefit obtained in reducing the spacing is much less for the practical cases where the soil is disturbed when compared with the ideal case of no disturbance.
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7 Design Considerations
Degree of Consolidation, U (%)
100
80
spacing = 1m×1m k hs/k ho = 0.2 smear = 2d transition = 12 d
7.1 Equivalent Single Smear Zone
60
40
20
0 0.001
0.01
0.1
1
10
100
Time Factor, T Mandrel = 125mm×50mm Mandrel = 150mm×50mm Mandrel = 120mm×120mm Mandrel = 150mm×150mm Fig. 13 Effect of the mandrel size on the rate of consolidation
A similar conclusion was reached for a 150 mm 9 150 mm mandrel, although the results are not shown. 6.7 Effect of Mandrel Size and Shape The effect of mandrel size is studied for two different unit cells of 1 m 9 1 m and 3 m 9 3 m. The U versus T curves obtained for the 1 m 9 1 m unit cell are plotted in Fig. 13. The smear and transition zones are assumed to extend to 2d and 12d (i.e., p = 2 and 12), respectively, with khs/kho = 0.2. Four different mandrel sizes: 125 mm 9 50 mm, 150 mm 9 50 mm, 120 mm 9 120 mm and 150 mm 9 150 mm are used in the study. For the 1 m 9 1 m unit cell and these mandrels, the values of T associated with U = 90% are 2.32, 2.4, 2.94 and 3.3, respectively. The curves obtained for a 3 m 9 3 m grid are not plotted in the figure, but the corresponding values of T are 2.68, 2.77, 3.49 and 3.69. It is clear from Fig. 13 that the rate of consolidation decreases with increasing mandrel sizes. The square mandrels disturb a much larger area than the rectangular mandrels. Therefore, PVDs installed with square mandrels are less effective.
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From a practical point of view, the transition zone is difficult to account for in design. One way of accounting for the transition zone in design is to replace the transition zone and the smear zone with a single equivalent smear zone with larger dimensions than the original smear zone. In order to validate this procedure, several combinations of PVD spacings (1 m 9 1 m, 2 m 9 2 m and 3 m 9 3 m) and mandrels (125 9 50, 150 9 50, 120 9 120 and 150 9 150, all dimensions in mm) are studied for smear zones of two different sizes (2d and 3d), transition zones of three different sizes (6d, 9d and 12d) and khs/kho values of 0.1, 0.2 and 0.3. It is found that such a replacement is possible for the cases where there is no overlap of adjacent transition zones. The extra length of smear zone required to replace the transition zone depends only on the khs/kho ratio and the dimensions of the transition zone itself. It is independent of the original dimensions of the smear zone, PVD spacings and mandrel size and shape. Table 5 gives the values for the extra length of smear zone per unit length of transition zone (f) required to replace the transition zone. For example, for an original domain consisting of a smear zone extending to 3d and a transition zone extending to 9d, a length equal to 6d of transition zone has to be replaced. For khs/kho = 0.1, f = 0.13 (see Table 5), and hence the extra length of smear zone required is 0.13 9 6d = 0.78d. Since the original extent of the smear zone is 3d, the new dimension of the equivalent smear zone is 3.78d. These guidelines are based on a trial and error procedure and no rational basis behind the results given in Table 5 can be outlined. Nevertheless, for all the cases studied, the match between the results Table 5 Extra length of smear zone required to replace transition zone khs/kho
Extra length of smear zone per unit length of transition zone (f)
0.1
0.13
0.2
0.20
0.3
0.25
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Degree of Consolidation, U (%)
100
80
60
spacing = 1m×1m mandrel = 125mm×50mm k hs/k ho = 0.3 original smear = 2d original transition = 12d new smear = 4.5d
40
spacing = 3m×3m mandrel = 150mm×150mm k hs/k ho = 0.2 original smear = 2d original transition = 12d new smear = 4d
20
0 0.001
0.01
0.1
1
10
100
Time Factor, T Equivalent Single Smear Original Domain
multiplied by the constant p of Eqs. (1) and (2) to obtain the equivalent smear zone diameter ds,eq. For a square mandrel, both methods yield the same smear zone diameter, while for rectangular mandrels, these diameters are different. The difference between the values of U obtained by considering square smear zones and the corresponding equivalent circles was negligible (\3%) for the different combinations of spacings, mandrels, smear and transition zone sizes and degrees of disturbance investigated. One such result for a 3 m 9 3 m unit cell with 300 mm 9 300 mm smear zone and khs/kho = 0.1 is shown in Fig. 15. For rectangular mandrels, the maximum difference in U was larger, amounting to about 10%. In all the cases investigated, the U versus T curves obtained for a rectangular domain always lie between those corresponding to equivalent circular domains with equivalent circular smear zones obtained by methods A and B. Moreover, method B always produced
Fig. 14 U versus T curves for the original domain with smear and transition zones and for the new domain with a single equivalent smear zone
7.2 Equivalent Circular Domain In order to use the available analytical solution of Hansbo (1981) in design, the rectangular domain needs to be converted to an equivalent circle. There can be two ways of obtaining an equivalent circular smear zone. One way (method A) is to first estimate the dimensions of the rectangular smear zone (lx 9 ly) from the mandrel dimensions (a 9 d) by using Eqs. (1) and (2), and then convert the rectangular shape to an equivalent circle by using Eq. (8) (with sx and sy replaced by lx and ly, respectively) to obtain the equivalent diameter ds,eq of the smear zone. Alternatively (method B), the rectangular mandrel with dimensions of a 9 d can be converted to an equivalent circle by using Eq. (8) (with sx and sy replaced by a and d, respectively) to obtain the equivalent mandrel diameter dm,eq, which is then
Degree of Consolidation, U (%)
considering the original and the new domains is almost perfect (Fig. 14 shows two such comparisons) and no exception to this was observed on any occasion. However, if there is an overlap of adjacent transition zones, then the values of Table 5 cannot be used.
