17486025 Analytical Solutions

8/3/2019 17486025 Analytical Solutions

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Analytical solutions for consolidation aided by vertical drains

D. BASU, P. BASU and M. PREZZI*School of Civil Engineering, Purdue University, West Lafayette, IN 47907, USA

(Received 3 November 2005; in final form 12 December 2005)

Soil disturbance caused during the installation of vertical drains reduces the in situ hydraulic conductivity of soft deposits in the immediate vicinity ofthe drains, resulting in a slower rate of consolidation than would be expected in the absence of disturbance. Experimental investigations have revealedthe existence of two distinct zones, a smear zone and a transition zone, within the disturbed zone around the vertical drain. The degree of change in thehydraulic conductivity in the smear and transition zones is difficult to assess without performing of laboratory tests. Based on the available literature,four different profiles of hydraulic conductivity versus distance from the vertical drain were identified. Closed-form solutions for the rate ofconsolidation for each of these four hydraulic conductivity profiles were developed. It is found that different variations of the hydraulic conductivityprofiles in the disturbed zone result in different rates of consolidation.

Keywords: Vertical drain; Analytical solution; Consolidation; Smear; Soil disturbance; Ground improvement

1. Introduction

Soft soil deposits are characterized by low shear strength, high

compressibility, and low hydraulic conductivity. In order to

improve the strength and stiffness of clayey soils, vertical

drains, combined with preloading, are often used in practice

(Johnson 1970, Holtz 1987, Bergado et al. 1993a). The installa-

tion of vertical drains reduces the water drainage path, speeding

up the dissipation of the excess pore pressure generated during

preloading. The fact that the flow is predominantly in the

horizontal direction (except near the top surface or close to ahighly permeable silt/sand seam) further helps the process

because, owing to depositional anisotropy, the hydraulic con-

ductivity is generally greater in the horizontal direction than in

the vertical direction.

Since the early 1970s, prefabricated vertical drains (PVDs)

have been successfully used in various sites throughout the

world (Bergado et al. 1993a, 1997, 2002, Lo and Mesri 1994).

PVDs consist of a plastic core surrounded by a filter sleeve with

typical cross-sectional dimensions of 100 mm 4 mm (Holtz1987). They are installed in rectangular or triangular patterns

with centre--centre distance ranging from about 1.0 m to 3.0 m

(Holtz 1987).

Closed-ended mandrels are generally used for the installation

of PVDs. Soil adjacent to the mandrel is displaced and dragged

down in order to produce a space for the PVD (Hird and

Moseley 2000). This creates a disturbed zone of soil around

the drain around the PVD in which the strength and the hydrau-

lic conductivity in the horizontal direction decrease, and the

compressibility increases (Holtz and Holm 1973, Lo and Mesri

1994). The reduction in hydraulic conductivity delays the con-

solidation process.

2. Disturbed zone characterization

The degree of change in the soil hydraulic conductivity kin the

disturbed zone with distance from the PVD is not known with

certainty. According to most researchers (Barron 1948,

Casagrande and Poulos 1969, Hansbo 1981, 1986, 1987,

1997, Bergado et al. 1991, 1993a), the soil within the disturbed

zone, which is also known as the smear zone, is completely

remoulded, resulting in a spatially uniform hydraulic conduc-

tivity ks which is smaller than the in situ hydraulic conductivity

kc (figure 1, case A). The degree of disturbance, quantified as

the ratio ks/kc of the hydraulic conductivities of the smear zone

and the undisturbed zone, and the extent of the smear zone have

been the subject of many investigations based on back analysis

of case histories, laboratory tests on samples collected from

field, model experiments, studies of pile driving, practical con-

siderations, and experience. According to Bergado et al.

(1993a,b), Hansbo (1986, 1997), and Hird and Moseley(2000), the degree of disturbance ks/kc varies between 0.1 and

0.33. However, Casagrande and Poulos (1969) proposed a

value as low as 0.001, while Bergado et al. (1991) suggested

that ks/kc ranges from 0.5 to 0.66.

The dimensions and shape of the disturbed zone depend on

many factors, such as the size and shape of the mandrel, the rate

of mandrel penetration, the type of mandrel shoe, and the soil

properties (Hird and Moseley 2000, Holtz et al. 1991). In the

Geomechanics and Geoengineering: An International Journal

Vol. 1, No. 1, March 2006, 63--71

*Corresponding author. Email: [email protected]

Geomechanics and Geoengineering: An International JournalISSN 1748-6025 print=ISSN 1748-6033 online 2006 Taylor & Francis

http:==www.tandf.co.uk=journalsDOI: 10.1080=17486020500527960


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case of circular sand drains, the smear zone is probably alsocircular. A variety of mandrels with different cross-sections

(circular, rectangular or diamond shaped) are used for PVDs

(Holtz et al. 1991, Bo et al. 2003). The disturbed zone of a non-

circular mandrel is likely to have a non-circular cross-section.

However, it is customary to convert the areas of non-circular

mandrels and the corresponding cross-sections of the smear

zones surrounding the PVDs to equivalent circular areas.

