grim163-237

Proceedings of the Institution ofCivil EngineersGround Improvement 163November 2010 Issue GI4Pages 237249doi: 10.1680/grim.2010.163.4.237

Paper 900048Received 31/10/2009Accepted 14/07/2010

Keywords: geotechnicalengineering/ground improvement/mathematical modelling

Prasenjit BasuAssistant Professor, Department ofCivil and Environmental Engineering,Pennsylvania State University,University Park, PA, USA

Dipanjan BasuAssistant Professor, Department ofCivil and Environmental Engineering,University of Connecticut, Storrs,CT, USA

Monica PrezziAssociate Professor, School of CivilEngineering, Purdue University,West Lafayette, IN, USA

Analysis of PVD-enhanced consolidation with soil disturbance

P. Basu PhD, D. Basu PhD and M. Prezzi PhD

Soil disturbance caused by the installation of

prefabricated vertical drains (PVDs) in soft soil deposits

has a detrimental effect on the rate of consolidation. The

current practice of accelerating consolidation using

PVDs captures the effect of soil disturbance typically by

reducing the in situ hydraulic conductivity in the

disturbed zone, and by assuming that the hydraulic

conductivity is spatially constant over the entire

disturbed zone and that preloading is instantaneous.

Through recent laboratory and field studies it has been

shown that the hydraulic conductivity varies spatially in

the disturbed zone surrounding a PVD. Based on the

data available in the literature, four possible profiles

were identified for the spatial variation of the hydraulic

conductivity in the disturbed zone. Analytical solutions

were developed for the rate of consolidation considering

these hydraulic conductivity profiles for instantaneous

and time-dependent preloading. This paper shows that

the consolidation rate depends not only on the hydraulic

conductivity profile in the disturbed zone but also on the

preloading rate.

NOTATION

ch coefficient of consolidation

j normalised (with respect to rd) radial distance of a point

within the transition zone where the bilinear profile of

hydraulic conductivity changes slope in case e

k soil hydraulic conductivity

kc in situ hydraulic conductivity

ks constant hydraulic conductivity in the disturbed zone

ksm hydraulic conductivity in the inner smear zone

ktr hydraulic conductivity in the transition (or outer smear)

zone

m normalised (with respect to rd) outer radius of the inner

smear zone

mv coefficient of volume compressibility

n normalised (with respect to rd) radius of the unit cell

p maximum value of the applied total stress due to

preloading

q normalised (with respect to rd) outer radius of the

transition zone

r radial distance measured from the centre of PVD

rc radius of unit cell

rc,eq equivalent radius of unit cell

rd radius of vertical drain

rd,eq equivalent radius of prefabricated vertical drain

rj radial distance at which the slope of the bilinear profile

of hydraulic conductivity within the transition zone

changes slope in case e

rm,eq equivalent radius of mandrel

rsm equivalent radius of inner smear zone

rtr equivalent radius of transition zone

sx , sy spacings of PVDs in two mutually perpendicular

directions

T time factor

t time

Tc time factor corresponding to the time tc of preload

construction

tc time of preload construction

u average excess pore pressure in the unit cell developed

due to preloading

uc excess pore pressure in the undisturbed zone

ui initial average excess pore pressure in the unit cell

usm excess por pressure in the inner smear zone

utr excess pore pressure in the transition (or outer smear)

zone

vc specific discharge in the undisturbed zone

vsm specific discharge in the inner smear zone

vtr specific discharge in the transition (or outer smear) zone

degree of soil disturbance within the inner smear zonejust adjacent to the PVD

j degree of disturbance (for use in case e) at radialdistance rj

t degree of soil disturbance at the boundary between theinner smear zone and the transition zone

w unit weight of waterv uniform vertical strain within the unit cell average total stress (due to preloading) in the unit cell9 average effective stress (due to preloading) in the unit cell