100
80
60
rectangular mandrel spacing = 1m×1m smear = 275mm×200mm k hs/k ho = 0.2
40 square mandrel spacing = 3m×3m smear = 300mm×300mm k hs/k ho = 0.1
20
0 0.001
0.01
0.1
1
10
100
Time Factor, T Square Mandrel Equivalent Circular Mandrel replacing Square Mandrel Rectangular Mandrel Equivalent Circular Mandrel replacing Rectangular Mandrel (method A) Equivalent Circular Mandrel replacing Rectangular Mandrel (method B)
Fig. 15 U versus T curves for square, rectangular and equivalent circular smear zone
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74
conservative results (for large time factors). The U versus T curves for a 1 m 9 1 m grid, 125 mm 9 50 mm mandrel size (with p of Eqs. (1) and (2) taken as 4), and khs/kho = 0.2 are also plotted in Fig. 15. Based on these results, it is recommended that, for rectangular mandrels, method B be used for estimating the equivalent circles because it always yields conservative results. 7.3 Example A practical example problem is worked out in this section to facilitate the understanding of the method outlined in the previous sections. A site with ch = 2 m2/year is considered, where a 125 mm 9 50 mm mandrel is to be used for installation of PVDs with dimensions 100 mm 9 4 mm (dw,eq = 66.2 mm). The degree of soil disturbance caused by the PVD installation is expressed by a khs/kho ratio of 0.2. The smear and transition zones are assumed to extend to 2d and 12d (i.e., p = 2 and 12), respectively. A transition zone of length 12d - 2d = 10d is to be replaced by adding an additional length to the original smear zone. Using Table 4, f = 0.2 for khs/kho = 0.2, which results in an additional length of smear zone of 0.2 9 10d = 2d. Thus, the new equivalent smear zone extends to 4d (2d ? 2d). For estimating the equivalent circular smear zone diameter, method B, as described in the previous section, is used. Using Eq. (8), the diameter dm,eq of the equivalent circular mandrel can be calculated as 89.2 mm and the diameter ds,eq of the equivalent smear zone is 4 9 89.2 = 356.8 mm. As a result, the ratio m = ds/dw is obtained as 5.39 [Table 4, Eq. (E)]. Assuming a spacing of 1 m 9 1 m, the diameter dc,eq [Eq. (8)] of the equivalent circular unit cell is 1.128 m, which results in n = dc/dw = 17.05 [Table 4, Eq. (D)]. This yields l = 8.82 [Table 4, Eq. (C)]. Assuming that 90% consolidation is to be achieved, T required is 2.54 [Table 4, Eq. (A)]. This corresponds to a time equal to 1.6 years, for ch = 2 m2/year and dc,eq = 1.128 m [Eq. (7)].