Assuming that the disturbed zone has a single value of

hydraulic conductivity ks (figure 1, case A), a number of

researchers (Holtz and Holm 1973, Jamiolkowski et al. 1983,

Hansbo 1986, 1997, Bergado et al. 1991, 1993b, Mesri et al.

1994, Chai et al. 2001) have concluded that the equivalent

smear zone radius (radius of the smear zone measured from

the centre of the drain) can be taken as approximately two to

four times the equivalent mandrel radius rm,eq.

The above discussion is based on the assumption that the

hydraulic conductivity remains constant within the disturbed

zone (case A). However, recent experimental investigationshave shown that the assumption of a single value for the

hydraulic conductivity in the disturbed zone is not valid

(Onoue et al. 1991, Madhav et al. 1993, Indraratna and

Redana 1998, Sharma and Xiao 2000). Madhav et al. (1993)

performed a field-scale study to investigate the variation of the

hydraulic conductivity profile in the disturbed zone. Soil sam-

ples were collected from soft ground in which PVDs were

installed and tested in the laboratory to obtain the hydraulic

conductivity profile. The results of Madhav et al. (1993) are

reproduced in figure 2, where the degree of disturbance

(expressed as the ratio k/kc) is plotted as a function of the

normalized distance from the drain (normalization is performed

with respect to the equivalent mandrel radius rm,eq). Based onthese results, both Madhav et al. (1993) and Miura et al. (1993)

suggested that the disturbed zone comprises of two distinct

zones: the smear zone and the transition zone. In the completely

remoulded smear zone immediately surrounding the drain, the

soil has a constant hydraulic conductivity ks. In the transition

zone, which surrounds the smear zone, the degree of distur-

bance gradually decreases as the distance from the drain

increases. Madhav et al. (1993) further suggested that the

hydraulic conductivity increases linearly (figure 1, case B)

from a value equal to ks at the smear zone boundary (i.e. the

boundary between the smear zone and the transition zone) to the

in situ value kc

at the transition zone boundary (i.e. the bound-

ary between the transition zone and the undisturbed zone).

ck k

ck k

ck k

Unit cell

rPervious boundary

Impervious boundary

rdSof t deposit

Undisturbed

zone

Transition zone

Smear zone

Vertical drain

rsmrtr

rc

(b)

ck

r

1Case A

r

c

1Case B

t

r

1

Case C

r

1Case D

rp

r

p

1Case E

(a)

k

k k

Figure 1. (a)Idealized domain: a unit cell with smear andtransitionzones.(b)Variation of the hydraulic conductivity with distance from the centre of thedrain for different cases.

0 4 8 12 16 20

Normalized distance, r/rm,eq

0

0.2

0.4

0.6

0.8

1

k/k

c

'= 118 kPa '= 235 kPa

Case B

Case C

Case E

Figure 2. Normalized hydraulic conductivity profiles from field samples.(Reproduced from Madhav et al. 1993.)

64 D. Basu et al.


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To our knowledge, only three laboratory model studies

(Onoue et al. 1991, Indraratna and Redana 1998, Sharma and

Xiao 2000) have been performed to investigate the variation of

hydraulic conductivity in the disturbed zone. The results of

these studies are reproduced in figure 3. Onoue et al. (1991)

used a circular steel drain, which acted as a mandrel, in their

experiments. Consequently, in figure 3, r/rm,eq starts from 1 for

the data of Onoue et al. (1991). Based on their study, Onoueet al. (1991) proposed a two-zone model for the disturbed zone.

However, unlike Madhav et al. (1993) and Miura et al. (1993),

Onoue et al. (1991) assumed a linear variation for the hydraulic

conductivity in the smear zone (figure 1, case C). This results in

a bilinear variation for the hydraulic conductivity; kincreases at

one rate from ks at the drain boundary (i.e. the drain--soil inter-

face) to ktat the smear zone boundary, and at another rate from

kt at the smear zone boundary to kc at the transition zone

boundary.

Case C fits the PVD data of Sharma and Xiao (2000) well

(figure 3). However, no information regarding the variation of

the hydraulic conductivity for the zone lying between r/rm,eq = 0

and r/rm,eq = 2 is available from their study (r/rm,eq for theirexperiment starts from zero since they performed tests with

PVDs of negligible thickness). Case C also fits the hydraulic

conductivity data of Madhav et al. (1993) reasonably well

(figure 2).

Holtz and Holm (1973) and Holtz et al. (1991) suggested that

the degree of disturbance decreases monotonically as the dis-

tance from the drain increases, and therefore there is no distin-

guishable smear zone (figure 1, case D). The data of Indraratna

and Redana (1998) (figure 3) appear to follow the profile of

case D, although a paucity of data immediately adjacent to the

drain makes it difficult to ascertain the actual variation of the

hydraulic conductivity in the smear zone.