1. INTRODUCTION

Prefabricated vertical drains (PVDs) have been successfully used

in conjunction with preloading to improve the mechanical

properties of soft soils since the early 1970s (Bergado et al.,

1993a; Holtz, 1987; Holtz et al., 1991; Johnson, 1970; Lo and

Mesri, 1994). The installation of PVDs facilitates the dissipation

of excess pore pressure generated during preloading by reducing

the drainage path within the ground. This speeds up the

consolidation process and thereby increases the strength and

stiffness of soft clayey soils. PVDs consist of a plastic core

surrounded by a filter sleeve and have typical cross-section

dimensions of 100 mm 3 4 mm. PVDs are installed (using

Ground Improvement 163 Issue GI4 Analysis of PVD-enhanced consolidation with soil disturbance Basu et al. 237

closed-ended mandrels) in square or triangular patterns (Figure

1) with a centre-to-centre spacing of 13 m (Holtz, 1987). Each

PVD collects water from a surrounding zone of influence known

as the unit cell. The dimension and shape of the unit cell depend

on the installation arrangement of the PVDs. For square pattern

of PVD installation, the unit cell is square in shape whereas for

triangular installation pattern, the unit cell is hexagonal in shape.

During installation, as the closed-ended mandrel (with the drain

in it) is pushed into the ground, the surrounding soil is

displaced making room for the PVD. Once the desired depth is

reached, the mandrel is withdrawn leaving the drain within the

ground. The installation of PVDs creates a disturbed soil zone

around them. Because of soil disturbance, the hydraulic

conductivity within the disturbed zone is less than the in situ

(undisturbed) hydraulic conductivity kc. This reduction in

hydraulic conductivity due to soil disturbance causes a decrease

in the consolidation rate. In this paper, the disturbed zone is

subdivided into two zones: the inner smear zone and the

transition zone (sometimes referred to as the outer smear zone).

Traditionally, however, the division of the disturbed zone into

two distinct zones is not done; the single disturbed zone is

commonly referred to in the literature as the smear zone.

Both theoretical and experimental research has been done on

PVDs to estimate the discharge rate of PVDs (i.e. the rate of

consolidation) and to determine and mitigate operational

problems associated with soil disturbance (Barron, 1948; Basu

and Madhav, 2000; Basu and Prezzi, 2007, 2009; Bo et al.,

2003; Chai et al., 1997; Hansbo, 1981, 1997; Indraratna and

Redana, 1997; Sathananthan and Indraratna, 2006; Walker and

Indraratna, 2006). Traditionally, the effect of soil disturbance is

accounted for by assuming a constant value for the hydraulic

conductivity, ks, over the disturbed zone (traditionally referred

to as the smear zone). The degree of disturbance in thissmear zone has been quantified in the literature in terms of the

ratio ks/kc (Table 1). The radial distance from the centre of the

PVD to the outer boundary of the smear zone has also been

defined by some authors in terms of the equivalent mandrel

radius rm,eq (Table 2); the assumption made in these estimations

is that both the mandrel and the smear zone cross-section can

be converted to equivalent circles.

Recent experimental investigations have shown that the

hydraulic conductivity in the disturbed zone is not spatially

constant (Indraratna and Redana, 1998; Onoue et al., 1991;

Madhav et al., 1993; Sharma and Xiao, 2000). Figure 2 shows

Authors Degree of disturbance ks/kc

Casagrande and Poulos (1969) 0.001Bergado et al. (1991) 0.500.66Onoue et al. (1991) 0.200.60Bergado et al. (1993a, 1993b) 0.10Madhav et al. (1993) 0.20Hansbo (1997) 0.250.30Hansbo (1986); Hird and Moseley (2000) 0.33Chu et al. (2004) 0.170.5

Table 1. Degree of disturbance in the soil adjacent to the drain boundary

(a)

(b)

(c)

bw

bt

Filter

Core

Mandrel

s

PVDs

Hexagonal unitcells

sy

sx

PVDs

Rectangular unitcells

Figure 1. (a) PVD and mandrel, (b) triangular arrangement and(c) square (when sx sy) arrangement

238 Ground Improvement 163 Issue GI4 Analysis of PVD-enhanced consolidation with soil disturbance Basu et al.

profiles for the hydraulic conductivity k (normalised with

respect to kc) in the vicinity of vertical drains, observed in

laboratory and field studies, as functions of normalised radial

distance r/rm,eq (r measured from the centre of the drain). Note

that the laboratory experiments of Onoue et al. (1991),

Indraratna and Redana (1998) and Sharma and Xiao (2000)

were performed using circular drains and circular mandrels,

while the field study reported by Madhav et al. (1993) was

done with PVDs installed using square mandrels. These

experimental investigations show that the assumption of a

single value for the hydraulic conductivity in the disturbed

zone is not strictly valid.

