Equivalent plane strain modeling of vertical drains in soft ground under embankment combined with...

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Equivalent plane strain modeling of vertical drains in soft groundunder embankment combined with vacuum preloading

Tuan Anh Tran *, Toshiyuki Mitachi

Hokkaido University, Graduate School of Engineering, Soil Mechanics Laboratory, Room A6-53, Building A, Sapporo 060-8628, Japan

Received 18 July 2007; received in revised form 10 November 2007; accepted 14 November 2007Available online 3 January 2008

Abstract

A conversion method is proposed to convert from an axisymmetric unit cell to an equivalent plane strain unit cell under embankmentloading combined with vacuum preloading. To verify the proposed method, we have conducted FE analyses for two cases of subsoil, oneis a subsoil having only one homogeneous clay layer, and the other one is a subsoil having two clay layers. The analyzed results showedthat the effects of both well resistance (in the vertical drain) and smear zone (around the vertical drain) are satisfactorily modeled by theproposed plane strain unit cell in both cases. 2007 Elsevier Ltd. All rights reserved.

Keywords: Analytical solution; Consolidation; Finite element analysis; Plane strain modeling; Vacuum-surcharge preloading; Vertical drain

1. Introduction

It is widely recognized that vertical drains are able tospeed up consolidation in soft clay subsoil that usuallyhave extremely slow natural drainage. In recent years, thevertical drains have been combined with vacuum preload-ing under embankment (see Fig. 1), and therefore a muchfaster consolidation in soft clay subsoil can be achieved;in short, we call this preloading technique vacuum-sur-charge preloading.

In vacuum-surcharge preloading, the vacuum pressurepropagates along the vertical drains, and a nearly isotropiccompression zone is created within the subsoil zone

beneath the embankment; therefore, this technique canenhance the stability of the subsoil during consolidationunder the embankment.

It is widely agreed that the performance of a verticaldrain under conventional embankment (or conventionalsurcharge) can be represented by an axisymmetric unit cell[2,6] as shown in Fig. 2. To simulate the performance of

multiple vertical drains under an embankment by FEM,

we have to carry out a 3D full-scale simulation, in whicha lot of cubic elements have to be employed [3,10]. As aresult, the time needed for computation becomes very long,and a powerful computer is needed.

However, if we assume that the performance of a verti-cal drain can be equivalently represented by a plane strainunit cell (see Fig. 2), then an equivalent full-scale planestrain simulation can be made. Consequently, the timeneeded for computing the full-scale plane strain simulationis much shorter than that needed in a full-scale 3D simula-tion. In fact, Chai et al. [3], Hird et al. [8], Indraratna andRedana [9,10], and Indraratna et al. [11] confirmed the fea-

sibility of this idea.In 2005, Indraratna et al. [12] proposed a conversionmethod for cases of single layer subsoil under vacuum-surcharge preloading condition. In this method, theydeveloped two analytical models for the axisymmetricand plane strain cells. Based on these models, theyproposed two solutions for development of the averageexcess pore water pressure with time within these cells.

Afterwards, by equating these two solutions, theyobtained a conversion expression of permeability fromthe axisymmetric cell to the equivalent plane strain cell.

0266-352X/$ - see front matter 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compgeo.2007.11.006

* Corresponding author. Tel.: +81 11 706 6194; fax: +81 11 706 7204.E-mail addresses: [email protected] (T.A. Tran), mitachi@

eng.hokudai.ac.jp (T. Mitachi).

www.elsevier.com/locate/compgeo

Available online at www.sciencedirect.com

Computers and Geotechnics 35 (2008) 655672
mailto:[email protected]:mitachi@mailto:mitachi@mailto:[email protected]


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As a result, by using this plane strain cell, a full-scale planestrain modeling of soft ground under vacuum-surchargepreloading can be conducted.Surcharge or embankment

Vacuum pump

Vertical drain Geomembrane

Fig. 1. Cross-section of subsoil improved by vertical drains under

vacuum-surcharge preloading.