A new case for the variation of hydraulic conductivity (figure

1, case E) may be identified for the data of Sharma and Xiao

(2000) if the hydraulic conductivity is assumed to be constant

(with a value equal to the value at r/rm,eq = 2) in the zone between

r/rm,eq = 0 and r/rm,eq = 2. For case E, the hydraulic conductivity

remains constant at ks within the smear zone and increases in the

transition zone following a bilinear curve with one slope from ksat the smear zone boundary to kp at any intermediate point within

the transition zone (at r = rp) and a different slope from kp (at

r = rp) to kc at the transition zone boundary. The hydraulic

conductivity profile (figure 2) obtained by Madhav et al. (1993)

can also be described by case E.

No definite conclusions regarding the variation of thehydraulic conductivity in the disturbed zone can be drawn

from the experimental studies of PVD disturbance (Onoue

et al. 1991, Madhav et al. 1993, Indraratna and Redana 1998,

Sharma and Xiao 2000) which take the transition zone into

account. These studies suggest that k/kc can be assumed to be

about 0.2 in the immediate vicinity of the drain ( r/rm,eq = 0) for

cases B,C, D, and E,and for caseC, k/kc can be assumedto vary

between 0.5 and 0.8 at the smear zone boundary. Based on the

studies by Onoue et al. (1991), Madhav et al. (1993) and

Sharma and Xiao (2000), the smear zone boundary can be

assumed to lie at a distance of 2rm,eq to 5rm,eq from the centre

of PVD, and the transition zone boundary (beyond which the

hydraulic conductivity does not vary with increasing distance

from the drain) can be assumed to vary between 6rm,eq and

15rm,eq. Jamiolkowski et al. (1983) suggested that the transition

zone radius can be up to 20rm,eq based on studies of pile driving

in clay. However, more laboratory and field studies are neces-

sary to determine the hydraulic conductivity profile and the

corresponding dimensions of the smear and transition zones

that are most likely to occur in the field.

3. Theoretical studies on soil disturbance

Theoretical studies on soil disturbance have generally been

restricted to case A. Analytical solutions for case A, assuming

a radial flow of water into the drain, were developed by Barron

(1948) and Hansbo (1981); their solutions can be used to

calculate the degree of consolidation as a function of time.

These formulations consider a vertical drain with a circular

cross-section. The solution obtained by Barron (1948) is

0 4 8 12 16 20

Normalized distance, r/rm,eq

0

0.2

0.4

0.6

0.8

1

k/k

c

Onoue et al. (1991)

Onoue et al. (1991)Indraratna and Redana (1998)

Indraratna and Redana (1998)





Sharma and Xiao (2000)



Onoue et al. (1991)

Indraratna and Redana(1998)

Sharma andXiao (2000)

Figure 3. Normalized hydraulic conductivity profiles from laboratory modelstudies.

Consolidation aided by vertical drains 65


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based on the Terzaghi--Rendulic theory of radial consolidation

(Terzaghi 1925, Rendulic 1935, 1936), while that obtained by

Hansbo (1981) is a simplified approach based on the continuity

of flow and Darcys law. The Hansbo (1981) solution matches

closely the rigorous solution obtained by Barron (1948) and is

widely used in practice. Leo (2004) developed analytical solu-

tions considering both radial and vertical flow. Numerical solu-

tions considering only smear (case A) also exist (Indraratna andRedana 1997; Basu and Madhav 2000).

Numerical studies of the variation of the hydraulic conduc-

tivity in the transition zone represented in cases B and C have

also been reported (Madhav et al. 1993, Hawlader et al. 2002,

Basu et al. 2005). Madhav et al. (1993) considered case B with

a simplified assumption that the hydraulic conductivity in the

transition zone is constant at a value equal to the average of ksand kc, and used finite-difference analysis to study the PVD

response. Basu et al. (2005) also considered case B but used

finite-element analysis, taking into account the actual linear

variation of the hydraulic conductivity in the transition zone.

Hawlader et al. (2002) considered case C and analysed the PVD

performance using an elasto-viscoplastic constitutive model.However, most of these numerical studies are case specific

and cannot be directly used in design. Chai et al. (1997)

obtained an analytical solution for consolidation by PVD for

case D; however, their expressions are too complex for use in

routine design.

3.1 Scope of the present study

In this paper, we develop analytical solutions for consolidation

by vertical drains, considering both the smear and the transition

zones, which are easy to use. Solutions are obtained for cases B,

C, D, and E using a methodology similar to that of Hansbo(1981).

In practice, a number of drains are installed in the ground,

and each drain has a zone of influence. This zone of influence is

called a unit cell because each cell behaves identically (for

homogeneous deposits), and water within one unit cell does not

flow into another unit cell. The analysis considers one such unit

cell with a circular cross-section. The cross-sections of the

drain and the disturbed zone are assumed to be circular.

4. Analysis

4.1 Definition of the problem and assumptions

It is assumed that a drain with a circular cross-section of radius

rd is installed in a saturated soft soil deposit. The length of the

drain spans the entire thickness of the soil deposit. An annular

cylinder of soil with inner and outer radii rd and rc (measured

from the centre of the drain) is considered as the unit cell (figure

1) (rd and rc are the drain radius and the unit cell radius,

respectively). The effect of the flow of water in the vertical

direction within the unit cell is negligible (Leo 2004). Therefore

the only pervious boundary of the unit cell is the interface

between the drain and the unit cell. This results in a radially

convergent horizontal flow of water into the drain. If a homo-

geneous deposit with no horizontal strain in the soil cylinder is

assumed, flow patterns are identical along any horizontal plane.