Authors Equivalent radius of smear zone

Holtz and Holm (1973); Hansbo (1986, 1987); Bergado et al. (1991, 1993b) 2rm,eqJamiolkowski et al. (1983) (2.53)rm,eqMesri et al. (1994) (24)rm,eqChai and Miura (1999); Hird and Moseley (2000) (23)rm,eqSathananthan and Indraratna (2006) 2.5rm,eq

Table 2. Extent of smear zone

0

0

4

4

8

8

12

12

16

16

20

20

Normalised distance, /r rm,eq

Normalised distance, /r rm,eq

0

0

02

02

04

04

06

06

08

08

10

10

k k/

ck

k/c

Case b

Case c

Case e

Field data(Madhav ., 1993)et al

(a)

(b)

Linear hydraulic conductivityprofiles based on results ofexperimental studies

Experimental studies

Onoue . (1991)et al

Sharma and Xiao (2000)

Indrarata and Redana (1998)

Figure 2. Hydraulic conductivity profiles from (a) field samples(Madhav et al., 1993) and (b) laboratory model studies

Case a

k/kc

k/kc

k/kc

k/kc

k/kc

r

1

Case b

r

1

Case c t

r

1

Case d

r

1

Case ej

1

(a)

Unit cell

rPervious boundary

Impervious boundary

rd

Soft deposit

Undisturbed zone

Transition zone

Inner smear zone

Vertical drain

rsmrtr

rc

rj(b)

r

Figure 3. (a) Idealised domain: a circular unit cell with circularinner smear zone and transition zone and (b) differentvariations of the hydraulic conductivity with radial distancefrom the centre of the drain


Figure 3 shows an idealised (circular) unit cell with a PVD

located centrally within it. The figure also shows different

possible variations of k within the unit cell. As can be seen

in this figure, there are three zones surrounding a PVD: the

inner smear zone (in which the hydraulic conductivity

remains constant at its least value ks), the transition zone or

outer smear zone (in which the hydraulic conductivity

increases with increasing radial distance from the drain) and

the undisturbed zone (where the hydraulic conductivity kcremains at its in situ value). According to Madhav et al.

(1993) and Miura et al. (1993), k increases approximately

linearly (case b, Figure 3(b)) from a value equal to ks at the

inner smear zone boundary (i.e. the boundary between the

inner smear zone and the transition zone) to the in situ

value kc at the transition zone boundary (i.e. the boundary

between the transition zone and the undisturbed zone).

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

inner smear zone (case c, Figure 3(b)). In their model, the

hydraulic conductivity k follows a bilinear curve; k increases

at a given rate (slope) from ks at the drain boundary (i.e. the

drainsoil interface) to kt at the inner smear zone boundary

and, at another rate, from kt at the inner smear zone

boundary to kc at the transition zone boundary. Holtz and

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

degree of disturbance decreases monotonically as the

horizontal distance from the drain increases and, therefore,

there is no distinguishable inner smear zone (case d, Figure

3(b)). The results of the laboratory experiments (Figure 2(b))

by Indraratna and Redana (1998) and Sharma and Xiao

(2000) support the approximate linear variation of k in the

transition zone; however, the experimental data immediately

adjacent to the drain are insufficient to ascertain the

variation of the hydraulic conductivity in the inner smear

zone. Based on the experimental data (Figure 2), a bilinear

variation of the hydraulic conductivity in the transition zone

(case e, Figure 3(b)) can also be assumed.

The transition zone radius (measured from the centre of the

drain to the transition zone boundary) has been found to be

(67)rm,eq by Onoue et al. (1991), more than 7rm,eq by

Indraratna and Redana (1998), (1014)rm,eq from the data

reported by Sharma and Xiao (2000) and approximately 12rm,eqby Madhav et al. (1993). Jamiolkowski et al. (1983), based on

studies on pile driving in clay, suggested that the transition

zone radius can be as large as 20rm,eq.

In this paper, closed-form solutions that can be used to

estimate the degree of consolidation in the unit cell

surrounding a PVD are presented. Both instantaneous and ramp

loading (the ramp loading is a simple form of time-dependent

preloading) are considered. The mathematical model considers

a circular PVD surrounded by a circular zone of influence (unit

cell) with circular inner smear, outer smear (transition) and

undisturbed zones. Flow in the vertical direction is neglected.