2D cell with smear zone 2D cell without smear zone

2B 2B

Axisymmetric cell

De

Fig. 2. Axisymmetric unit cell and its equivalent plane strain unit cells.

Nomenclature

a width of the prefabricated vertical drainb thickness of the prefabricated vertical drainB half-width of the plane strain unit cell

bs half-width of the smear zone of the plane strainunit cellbw half-width of the drain-wallCha coefficient of consolidation for horizontal drain-

age in axisymmetric caseChp coefficient of consolidation for horizontal drain-

age in plane strain caseds smear zone diameterdw equivalent drain diameterk1 maintaining factor of vacuum pressure

(0 6 k1 6 1)kha horizontal permeability coefficient in undis-

turbed zone of the axisymmetric unit cell

khp horizontal permeability coefficient in undis-turbed zone of the plane strain unit cell

ksa horizontal permeability coefficient in smear zoneof the axisymmetric unit cell

ksp horizontal permeability coefficient in smear zoneof the plane strain unit cell

l The length of drainage path of the drain insidethe axisymmetric unit cell

na ratio R/rw of the axisymmetric unit cellp0 vacuum pressure applied to the top of the drain

in axisymmetric case, and to the top of thedrain-wall in plane strain case

qwa discharge capacity of the drain of the axisym-metric unit cell

qwp discharge capacity of the drain-wall of the planestrain unit cell

R radius of the axisymmetric unit cell

rs radius of the smear zone of the axisymmetricunit cellrw radius of the drain of the axisymmetric unit cellsa ratio rs/rw of the axisymmetric unit cellTha dimensionless time factor for horizontal drain-

age in the axisymmetric unit cellThp dimensionless time factor for horizontal drain-

age in the plane strain unit cellu average excess pore water pressure of the unit cellr1 initial overburden pressure due to surcharge

preloadinglza parameter depending on depth z, including well

resistance, smear effect, and the geometry of the

axisymmetric unit celllzp parameter depending on depth z, including well

resistance, smear effect, and the geometry of theplane strain unit cell

Subscriptsa axisymmetricp plane strains smear zoneh horizontalz depth z

656 T.A. Tran, T. Mitachi / Computers and Geotechnics 35 (2008) 655672


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We see that, in their method, the inclusion of theplane strain smear zone in the plane strain cell is deemedto be needed (see Figs. 2 and 4a). However, in our expe-rience, the inclusion of plane strain smear zones in planestrain finite element simulation increases the number ofelements and material parameters. In particular, for afull-scale simulation of a multi-layer subsoil incorporat-

ing a large number of small vertical drain elements, thenumber of elements and the material parameters forthe plane strain smear zones become very large. In addi-tion, for determining the equivalent permeability by theirmethod, the determination of the length of the drainagepath of the drain (l) in each soil layer of that multi-layersubsoil (where the drain is driven through) is required(see Fig. 3a).

In 2006, Chai et al. [5] presented a plane strain numeri-cal modeling of a soft ground improved by vertical drainsunder vacuum-surcharge preloading. In their numericalmodeling, they used a conversion method from the axisym-metric cell to the plane strain cell proposed by Chai et al.[4].

We have observed that, this conversion method is devel-oped under the condition of conventional surcharge load-ing, and the boundary condition of excess pore pressureof the vertical drain defined in this method is that the excesspore pressure at the top of the drain is equal to zero, only.Therefore, this method might not apply well in the case ofvacuum-surcharging.

In reality, the subsoil usually has many layers, and, insome cases, there is a sandy silt layer having high perme-ability just below an upper clay layer having low perme-ability. In this case, the vacuum pressure propagates from

the vertical drain to the surrounding soil in the sandy silt

layer will be much faster than the propagation of the vac-uum pressure in the surrounding soil of the upper claylayer. This means that the vacuum pressure in the sandy siltwill reach the maximum value much sooner than that in theupper clay layer.