Consideration of only one such horizontal plane with axisym-

metric flow is sufficient to solve this problem. In addition, the

flow of water is assumed to follow Darcys law. It is further

assumed that the vertical strain within the unit cell is spatiallyuniform. This represents the case of equal strain consolidation

(Richart 1959).

For cases B, C, and E, the smear and transition zones are

assumed to have annular cross-sections with outer radii (as

measured from the centre of the drain) rsm and rtr, respectively

(rsm and rtr are the smear zone radius and the transition zone

radius, respectively). As canbe seen in figure 1, rd, rsm, rtr, rc.

For case D, no smear zone is considered (figure 1). For all

cases (B, C, D and E), the undisturbed zone lies between rtrr rc with r measured radially outward from the centre of thedrain.

4.2 Average excess pore pressure

4.2.1 Case B. A radial coordinate system, where rrepresents

the radial distance from the centre of the drain, is used in the

analysis. In this case, the hydraulic conductivity ksm(r) within the

smear zone(i.e. for rd r rsm) is assumedto be a constant equalto ks. In the transition zone (i.e. for rsm r rtr), the hydraulicconductivity ktr(r) increases linearly from ks at the smear zone

boundary (r= rsm) to kc at the transition zone boundary (r= rtr).

The hydraulic conductivity kc remains constant in the

undisturbed zone (i.e. for rtr r rc). The linear variation ofktr(r) can be expressed mathematically as

ktrr ks r rsmrtr rsm kc ks for rsm r rtr: 1a

which can be rearranged as

ktrr A Br 1b

where

A ksrtr kcrsmrtr rsm 2

B kc ksrtr rsm : 3

The specific discharge vc in the undisturbed zone can be

written as

vc kcw

@uc@r

for rtr r rc 4a

where w is the unit weight of water and uc is the excess porepressure at a distance r in the undisturbed zone. Similarly, the

66 D. Basu et al.


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specific discharges within the transition and smear zones can be

written as

vtr ktrw

@utr@r

for rsm r rtr 4b

vsm ksmw

@usm@r

for rd r rsm 4c

The total volume of water entering a cylinder of arbitrary radius r

(r, rc) within the unit cell from the outer hollow cylinder (of

thickness rc -- r) must be equal to the change in volume of the

outer hollow cylinder. Using this concept, the pore pressure at any

distance rwithin the unit cell can be related to the vertical strain ev(whichis assumed to be uniform throughout theunit cell) as follows:

2rvc r2c r2 @"v

@tfor rtr r rc 5a

2rvtr r2c r2 @"v

@tfor rsm r rtr 5b

2rvsm

r2c

r2 @"v

@t

for rd

r

rsm

5c

where tis time.

Replacing vc, vtr and vsm in equations (5a), (5b), and (5c) by

equations (4a), (4b), and (4c), respectively, we obtain

@uc@r

w2kc

r2cr r

8>>: 9>>; @"v@t

for rtr r rc 6a

@utr@r

w2ktr

r2cr r

8>>: 9>>; @"v@t

for rsm r rtr 6b

@usm@r

w2ks

r2cr r

8>>:

9>>;

@"v@t

for rd r rsm: 6c

Integrating equation (6c) and applying the boundary condi-tion that the excess pore pressure is fully dissipated at the drain

boundary (i.e. usm = 0 at r = rd), we obtain

usm w2ks

r2c lnr

rd

8>: 9>; 12

r2 r2d ! @"v

@t: 7a

Integrating equation (6b) and using the continuity condition

utr = usm at r= rsm, we obtain

utrw2

r2cA

lnksr

ABrrsm

& ' 1

B2

ABrksAln ABr

ks

8>:

9>;

& '

1ks

r2c lnrsm

rd

8>: 9>;12

r2smr2d & '!

@"v@t

: 7a

Similarly, integrating equation (6a) and using the continuity

condition uc = utr at r = rtr, we obtain

ucw2

1

kcr2c ln

r

rtr

8>: 9>;12

r2r2tr & '

1ks

r2c lnrsm

rd

8>: 9>;12

r2smr2d & '

r2c

Aln

rtrks

rsmkc

8>: 9>; 1B2

kcksAln kcks

8>: 9>;& '!@"v@t

: 7c

Let u be the average excess pore pressure throughout the unit

cell. Then we can write the following equation:

r2c r2d

u rsmrd

2rusmdrrtr

rsm

2rutrdrrcrtr

2rucdr: 8

Substituting usm, utr, and uc from equations (7a), (7b), and (7c),

respectively, in equation (8) and rearranging terms we obtain

u wr2c

2kc

@"v@t

9

where

r2c

r2cr2dln

rc

rtr8>: 9>;

kc

ksln

rsm

rd8>: 9>;

kcrtrrsmksrtrkcrsm

lnksrtr

kcrsm8>: 9>;

3

4

1r2cr2d

kc

ksr2smr2d r2trkcrtrrd r2trr2d

ksrtrkcrsm

!