8 Comparison with Experimental Study As far as the authors know, there is no field case study available in which data regarding both the spatial variation of the hydraulic conductivity and the
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degree of consolidation (or settlement) with time have been provided. Thus, a direct comparison of this analysis with field results is not yet possible without making assumptions on the spatial variation of the hydraulic conductivity (which can vary greatly depending on the field conditions and the method of drain installation). However, Walker and Indraratna (2006) reported the settlement versus time data obtained from the large-scale consolidometer experiment performed by Indraratna and Redana (1998) in which the performance of a PVD was studied by considering the spatial soil disturbance variability. A 75 mm 9 4 mm PVD was installed in a consolidometer of height (H) 950 mm and diameter (dc) 450 mm with a mandrel of dimension 80 mm 9 10 mm. The coefficients of consolidation cv and ch for vertical and horizontal flow were equal to 1.5 9 10-8 m2/s and 2.4 9 10-8 m2/s, respectively. The compression and recompression indices Cc and Cr were reported to be 0.34 and 0.14, respectively. The soil in the consolidometer was subjected to a preconsolidation pressure rp0 = 35 kPa. After PVD installation, an overconsolidated state was induced by maintaining an initial pressure ro0 = 20 kPa, which corresponds to an initial void ratio e0 = 0.95. Then the pressure was increased in steps to 50, 100 and 200 kPa, respectively. As shown Fig. 2, in the experiment of Indraratna and Redana (1998), the outer diameter of the smear and transition zones are approximately 2d and 7d (d is the equivalent mandrel diameter dm,eq), respectively, with khs/kho = 0.55. Since Table 5 does not provide the value of f corresponding to khs/kho = 0.55 required to replace the transition zone, we extrapolated the data available in Table 5 by fitting a power 0:5981 law equation: f ¼ 0:5175 khs=kho , and obtained a value of f = 0.36. Accordingly, the distance of 5d (7d - 2d) of the transition zone is to be replaced by (0.36 9 5d) = 1.8d. The expanded equivalent smear zone is (2d ? 1.8d) = 3.8d, and the diameter dm,eq of the equivalent circular mandrel is 31.9 mm. Therefore, the diameter ds,eq of the equivalent smear zone is (3.8 9 31.9) = 121.3 mm, and the equivalent drain diameter dw,eq is 50.3 mm. This results in m = ds/dw = 2.16, and n = dc/dw = 8.95, which yields l = 2.16. Using these values, we calculated the degree of consolidation. The well resistance was neglected, as suggested by Walker and Indraratna
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(2006), because of the small length of the PVD. The total settlement qc was calculated using the following expressions: 0 8 HCr r 0 < log 10 1þe0 r0o ; r\rp 0 0 qc ¼ : HCr log rp0 þ HCc log r0 ; r0 [ r0 10 r 10 r p 1þe0 1þe0 o
p
ð12Þ
Stress (kPa)
200
Settlement (mm)
where r0 is the applied stress. Combining the degree of consolidation with the total settlement values for different applied stresses, the settlement versus time relationship is obtained and plotted in Fig. 16. Also plotted in Fig. 16 are the settlement versus time data measured by Indraratna and Redana (1998) and the settlement versus time data calculated without considering the transition zone (i.e., with only a smear zone of thickness 2dm,eq). The analysis results with expanded smear zone are in reasonable agreement with the measured data. Figure 16 also shows that if the transition zone is neglected, then proper values of the degree of consolidation (or settlement) may not be obtained.
40
100
9 Conclusions In this paper, the effect of soil disturbance caused by PVD installation on the rate of PVD-enhanced consolidation is studied through a two-dimensional finite element analysis. A rectangular pattern of PVD installation is assumed. The zone of disturbance is assumed to comprise of a smear zone immediately surrounding the PVD, where the disturbance is most severe, and a transition zone surrounding the smear zone, where the disturbance gradually decreases with distance from the PVD. The hydraulic conductivity in the transition zone is assumed to increase linearly from a low value in the smear zone to the original in situ value in the undisturbed zone. In particular, the effects of the degree of soil disturbance, size of the smear and the transition zones, PVD spacing and mandrel size and shape are investigated. The results of this analysis were compared with the results of an experimental study. It was found that the results of this analysis match the experimental results reasonably well. It was also observed that neglecting the transition zone may lead to erroneous estimation of the degree of consolidation and the resulting settlement. The following conclusions can be made from this study: (1)
0
(2)
80 120 160
(3) 0
10
20
30
40
50
Time (days) Measured (Indraratna & Redana 1998) Present Analysis (Smear Zone = 2d,Transition zone = 5d,k hs/k ho = 0.2)
(4)
Analysis Without Transition Zone (Smear Zone = 2d,k hs/k ho = 0.2)
Fig. 16 Comparison of settlement versus time data obtained from our analysis with that observed in the experimental study of Indraratna and Redana (1998)
(5)
The transition zone has a definite role in slowing down the consolidation rate and should not be neglected in design. The error in estimating the degree of consolidation caused by converting rectangular unit cells to equivalent circles can be [ 5%. However, accounting for the presence of soil disturbance reduces the error. The factors related to drain installation that affect the rate of consolidation are: soil disturbance, the extent of the disturbed zone and the mandrel size and shape. Of these, soil disturbance (resulting in reduced hydraulic conductivity) is the factor that most affects the rate of consolidation. For a particular degree of consolidation, the time factor is related to the ratio of the conductivities of the smear and the undisturbed zones by a power law. The shape of the mandrels affects the consolidation rate. The square mandrels cause more
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(6)
(7)
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disturbance to the soil than rectangular mandrels. The transition zone can be replaced by an equivalent smear zone with increased dimensions. The extra length of smear zone required depends only on the degree of disturbance and the size of the transition zone. This facilitates the design, because analytical solutions for a single smear zone are available. Converting domains with square smear zone to equivalent circles leads to minimal error. A method of converting rectangular smear zones to equivalent circles has been proposed that will always produce conservative designs.
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