Well resistance is also neglected and, hence, the solutions

presented in this paper are generally applicable at any depth

along a PVD. The vertical strain (due to consolidation) is

assumed to be constant at different radial distances measured

from the centre of the PVD. Different hydraulic conductivity

profiles (cases b to e in Figure 3(b)) are assumed in the

disturbed zone. The use of the proposed solutions is illustrated

through a practical example.

2. PREVIOUS THEORETICAL STUDIES

Analytical solutions that can be used to calculate the degree

of consolidation as a function of time were developed by

Barron (1948) and Hansbo (1981) for case a. These

formulations consider only a single disturbed (smear) zone

with circular cross-sections for both the vertical drain and the

smear zone, and assume axisymmetric flow of water into the

drain. The hydraulic conductivity was assumed to be a

constant in these analyses; thus, no transition zone was

considered in the derivation of these solutions. The solution

by Barron (1948) is based on the TerzaghiRendulic theory of

radial consolidation, whereas that by Hansbo (1981) is a

simplified approach based on continuity of flow and Darcys

law. The solution by Hansbo (1981) matches closely the

rigorous solution by Barron (1948) and is widely used in

practice. Leo (2004) developed analytical solutions for case a

considering both radial and vertical flow.

Chai et al. (1997) obtained analytical solutions for the degree

of consolidation and the average excess pore pressure

assuming linear and bilinear variations for the hydraulic

conductivity in the disturbed zone, as represented in case d and

case c, respectively. Walker and Indraratna (2006) provided an

analytical solution for the case of a parabolic variation of the

hydraulic conductivity in the disturbed zone.

Numerical solutions considering only a single smear zone (case

a) also exist (Basu and Madhav, 2000; Indraratna and Redana,

1997). Numerical studies considering the variation of the

hydraulic conductivity in the transition zone, described by

cases b and c, have also been done (Basu and Prezzi 2007;

Hawlader et al., 2002; Madhav et al., 1993).

The above-mentioned studies considered that preloading is

applied instantaneously. However, for most practical cases,

preloading is applied through several lifts placed on the

ground over a finite period of time. Studies that considered

time-dependent preloading are rather few (Conte and Trocone,

2009; Lekha et al., 1998; Leo, 2004; Olson, 1977; Tang and

Onitsuka, 2000; Zhu and Yin, 2004). These studies did not

separately consider the inner smear and the transition zones;

rather, a single smear zone with constant hydraulic

conductivity was used. Moreover, two broad categories of

analysis assuming equal vertical strain (e.g. Leo, 2004) and

free vertical strain (e.g. Zhu and Yin, 2004) are present in the

literature.

3. PROBLEM DEFINITION AND ASSUMPTIONS

A drain, installed in a saturated clay deposit, is assumed to

have a circular cross-section with radius rd. The length of the

drain spans over the entire thickness of the soft clay layer. 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 3(a)); rd and rc are the radii of the drain and

unit cell, respectively. The flow of water is assumed to be

horizontal within the unit cell. Therefore, the only pervious

boundary is the interface between the drain and the unit cell.

This results in radially convergent horizontal flow of water

towards the drain. Note that the assumption of purely

horizontal flow implies that the consolidation achieved through

vertical flow is neglected. According to Carrillo (1942),

irrespective of the value of the vertical degree of consolidation,


the degree of consolidation due to combined radial and vertical

drainage approaches that due to radial drainage alone only

when the latter approaches 100%. For a homogeneous deposit

with no horizontal strain in the soil cylinder, flow patterns are

identical along any horizontal plane. Hence, analysis

considering only one such horizontal plane with axisymmetric

flow is sufficient to solve the problem. It is also assumed that

the vertical strain within the unit cell (due to consolidation) is

spatially uniform. This represents the case of equal-strain

consolidation (Richart, 1959). Additionally, flow of water is

assumed to follow Darcys law.

For cases b and c, the inner smear and transition zones are

assumed to be annular in cross-section with outer radii

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

with rd , rsm , rtr , rc (Figure 3). For case d, no distinctive

inner smear zone is considered; hence, rsm does not exist. Case

e is a combination of cases b, c and d. For all the cases, the

undisturbed zone lies between rtr < r< rc, with r measured

radially outward from the centre of the drain. Three

dimensionless terms (n, m and q) are defined to normalise

radial distances from the centre of the PVD with respect to the

radius of the drain as n rc/rd, m rsm/rd, q rtr/rd. Note thatno overlapping of disturbed zones surrounding two adjacent

drains was considered. Thus, the solution presented herein is

valid for m , q , n.