1r2c r

2cr2d

kc4ks

r4smr4d kc

3kcks r3trr3sm

rtrrsm

kcksrtrkcrsmrtrrsm2

2kcks35ksrtrkcrsmrtrkcksrsmf g

kcrtrrsmksrtrkcrsm3

kcks4ln

kc

ks

8>: 9>;r4tr4

#:

10

We now define the following dimensionless terms:

nrcrd

11

mrsmrd

12

qrtrrd

13

ks

kc: 14Equation (10) can then be rewritten in terms of these quantities

as

n2

n21 lnn

q

8>>: 9>>;1

ln m qmqmlnq

m

8>: 9>;34

!

1n2 1

1

m2 1 q2 q m q2 m2 q m

!



6/9

1n2 n2 1

1

4

m4 1

131 q

3 m3 q m

q mq m2

21 3 5q m q mf g

q mq m3

1 4 ln1

8>: 9>; q44

#: 15a

Equation (15a) is too cumbersome for use in routine design.

However, a number of terms on the right-hand side make a

negligible contribution to the value of m. If we neglect these

terms, equation (15a) simplifies to

lnn

q8>>: 9>>;

1

ln m q

m

q m lnq

m8>: 9>;

3

4 : 15b

The ratio n2/(n2 -- 1) is close to unity for the typical unit cell and

drain diameters used in practice, and is not included in equation

(15b).

4.2.2 Case C. In this case, the hydraulic conductivity

ksm(r) in the smear zone varies from ks at the drain--soil

interface (r = rd) to kt at the smear zone boundary (r = rsm),

and is given by

ksm

r

ks

r rdrsm rd

kt

ks

for rd:

9

>>; m 1

m t lnm

t

8>:

9>; q m tq m ln

tq

m8>: 9>; 34 : 17

The dimensionless term bt is defined as follows:

t ktkc

: 18

4.2.3 Case D. In this case the disturbed zone consists of the

transition zone of radius rtr, and the hydraulic conductivity

ktr(r) varies from ks at the drain boundary (r = rd) to kc at the

transition zone boundary (r = rtr). The expression for ktr(r) can

be obtained from equation (1a) by replacing rsm by rd. As

before, the hydraulic conductivity kc in the undisturbed zone

is a constant. The expression for m (associated with equation

(9)) is derived following the same procedure as outlined for

case B. After eliminating the terms which make a negligible

contribution, the following equation is obtained for m:

ln nq

8>>: 9>>; q 1 q 1 ln q

3

4: 19

4.2.4 Case E. In this case, the hydraulic conductivity ksm(r)

has a constant value ks within the smear zone (i.e. for rd rrsm) and increases in the transition zone, following a bilinear

curve with one slope between ks (at r= rsm) and kp (at r= rp, say)

and another slope between kp (at r = rp) and kc (at r = rtr).

Thereafter, the hydraulic conductivity in the undisturbed zone

remains constant at kc. This variation can be described

mathematically as follows:

ln nq

8>>: 9>>; 1

ln m p mp pm lnp

pm

8>>: 9>>; q p

pq p ln pq

p

8>>: 9>>; 34

20

where the dimensionless terms p and bp are defined as

p rprd

21

p kpkc

22

where rd, rsm , rp , rtr, rc and ks , kp , kc.

4.3 Degree of consolidation

If we assume that all the excess pore pressure due to preloading

is developed instantly, we can write the following relationship:

@"v@t

mv @0

@t mv @u

@t23

where 0is the average effective stress in the unit cell due topreloading at the end of consolidation, u is the average excess

pore pressure at the time of load application, and mv is the

coefficient of volume compressibility.

The coefficient of consolidation ch in the horizontal direction

and the time factor Tare defined as follows:

ch kcmvw

24

T cht4r2c

25

68 D. Basu et al.


7/9

Substituting equation (9) into equation (23) we obtain the linear

differential equation

du

dt 2kc

mvwr2cu 0: 26

Solving equation (26) using the initial condition u

u0 at t= 0,

where u0 is the initial average excess pore pressure, and using

the dimensionless terms defined in equations (24) and (25), we

obtain the change in average excess pore pressure with time:

u u0e8T : 27

The degree of consolidation U at a particular time t(or time

factor T) is the ratio of the excess pore pressure dissipated to the

excess pore pressure induced at that time. U can be expressed

mathematically as follows:

U 1 uu0

: 28

Substituting equation (27) in equation (28) gives the following

expression for the degree of consolidation:

U 1 e8T : 29

5. Results

5.1 Consolidation rates for different cases

In order to determine the influence of the various hydraulic

conductivity profiles described above on the consolidation rate,

the solution for case A given by Hansbo (1981) is reproduced

here so that a comparison can be made:

ln nm

8: 9; 1

lnm 34: 30

Hansbo (1981) suggested that, by using an equivalent radius

rd,eq, the analytical solutions can also be applied to PVDs. The

equivalent radius is calculated as follows:

rd;eq 1bw bt 31

where bw and btare the width and thickness, respectively, of the

PVD. Rectangular or hexagonal unit cells are obtained when

PVDs are installed in rectangular or triangular patterns (Holtz

et al. 1991). In order to use the analytical solutions, these shapes

need to be replaced by equivalent circles which have the same

area as the rectangular or hexagonal unit cell. The equivalent

radius rc,eq of the unit cell for a rectangular installation pattern is

rc;eq ffiffiffiffiffiffiffiffi

sxsy

r32

where sx and sy are the spacings of the PVDs in two mutually

perpendicular directions. For a triangular pattern, the equivalent

radius is given by

0.001 0.01 0.1 1 10

Time factor, T

0

20

40

60

80

100

Degreeofconso

lidation,

U

(%)

Case A

Case B

Case C

Case D

n = 17.05m = 2.69q = 16.17

= 0.2

t= 0.6

(a)

0.001 0.01 0.1 1 10

Time factor, T

0

20

40

60

80

100

Degreeo

fconsolidation,

U

(%)

Case A

Case B

Case C

Case D

n = 51.14m = 5.11q = 30.67

= 0.2

t= 0.6

(b)

Figure 4. Plots of degree of consolidation versus time factor for differenthydraulic conductivity profiles: (a) spacing of 1 m; (b) spacing of 3 m. Thecurve for case D is just to the right of the curve for case A. The curve for case Acorresponds to the solution obtained by Hansbo (1981).



8/9

rc;eq ffiffiffiffiffiffiffiffiffiffi

3p

2

ss 33

where s is the PVD spacing.

Figures 4(a) and 4(b) show plots of the degree of con-

solidation U versus time factor T for PVDs installed in a

rectangular arrangement with centre--centre spacings of 1 m

(rc,eq = 564.2 mm) and 3 m (rc,eq = 1692.6 mm), respec-

tively. Four hydraulic conductivity profiles (cases A, B, C

and D) are considered. The PVDs are assumed to have a cross-

section of 100 mm 4 mm (rd,eq = 33.1 mm). Mandrels with arectangular cross-section (a b) are considered, with dimen-sions 125 mm 50 mm (Saye 2003) (rm,eq = 44.6 mm) for aspacing of 1 m and 150 mm 150 mm (Bergado et al. 1993b)(rm,eq = 84.6 mm) for a spacing of 3 m. The equivalent mandrel

radii are obtained from equation (32) by replacing sx and syby a and b, respectively. The extent of the disturbed zone is

defined by rsm = 2rm,eq (except for case D) and rtr = 12rm,eq.

The degree of disturbance b at the drain surface is taken as

0.2. For case C, bt = 0.6 is assumed.

Figures 4(a) and 4(b) indicate that the hydraulic conductivity

profile in the disturbed zone has a definite impact on the rate of

consolidation. In figure 4(a), the time factors T at U = 90%

corresponding to cases A, B, C, and D are 1.74, 2.54, 1.37, and2.09, respectively. For ch = 1 m

2/year, the corresponding actual

times are 2.2, 3.2, 1.7 and 2.7 years. With respect to case A

(Hansbo 1981), the increase in time (or time factor) required for

90% consolidation is 46% and 20% for cases B and D, respec-

tively; for case C, the time required for 90% consolidation

decreased by 21%. The time factors corresponding to U= 90%

in figure 4(b) are 2.79, 3.59, 2.13, and 2.95 for cases A, B,

C, and D, respectively. The increases in T for cases B and D

compared with case A are 29% and 6%, respectively, while

for case C the decrease in T relative to case A is 24%.

It is clear from these results that a proper knowledge of the

hydraulic conductivity profile in the disturbed zone is needed

for accurate design. In addition, neglecting the transition zonein design may lead to errors in the estimation of the consolida-

tion rate. Knowledge of the degree of soil disturbance in the

immediate vicinity of the drain is of utmost importance for

predicting drain performance. This is evident by comparing

the curves for cases B, C, and D. For cases B and D, k/kc is

approximately 0.2 in the vicinity of the drain. However, for case

C this ratio increases from 0.2 to 0.6 in the vicinity of the drain.

Consequently, the difference in response between cases C and

B or cases C and D ismore than that observed when cases B and

D are compared.

5.1.2 Example In order to understand the impact of the

various hydraulic conductivity profiles on the rate of

consolidation, a practical example is analysed for all the

hydraulic conductivity profiles of figure 2. It is assumed that

the PVDs were installed with a mandrel of cross-section 120

mm 120 mm (rm,eq = 67.7 mm), the PVDs have a cross-section of 100 mm 4 m m (rd,eq = 33.1mm), and the clay at thesite has ch = 10 m

2/year.