However, in Chai et al. [4] method, their plane strainunit cell has only an equivalent vertical permeability and

has no drain-wall. Therefore, even though the lower soillayer has a much higher permeability than that of the upperclay layer, and if the vacuum pressure is applied to the topof the cell, then the vacuum pressure would only able topropagate gradually from the upper layer to the lower layerin their plane strain cell. This means that, in their planestrain cell, there is no way to let the vacuum pressure inthe lower clay layer reach the maximum value sooner thanthat in the upper clay layer.

It is observed that there are various ways to develop aconversion method. In this paper, by modifying Indraratnaet al. [12] method (developed for single layer cases undervacuum-surcharge preloading), a new conversion methodhas been proposed, which is different to those of Hirdet al. [8] and Indraratna and Redana [9,10].

2. The proposed conversion method under vacuum-surcharge

preloading condition

To find the conversion expression of permeability fromthe axisymmetric unit cell to the equivalent plane strainunit cell under vacuum-surcharge preloading condition,we conducted the mathematical formulation as follows.

Firstly, based on the analytical model of Indraratnaet al. [12] for the axisymmetric unit cell, we used analytical

mathematics to find a function of development of excess

(a) Analytical model of the axisymmetric unit cell

R

- k1p

0

C

undisturbedzone

undisturbedzone

smearzone

s

mearzone

0- p

-p [1 - (1 - k )z /l]0 1

vacuum pressureAssumed distribution of

impermeable

1

(b) Analytical model of the plane strain unit cell

vacuum pressureAssumed distribution of

equivalentzone

- k1p

0

0- p

-p [1 - (1 - k )z /l]0 1

eq

uivalentzone

rw

sr

B

bw

impermeable

Thelengthofthedrainagepathofthedrain(l)

impermeable

1

impermeable

impermeable

impermeable

Thelengthofthedrainagepatho

fthedrain(l)

L CL

Fig. 3. Analytical models of the axisymmetric and plane strain cells.

T.A. Tran, T. Mitachi/ Computers and Geotechnics 35 (2008) 655672 657


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pore water pressure with time at any given depth z withinthis cell.

Subsequently, we built another analytical model for theplane strain unit cell excluding plane strain smear zone,and then the function of development of excess pore waterpressure for this cell was developed.

Finally, by equating two obtained functions of excesspore water pressure, we found the conversion expressionof permeability from the axisymmetric unit cell to theequivalent plane strain unit cell under vacuum-surchargepreloading condition.

2.1. Analytical models of the axisymmetric and plane strain

unit cells under vacuum-surcharge preloading

In general, these two models are depicted in Fig. 3. Inthese figures, r1 denotes the surcharge; p0 is the vacuumpressure applied to the top of the drain as well as the

drain-wall; k1 is the maintaining factor of vacuum pressure(0 6 k1 6 1); z the depth; l the length of the drainagepath of the drain under the condition that the bottom ofthe drain is undrained, and the excess PWP at the top ofthe drain is equal to p0; therefore, l is also equal tothe length of the unit cell in this ideal case (one homoge-neous soil layer); rw the equivalent drain radius; rs thesmear zone radius; R the equivalent radius of the influencezone; bw the half-width of the drain-wall; B is the half-width of the plane strain unit cell.

Main assumptions of these models are:

1. The soil within the cell is fully saturated andhomogeneous.

2. The permeability of the soil is assumed to be constantduring consolidation.

3. The vertical flow within the soil of the relatively longunit cell is insignificant, i.e. it is assumed that only radialflow occurs within the soil.

4. Equal strain hypothesis of Kjellman [14] is followed, i.e.the horizontal sections of the axisymmetric and planestrain unit cells remain horizontal during the consolida-tion process.

5. The displacement at outer boundaries of the verticaldrain and the cell are fixed in horizontal direction, i.e.only vertical displacement is allowed at theseboundaries.