4. CALCULATION OF AVERAGE EXCESS PORE

PRESSURE

4.1. Case b

An axisymmetric coordinate system, with r representing the

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

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

the inner smear zone (i.e. for rd < r < rsm) is assumed to be

constant at ks. In the transition zone (i.e. for rsm < r < rtr), the

hydraulic conductivity ktr(r) increases linearly from ks at the

inner smear zone boundary (r rsm) to kc at the transition zoneboundary (r rtr). The hydraulic conductivity in theundisturbed zone (i.e. for rtr < r < rc) remains constant at its in

situ value kc. The linear variation of ktr(r) within the transition

zone can be expressed mathematically as

ktr(r) ks rtr kc rsmrtr rsm

kc ksrtr rsm r

A Br; for rsm < r < rtr1a

This expression of hydraulic conductivity can be rearranged to

make it dimensionless by dividing it by the hydraulic

conductivity of the undisturbed zone. In the dimensionless

form, the variation of the hydraulic conductivity k(r*) at any

radial distance r* ( r/rd; r measured from the centre of PVD)within the unit cell can be expressed as

k(r)kc

for 1< r< mAb Bb r for m< r< q1 for q< r < n

8 Tc), as

illustrated in Figure 5(b).

Parameters Assumed/calculated values fordifferent parameters

Mandrel dimension 120 mm 3 120 mmbw 100 mmbt 4 mmsx , sy 2 mch 10 m

2/yearp 500 kParm,eq (calculated) 67.7 mmrc,eq (calculated) 1128.4 mmrd,eq (calculated) 33.1 mmrsm 2.5rm,eqrtr 6rm,eqrj 4rm,eqm 5.11n 34.09q 12.27j 8.18 0.20t 0.75j 0.90

Table 3. Parameters used in the practical example


Figure 6 shows that, at any point of time, the degree of

consolidation U attained due to instantaneous preloading was

greater than that attained with gradual preload application. The

difference in U between these two cases became a maximum at

T Tc. For case c, U at T Tc for instantaneous preloadingwas 47% higher than that achieved with ramp loading (Figure

6). Thus, the assumption of instantaneous application of

preloading always led to overestimation of the degree of

consolidation.

Figures 7 and 8 show u=p plotted against T and U plotted

against T, respectively, for the different hydraulic

conductivity profiles considered in this paper and for both

instantaneous preloading and ramp loading. Furthermore, the

results obtained for cases a to e were compared (see Figures

7(a) and 8(a)) with those obtained for a parabolic hydraulic

conductivity profile within the disturbed zone, as proposed

by Walker and Indraratna (2006). The results obtained for

cases b and e were almost identical. For ramp loading, the

average excess pore pressure u at any given time within the

construction period was almost unaffected by the different

profiles of hydraulic conductivity considered in the disturbed

zone (see Figure 7(b)). This happened because relatively less

excess pore pressure was dissipated within the construction

time tc; at t tc, the degree of consolidation was only about

0

0

0

0

04

04

04

04

08

08

08

08

12

12

12

12

Nor

mal

ised

exc

ess

pore

pre

ssur

eu

p/N

orm

alis

ed e

xces

s po

re p

ress

ure

u p/

Nor

mal

ised

incr

ease

in to

tal s

tres

s/

pN

orm

alis

ed in

crea

se in

tota

l str

ess

/p

0

0

02

02

04

04

06

06

08

08

1

1

Time factor T

Time factor T

Increase in total stress due toinstantaneous preloading

Increase in total stress dueto ramp loading

Parameters used:

Parameters used:

u p/ for instantaneous preloading

u p/ for ramp loading

m nq

511, 3409,1227, 02,075

t

T mn q

c

t

027, 511,3409, 1227,02, 075

A1

A3

A4

A2

A5

A6

(a)

(b)

Figure 5. Variation of normalised average excess porepressure u with time factor T for hydraulic conductivityprofile c: (a) instantaneous preloading, and (b) ramp loading

0

20

40

60

80

100

0001 001 01 1 10TcTime factor T

Deg

ree

ofco

nsol

ida

tion

: %U T m n

q

Parameters used:027, 511, 3409

1227, 02, 075c

t

For instantaneouspreloading

For ramp loading

Results are for Case c

Figure 6. Effect of instantaneous preloading and ramp loadingon the degree of consolidation U for hydraulic conductivityprofile c