For a hydraulic conductivity profile corresponding to case B,

the smear zone extends to 2rm,eq and the transition zone extends

to 11rm,eq (figure 2). If the hydraulic conductivity profile cor-

responds to case C, rsm and rtr are 4.5rm,eq and 13rm,eq, respec-tively. However, if the hydraulic conductivity profile

corresponds to case E, rsm, rp, and rtr are equal to 2rm,eq,

7rm,eq, and 15rm,eq, respectively. The degree of disturbance b

near the drain can be taken as 0.2 for all the cases (figure 2). For

case C, bt = 0.75 and for case E, bp = 0.9 (figure 2). A square

arrangement of PVDs with a centre--centre spacing of 2 m

(rc,eq = 1128.4 mm) is chosen. The values of m calculated

for cases B, D, and E (table 1) are 11.00, 7.50, and 10.32,

respectively. The value of T for U = 90% is calculated

from equation (29) as 3.17, 2.16, and 2.97 for cases B, C,

and E, respectively. For ch = 10 m2 /year, the actual times

required for 90% consolidation are 1.6 years, 1.1 years, and

1.5 years for cases B, C, and E, respectively.

6. Conclusions

Installation of vertical drains disturbs the soil around the drain.

The hydraulic conductivity of the disturbed soil is less than that of

the original soil, reducing the acceleration of the consolidation

process caused by the presence of the drains to less than it would

be in the absence of disturbance. A number of researchers have

proposed various hydraulic conductivity profiles in the disturbed

zones. Five possible hydraulic conductivity profiles (cases A, B,

C, D, and E) have been considered in this paper. An analyticalsolution for the rate of consolidation, corresponding to case A, is

already available in the literature (Hansbo 1981). Analytical solu-

tions for the remaining cases have been developed in this paper.

Our analyses showed that the transition zone has a definite

impact in slowing down the consolidation process and therefore

must be considered in design. Moreover, the rateof consolidation

can vary greatly depending on how the hydraulic conductivity

varies within the transition zone. Hence, proper identification of

the hydraulic conductivity profile around a vertical drain is

necessary for accurate prediction of the rate of consolidation.

Table 1. Solution of examplea

Case rsm (mm) rtr (mm) rp (mm) m q p n b bt bp m

B 135.4 744.7 -- 4.09 22.50 -- 34.09 0.2 -- -- 11.00C 304.7 880.1 -- 9.20 26.59 -- 34.09 0.2 0.75 -- 7.50E 135.4 1015.5 466.9 4.09 30.68 14.11 34.09 0.2 -- 0.9 10.32

ard = 33.1 mm; rc = 1128.4 mm.

70 D. Basu et al.


9/9

The experimental data available in the literature concerning the

variation of the hydraulic conductivity within the transition zone

was collected and analysed. Definite conclusions regarding the

most likely hydraulic conductivity profile could not be reached

because of the limited amount of experimental data. Until more

information regarding this issue becomes available, all possible

hydraulic conductivity profiles, as outlined in this paper, should

be considered before final design decisions are made.

References

Barron, R.A., Consolidation of fine-grained soils by drain wells. Trans. ASCE,1948, 113, 718--742. Re printed in A History of Progress, Vol. 1, pp. 324-348, 2003 (ASCE: Reston, VA).

Basu, D. and Madhav, M.R., Effect of prefabricated vertical drain clogging onthe rate of consolidation: a numerical study. Geosynth Int., 2000, 7(3),189--215.

Basu, D., Basu, P., and Prezzi, M. Study of consolidation by prefabricatedvertical drain. Internal Geotechnical Report 2005-01, 2005 (PurdueUniversity: West Lafayette, IN).

Bergado, D.T., Asakami, H., Alfaro, M.C. and Balasubramaniam, A.S., Smeareffects on vertical drains on soft Bangkok clay. J. Geotech. Eng.--ASCE,1991, 117(10), 1509--1530.

Bergado, D.T., Alfaro, M.C. and Balasubramaniam, A.S., Improvement of softBangkok clay using vertical drains. Geotext Geomembranes, 1993a, 12,615--663.

Bergado, D.T., Mukherjee, K., Alfaro, M.C. and Balasubramaniam, A.S.,Prediction of vertical-band-drain performance by the finite-elementmethod. Geotext Geomembranes, 1993b, 12, 567--586.

Bergado, D.T., Balasubramaniam, A.S., Fannin, R.J., Anderson, L.R., andHoltz, R.D., Full scale field test of prefabricated vertical drain (PVD) onsoft Bangkok clay and subsiding environment. in Ground Improvement,Ground Reinforcement, Ground Treatment: Developments 1987--1997,edited by V.R. Schaefer, pp. 372--393, 1997 (American Society of CivilEngineers: New York).

Bergado, D.T., Balasubramaniam, A.S., Fannin, R.J. and Holtz, R.D.,Prefabricated vertical drains (PVDs) in soft Bangkok clay: a case study of

the new Bangkok International Airport project. Can. Geotech. J., 2002, 39,304--315.Bo, M.W., Chu, J., Low, B.K., and Choa, V., Soil Improvement: Prefabricated

Vertical Drain Techniques, 2003 (Thomson Learning: Stamford, CT).Casagrande, L. and Poulos, S., On the effectiveness of sand drains. Can.

Geotech. J., 1969, 6(3), 287--326.Chai, J.C., Miura, N. and Sakajo, S., A theoretical study on smear effect around

vertical drain, in Proceedings of the 14th International Conference on SoilMechanics and Foundation Engineering, Hamburg, 1997, pp. 1581--1584.