6. Darcys law is considered to be valid, and the solutionsare based on the Darcys law.

7. The change in volume corresponds to the change in voidratio, and coefficient of volume compressibility, mv, isconstant during consolidation process.

8. Indraratna et al. [12] assumption about the loss of vac-uum pressure along the vertical drain is employed, i.e.the vacuum loss is a linear increase with depth (seeFig. 3), in which p0 is the vacuum pressure at thetop of the drain, and k1p0 is the corresponding value

at the bottom of the drain.

2.1.1. Analytical solution for the axisymmetric unit cell

(modified by the present authors after Indraratna et al.

[12])

It is observed that Indraratna et al.s [12] function of theexcess pore water pressure (PWP) is an average functionfor the whole axisymmetric cell at a given time. Therefore,

we modified it to become the following function, which canshow the average value of the excess PWP throughout ahorizontal cross-section of the cell at any given depth zfor a given time t

u r1 p0 1 1 k1z

l

h i exp

8Thalza

p0 1 1 k1z

l

h i1

The detailed analytical formulation of Eq. (1) is given inAppendix A.

In Eq. (1)

lza lnna

sa

kha

ksalnsa

3

4 pz2l z

kha

qwa2

in which

na R

rw; sa

rs

rw

where kha and ksa are horizontal permeability coefficients ofthe axisymmetric unit cell in undisturbed zone and in smearzone, respectively; qwa the discharge capacity of the drain;Tha is the dimensionless time factor for horizontal drainage

and

Tha Chat

4R2

kha

mvcw

t

4R23

where Cha is the coefficient of consolidation for horizontaldrainage; mv the coefficient of volume compressibility forone-dimensional compression; t the time; cw is the unitweight of water.

2.1.2. The proposed analytical solution for the plane strain

unit cell excluding the plane strain smear zoneIn 1992, Hird et al. [8] introduced a plane strain unit cell

excluding plane strain smear zone under conventionalembankment loading condition. By adapting Hird et al.s[8] unit cell for the case subjected to embankment com-bined with vacuum preloading, we developed the followingsolution for the average excess PWP within the adaptedunit cell

u r1 p0 1 1 k1z

l

h i exp

8Thplzp

p0 1 1 k1z

l

h i4

The detailed analytical formulation of Eq. (4) is given in

Appendix B.

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In Eq. (4)

lzp 2

3

2khpBqwp

2lz z2 5

Thp Chpt

4B2

t

4B2khp

mvcw6

in which khp is horizontal permeability coefficient in theequivalent zone of the plane strain cell; qwp is dischargecapacity of the drain-wall; Thp and Chp are dimensionlesstime factor and coefficient of consolidation for horizontaldrainage, respectively.

2.2. The proposed conversion expression of permeability for

the proposed plane strain unit cell

By adapting the matching procedure of Hird et al. [8]under conventional surcharge preloading condition forthe proposed plane strain cell under vacuum-surcharge pre-

loading condition, the following steps are conducted.Equating Eqs. (1) and (4), the following equation can be

obtained

Tha

lza

Thp

lzp7

Then, substituting Eqs. (3) and (6) into Eq. (7), and aftersome rearrangements, we can obtain

kha

R2lza

khp

B2lzp8

Subsequently, substituting the expressions for lza and lzp

into Eq. (8), and the terms rearranged to give

2

3khaB

2 lnna

sa

3

4

kha

ksalnsa

khpR

2

khpkhapR

2

qwa

khpkha2B

qwp

2lz z2 9

If setting

khpkhapR2

qwa

khpkha2B

qwp 0 10

then, the effect of well resistance is matched independently,

as follows

qwp 2B

pR2qwa 11

And, by using Eq. (10) as a condition for Eq. (9), the con-version expression of permeability for the equivalent planestrain unit cell can be obtained

khp 2B2

3R2kha

ln nasa

34

khaksa

lnsa12

In conclusion, our proposed method to convert from theaxisymmetric unit cell to the plane strain unit cell undervacuum-surcharge preloading condition includes the com-bined use of both Eqs. (11) and (12).