0

0

02

02

04

04

06

06

08

08

1

1

Nor

mal

ised

exc

ess

pore

pre

ssur

eu

p/N

orm

alis

ed e

xces

s po

re p

ress

ure

u p/

Instantaneous preloading

Ramp loading

Case a

Case a

Case b

Case b

Case c

Case c

Case d

Case d

Case e

Case e

Parabolic variation ofwithin the disturbedzone (Walker andIndraratna, 2006)

k

Parameters used:511, 3409,1227, 818,02, 075, 09

m nq j t j

Parameters used:027511, 3409,1227, 818,02, 075, 09

Tm nq j

c

t j

0001

0001

001

001

01

01

1

1

10

10

Time factor T

Time factor T

(a)

(b)

Tc

Figure 7. Variation of normalised excess pore pressure u withtime factor T for different hydraulic conductivity profiles:(a) instantaneous preloading and (b) ramp loading


816% (U 8.0, 9.7, 16.0, 13.6, and 10.1% for the hydraulicconductivity profiles a, b, c, d and e, respectively) (see Figure

8(b)). The effect of the different hydraulic conductivity

profiles became prominent for t. tc as more and more

excess pore pressure dissipated.

To consider different hydraulic conductivity profiles, a

parametric study was performed to investigate the effects of

possible variations of m, q and on the degree ofconsolidation U achieved at a particular time. Values of U were

calculated for a time factor T 1.0 and for some extremevalues of m, q and . One parameter was set to its minimum ormaximum probable value while keeping the values of all other

parameters equal to the values reported in Table 3. The

calculated extreme values of U were compared (see Figure 9)

with values obtained from the practical example described

above and presented in Figure 8(a). Figure 9(a) shows that the

difference in U was a maximum ( 9%) for case c when m wasvaried from 4.01 to 8.18 (i.e. rsm was varied from 2 to 4rm,eq).

Note that U calculated for cases a and d did not change with

m. Similar variation of U was observed when two possible

extreme values of q were used. Differences in U (calculated

with q equal to 10.23 and 20.45, i.e. rtr was varied from 5 to

10rm,eq) were equal to 6.9, 4 and 8.6%, respectively, for cases a,

b and d (Figure 9(b)). For all other cases, U was insensitive (1.1

and 0.3% difference for cases c and e, respectively) to the

variation of q. U was most sensitive to changes in the degree of

disturbance . The differences in U were equal to 50.8, 47,22.1, 30.7 and 46.5%, respectively, for cases a, b, c, d and e, for

varying from 0.1 to 0.5.

Figure 10 shows the effect of preloading time tc on the degree

of consolidation. Note that, for the results shown in Figure 10,

p was kept constant at 500 kPa; thus, the rate of application of

preload R was varied in the simulations to obtain different

values of tc. Time-dependent preloading prolonged the

Deg

ree

ofco

nsol

ida

tion

: %U

Deg

ree

ofco

nsol

ida

tion

: %U

0

0

20

20

40

40

60

60

80

80

100

100


Ramp loading

Case a

Case a

Case b

Case b

Case c

Case c

Case d

Case d

Case e

Case e

Parabolic variation ofwithin the disturbedzone (Walker andIndraratna, 2006)

k

Parameters used:511, 3409,1227, 818,02, 075, 09

m nq j t j

Parameters used:027511, 3409,1227, 818,02, 075, 09

Tm nq j

c

t j

0001

0001

001

001

01

01

1

1

10

10

Time factor T

Time factor T

(a)

(b)

Tc

Figure 8. Degree of consolidation plotted against time factorfor different hydraulic conductivity profiles: (a) instantaneouspreloading and (b) ramp loading

100

80

60

40

20

0

Deg

ree

ofco

nsol

ida

tion

: %U

m 401m 511m 818

Case a Case b Case c Case d Case e(a)


n q j

T

3409, 1227, 818,02, 075,

09, 10

t

j

100

80

60

40

20

0

Deg

ree

ofco

nsol

ida

tion

: %U

q 1023q 1227q 2045

Case a Case b Case c Case d Case e(b)


n m j

T

3409, 511, 818,02, 075,

09, 10

t

j

100

80

60

40

20

0

Deg

ree

ofco

nsol

ida

tion

: %U

01 02 05

Case a Case b Case c Case d Case e

(c)

Instantaneous preloadingn m q

j T

3409, 511, 1227,

818, 075, 09, 10 t j

Figure 9. Effects of m, q and on the degree of consolidationU for different hydraulic conductivity profiles; (a) variation ofU with m, (b) variation of U with q and (c) variation of Uwith


consolidation process; for a given p, the greater the

construction time tc, the less was the degree of consolidation U

at any given time t.