Chai, J.-C., Shen, S.-L., Miura, N. and Bergado, D.T., Simple method ofmodeling PVD-improved subsoil. J. Geotech. Geoenviron. Eng., 2001,127(11), 965--972.

Hansbo, S., Consolidation of fine-grained soils by prefabricated drains, inProceedings of the 10th International Conference on Soil Mechanics and

Foundation Engineering, Stockholm, 1981, pp. 677--682.Hansbo, S., Preconsolidation of soft compressible subsoil by the use of pre-

fabricated vertical drains. Ann Trav Publics Belg, 1986, 6, 553--563.

Hansbo, S., Design aspects of vertical drains and lime column installations, inProceedings of the 9th Southeast Asian Geotechnology Conference,Bangkok, 1987, pp. 1--12.

Hansbo, S., Practical aspects of vertical drain design, in Proceedings of the 14thInternationalConference on SoilMechanics and Foundation Engineering,Hamburg, 1997, pp. 1749--1752.

Hawlader, B.C., Imai, G. and Muhunthan, B., Numerical study of the factorsaffecting the consolidation of clay with vertical drains. GeotextGeomembranes, 2002, 20, 213--239.

Hird, C.C. and Moseley, V.J., Model study of seepage in smear zones aroundvertical drains in layered soil. Geotechnique, 2000, 50(1), 89--97.

Holtz, R.D., Preloading with prefabricated vertical strip drains. GeotextGeomembranes, 1987, 6, 109--131.

Holtz, R.D. and Holm, B.G., Excavation and sampling around some sand drainsin Ska-Edeby, Sweden. Sartryck och Preliminara Rapporter, 1973, 51,79--85.

Holtz, R.D., Jamiolkowski, M.B., Lancellotta, R., and Pedroni, R.,Prefabricated Vertical Drains: Design and Performance, 1991(Butterworth Heinemann: Oxford).

Indraratna, B. and Redana, I.W., Plane-strain modeling of smear effects asso-ciated with vertical drains. J. Geotech. Geoenviron. Eng., 1997, 123(5),474--478.

Indraratna, B. andRedana, I.W., Laboratory determination of smear zone duetovertical drain installation. J. Geotech. Geoenviron. Eng., 1998, 124(2),180--184.

Jamiolkowski, M., Lancellotta, R. and Wolski, W., Precompression and speed-ing up consolidation, in Proceedings of the 8th European Conference onSoilMechanics and Foundation Engineering, 1983, Vol. 3, pp. 1201--1226(A.A. Balkema: Rotterdam).

Johnson, S.J., Foundation precompression with vertical sand drains.J. Soil Mech.Fdn. Div., 1970, 96(SM1), 145--175.Leo, C.J., Equalstrain consolidation by vertical drains.J. Geotech.Geoenviron.

Eng., 2004, 130(3), 316--327.Lo, D.O.K., and Mesri, G., Settlement of test fills for Chek Lap Kok airport. in

Vertical and Horizontal Deformations of Foundations and Embankments,edited by A.T. Yeung and G. Feaalio, pp. 1082--1099, 1994 (AmericanSociety of Civil Engineers: New York).

Madhav, M.R., Park, Y.-M. and Miura, N., Modelling and study of smear zonesaround band shaped drains. Soils Found, 1993, 33(4), 135--147.

Mesri, G., Lo, D.O.K., and Feng, T-W., Settlement of embankments on softclays. in Vertical and Horizontal Deformations of Foundations andEmbankments, edited by A.T. Yeung and G. Feaalio, pp. 8--56, 1994(American Society of Civil Engineers: New York).

Miura, N., Park, Y. and Madhav, M.R., Fundamental study on drainage perfor-mance of plastic-board drains. J. Geotech. Eng.--JSCE, 1993, 481(III-25),31--40. (in Japanese).

Onoue, A., Ting, N.-H., Germaine, J.T. and Whitman, R.V., Permeability ofdisturbed zone around vertical drains, in Geotechnical EngineeringCongress, Proceedings of the Congress of the Geotechnical Engineering

Division, 1991, pp. 879--890 (American Society of Civil Engineers: NewYork).

Rendulic, L., Der hydrodynamische Spannungsausgleich in zentral entwasser-ten Tonzylindern. Wasserwirtsch-Wassertech , 1935, 2, 250--253.

Rendulic, L., Porenziffer und Porenwasserdruck in Tonen. Bauingenieur, 1936,17, 559--564.

Richart, F.E., Review of the theories for sand drains. Trans. ASCE, 1959, 124,709--736.

Saye, S.R., Assessment of soil disturbance by the installation of displacementsand drains and prefabricated vertical drains. in Soil Behavior and SoftGround Construction, pp. 372--393, 2003 (American Society of CivilEngineers: New York).

Sharma, J.S. and Xiao, D., Characterization of a smear zone around verticaldrains by large-scale laboratory tests. Can. Geotech. J., 2000, 37,

1265--1271.Terzaghi, K., Erdbaumechanik auf bodenphysikalischer Grundlage, 1925(Deuticke: Vienna).


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