2.3. The way to use our proposed conversion method

By inputting R, B, na, sa, kha, and the ratio kha/ksa intoEq. (12), we can obtain the horizontal permeability coeffi-

cient for the equivalent zone between two vertical drainsin the full-scale plane strain simulation.

In the next step, usually, the information on dischargecapacity, qwa, of the vertical drain is available; therefore,by inputting parameters qwa, B, and R into Eq. (11), thevalue of qwp for the drain-wall of the plane strain cell canbe obtained.

Finally, inputting this qwp into the following equation,we can obtain kwp for the drain-walls in the full-scale planestrain simulation

kwp qwp

2bw13

In order to make clear the difference between our conver-sion expression of permeability and that of Indraratnaet al. [12], we made the Table 1 as follows.

In Table 1, the meaning of symbols is

na R

rw; sa

rs

rw; a

2

3

np sp3

n2pnp 1;

b 2sp 1

n2pnp 1npnp sp 1

1

3s2p sp 1

!

h 4khp

3Bqwp

1 1

np l2; np

B

bw

; sp bs

bw

Table 1Comparison of the conversion expressions of permeability between Indraratna et al. [12] method and the proposed method under vacuum-surchargepreloading condition

The conversion expression of permeabilityproposed by Indraratna et al. [12]

The conversion expression of permeabilityproposed by the present authors

khp

kha

a khp

kspb h

h iln na

sa

kha

ksa

lnsa 34 p

2kha3qwa

l2h i khp 2B

2

3R2kha

ln nasa

34

khaksa

lnsa12

and

qwp 2B

pR2qwa 11



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khp and ksp are horizontal permeability coefficients in the

undisturbed and smear zones of Indraratna et al. [12] planestrain unit cell, respectively; in our proposed method, khp ishorizontal permeability coefficient in the equivalent zone ofour plane strain unit cell; bs is half-width of the plane strainsmear zone in the plane strain unit cell of Indraratna et al.[12] method (see Fig. 4). The meaning of other symbols waspreviously mentioned and is shown in Figs. 4 and A1 inAppendix A.

3. Verification of the proposed method via finite element

method

To verify the proposed method, we have conducted FEanalyses for two cases under vacuum-surcharge preloading

condition: Case 1 is a homogeneous clay layer that has the

thickness being 10 m, and Case 2 is the case having two claylayers, in which the upper clay layer and the lower clay layerhave the thickness being 4.95 and 5.05 m, respectively.

In general, the illustration of these two cases is shownschematically in Fig. 5. In Fig. 5a, k1ha and k1sa are the hor-izontal permeability in the undisturbed and smear zones ofthe axisymmetric cell, respectively. k1va and k1vsa are thevertical permeability in the undisturbed and smear zones,respectively.

In Fig. 5b, the horizontal and vertical permeability ofthe upper clay layer is the same as those of the homoge-neous clay layer in Fig. 5a, but the horizontal and vertical

permeability of the lower clay layer is five times higher thanthose of the upper clay layer.

a b

dy = 1

Equivalent zone

dz

B

dz

wb

qwp

Plane-strain undisturbed zone

dy = 1

wpq

sb

B

wb

Plane-strain smear zone

Drain wall Drain wall

Fig. 4. (a) A horizontal cross-sectional slice of Indraratna et al. [12] plane strain unit cell, and (b) that of our proposed plane strain unit cell.