8. CONCLUSIONS

Installation of prefabricated vertical drains causes soil

disturbance. The hydraulic conductivity of the disturbed soil is

less than that of the in situ (undisturbed) soil; this causes a

reduction in the rate of consolidation. In this paper, the

experimental data available in the literature concerning the

variation of the hydraulic conductivity in the disturbed zone

were collected and analysed. Four possible variations for the

hydraulic conductivity in the disturbed zone (cases b, c, d and

e) were identified from field and laboratory experiments

performed on PVDs. Analytical solutions describing the rate of

consolidation considering these different hydraulic

conductivity profiles were developed for both instantaneous

and time-dependent preloading. A practical example is

presented to illustrate how the analytical solutions can be used

in these different cases to calculate the degree of consolidation

achieved at any given time.

Our analyses showed that the rate of consolidation depends not

only on the degree of disturbance of the soil adjacent to the

drain but also on how the hydraulic conductivity varies within

the disturbed zone. Hence, proper identification of the

operative hydraulic conductivity profile in the vicinity of the

drain is necessary for accurate prediction of the rate of

consolidation. Moreover, time-dependent preloading has an

impact on the rate of consolidation that needs to be accounted

for in design.

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Results are for Case c


Tc 005

Tc 011

Tc 016

Tc 027

Tc 054

Parameters used:m nq

511, 3409,1227, 500 kPa02, 075

p t

Deg

ree

ofco

nsol

ida

tion

: %U

001 01 1 10Time factor T

0

20

40

60

80

100

Figure 10. Effect of construction time on the degree ofconsolidation U


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NOTATION1. INTRODUCTIONTable 1Figure 1Table 2Figure 2Figure 3

2. PREVIOUS THEORETICAL STUDIES3. PROBLEM DEFINITION AND ASSUMPTIONS4. CALCULATION OF AVERAGE EXCESS PORE PRESSURE4.1. Case bEquation 1aEquation 1bEquation 2aEquation 2bEquation 2cEquation 3aEquation 3bEquation 3cEquation 4aEquation 4bEquation 4cEquation 5aEquation 5bEquation 5cEquation 6Equation 7Equation 8Equation 94.2. Case cEquation 10aEquation 10bEquation 114.3. Case dEquation 12Equation 134.4. Case eEquation 144.5. Case aEquation 15

5. DEGREE OF CONSOLIDATION5.1. Consolidation due to instantaneous preloadingEquation 16Equation 17Equation 18Equation 19Equation 20Equation 21Equation 225.2. Consolidation due to multistage preloadingEquation 23Equation 24Equation 25Equation 26Equation 27Figure 4Equation 28aEquation 28b

6. CALCULATION OF EQUIVALENT RADIUSEquation 29Equation 30Equation 31Equation 32

7. RESULTSTable 3Figure 5Figure 6Figure 7Figure 8Figure 9

8. CONCLUSIONSFigure 10

REFERENCESBarron, 1948Basu and Madhav, 2000Basu and Prezzi, 2007Basu and Prezzi, 2009Bergado et al., 1991Bergado et al., 1993aBergado et al., 1993bBo et al., 2003Carrillo, 1942Casagrande and Poulos, 1969Chai and Miura, 1999Chai et al., 1997Chu et al., 2004Conte and Trocone, 2009Hansbo, 1981Hansbo, 1986Hansbo, 1987Hansbo, 1997Hawlader et al., 2002Hird and Moseley, 2000Holtz, 1987Holtz and Holm, 1973Holtz et al., 1991Indraratna and Redana, 1997Indraratna and Redana, 1998Jamiolkowski et al., 1983Johnson, 1970Lekha et al., 1998Leo, 2004Lo and Mesri, 1994Madhav et al., 1993Mesri et al., 1994Miura et al., 1993Olson, 1977Onoue et al., 1991Richart, 1959Rixner et al., 1986Sathananthan and Indraratna, 2006Sharma and Xiao, 2000Tang and Onitsuka, 2000Walker and Indraratna, 2006Zhu and Yin, 2004

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