The case of one clay layer

ThicknessH=

10m

One homogeneousclay layer

C

H

=4.9

5m

Lower clay layer

The case of two clay layers

H

=5.0

5m

k1ha

= k1va 1vsa

= k = 4.30 E-4 (m/day)

1sak

1ha= (1/5)k

k1sa= (1/5)k1ha

= k= kk2vsa2va 2ha

2ha= (1/5)k2sak

= 5 (k )1ha

Upper clay layer

a b

EmbankmentEmbankment

k = k = k = 4.30 E-4 (m/day)1ha1vsa1va

L CL

Fig. 5. Schematic description of (a) the case of one homogeneous clay layer and of (b) the case of two clay layers, together with the permeability coefficient

of each soil layer.

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3.1. Detailed description of the case of one clay layer

Regarding the equivalent radius of the band-shapeddrain, rw, FE analyses performed by Rixner et al. [17]and supported by Hansbo [7] indicated that the equivalentdiameter of the band-shaped drain for use in practice can

be determined by

dw a b

214

where a and b are the width and the thickness of the rect-angular cross-section of the band-shaped drain,respectively.

Commonly, the band-shaped drain or PVD has dimen-sions being 10 cm 0.4 cm. Using Eq. (14), we can obtaindw ffi 5 cm and therefore rw ffi 2.5 cm or 0.025 m.

Concerning the smear zone radius, rs, according to Jam-iolkowski and Lancellotta [13], the diameter of the smearzone, ds, can be in the range of 2.5dm to 3dm, where dm is

the equivalent diameter of the mandrel used for drivingPVD into the soft ground. Referring to the practice of vac-uum-surcharging in Japan, the mandrel usually has anequivalent diameter dm = 12 cm, and, in this study, wechose ds = 3dm. This leads to the smear zone diameter usedin this study is 36 cm or 0.36 m; therefore, rs = 0.18 m.

PVDs are commonly driven on a square grid; therefore,the equivalent radius of the influence zone, R, needs to becalculated. Logically, it can be defined to be the radius of acircle having the same area as that of a square. Therefore,the following expression is obtained

R 1:

128S

2

15

where S is the spacing between two vertical drains.We observed that, in Japan, S is usually 0.8 m. Hence,

by using Eq. (15), the equivalent radius R was determinedto be 0.45 m for FE analyses in this paper.

Regarding Bfor FE analyses in this paper, we made twoplane strain unit cells, one has B= 0.45 m and the otherone has B= 0.75 m, in which their permeability was con-verted by our conversion method from the axisymmetricunit cell having R = 0.45 m. For convenience, bw of thesetwo plane strain cells was chosen equal to rw (i.e.

0.025 m) of the axisymmetric unit cell.Concerning well resistance of vertical drains, we have

tested our proposed conversion method with various valuesof well resistance that are selected on the basis of Mesri andLo [16] discharge capacity factor as follows

Fd qwa

khal2

pkwa

kha

rw

l

216

where kha is the horizontal permeability coefficient inundisturbed zone of the axisymmetric unit cell; qwa the dis-charge capacity of the drain; l the drainage length of thecell; rw the drain radius; kwa is the equivalent vertical per-

meability coefficient of the drain.

Mesri and Lo [16] reported that well resistance is consid-ered to be insignificant ifFd is larger than 5. Therefore, wehave tested our conversion method for two values of Fd,which are 0.1 (i.e. very high well resistance) and 20 (i.e.no well resistance), respectively.

3.1.1. Boundary conditions and computed casesBecause one of the basic assumptions of our analyticalsolutions is based on equal strain hypothesis, therefore,we have conducted FE analyses as follows:

(1) All the drain, the smear and undisturbed zones of theaxisymmetric cells, and both the drain-wall and theequivalent zone of the plane strain cells convertedby our method were simulated by linear elastic mod-els having the same elastic modulus (E= 1000 kN/m2) and zero Poissons ratio.

(2) In the case of the axisymmetric cell, to avoid horizon-tal displacement from the smear zone to the drain or

vice versa, the nodes on the boundary between thedrain and the smear zone are allowed to move inthe vertical direction only. Similarly, the nodes onthe boundary between the drain-wall and the equiva-lent zone of the plane strain cell are also allowed tomove in the vertical direction only.

(3) An undrained rigid plate was put on the surface of eachof these axisymmetric and plane strain cells to ensurethe uniform settlement at the surface of the cells (seeFig. 7). In addition, in accordance with the boundarycondition of the analytical solutions, the vacuum pres-sureis applied to the top of the drain, not tothe surface

of the cell; besides, all the vertical permeability of boththe axisymmetric and plane strain cells are set equal tozero. This case, we named it VTD-ES (vacuum at thetop of the drain with equal strain).

In addition, we carried out other FE analyses to checkthe applicability of our method in the case of free-strainis allowed at the surface of the cell; in this case, theundrained rigid plate in Fig. 7 is removed from the surfaceof the cells, and the vacuum pressure is applied to both thetop of the drain and the surface of the cell. We named thiscase VS-FS (vacuum pressure applied to the surface withfree strain).

In 1986, Rixner et al. [17] reported that, of clay soil withno or slightly developed macrofabric, essentially homoge-neous deposits, the ratio of the horizontal permeability tothe vertical permeability, kh/kv, can be in the range of 11.5. Based on this report, for the case VS-FS, the verticalpermeability of the axisymmetric and plane strain cells ischosen as follows:

(1) We assume the ratio, kha/kva, equal to 1 for the undis-turbed zone of the axisymmetric unit cell.

(2) For the smear zone of the axisymmetric unit cell, weassume its vertical permeability, kvsa, is equal to the

vertical permeability of the undisturbed zone, kva.



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(3) For the plane strain cells converted by our method,the vertical permeability of the equivalent zone, kvp,is assumed to be equal to kva.

According to Kobayashi et al. [15], the horizontal per-meability of the smear zone of the clay ksa can be decreased

to 1/5 of that of the undisturbed zone kha. Therefore, forthe axisymmetric cell, we assumed the ratio ksa/kha, to beequal to 1/5.

In summary, all computed cases are illustrated in Fig. 6,and all input parameters used for the case of one clay layerare tabulated in Table 2. And, the boundary conditions andmeshes of the cells are shown in Fig. 7.

Note that, in engineering practice, the value of qwa isavailable and then qwp can be calculated by using Eq.(11). In this paper, for the purpose of investigating theeffect of well resistance on each of cases computed byIndraratna et al. [12] method and by our method, thedischarge capacity factor Fd was firstly assumed as

shown in Table 2, and then qwa was calculated basedon Eq. (16).

Regarding the FEM program used, the Sage Crisp pro-gram developed by the CRISP Consortium Ltd. and SAGEEngineering Ltd. [18] on the basis of consolidation theoryof Biot [1] was employed; we used linear strain quadrilat-eral elements that incorporate quadratic displacementnodes together with linearly interpolated pore pressurenodes in this program (see Fig. 8).

3.2. Detailed description of the case of two clay layers

We observed that the conversion method proposed byIndraratna et al. [12] showed very good matching resultsfor the case of one clay layer, in their paper. Therefore,a check on the applicability of both our proposed

method and Indraratna et al. [12] method for the caseof two clay layers would be worthwhile; for this reason,FE analyses for the case of two clay layers were con-ducted in this study.

In this two-clay-layer case, the geometric parameters ofthe axisymmetric unit cell, R, rs, rw, were chosen the sameas those of the axisymmetric unit cell in the case of one claylayer listed in Table 2. In general, the boundary conditionsand the mesh of the cells are shown in Fig. 9.

With regard to the soil properties and material param-eters of the axisymmetric unit cell in this two-clay-layercase.

For the vertical drain, all material parameters were

assumed to be the same as those in the case of one homo-geneous clay layer.

For the upper and lower clay layers, all soil propertiesexcept permeability were assumed to be the same as thoseof the case of one homogeneous clay layer. For the perme-ability of the smear and undisturbed zones in each layer,the assumption is illustrated in Fig. 5b.

Note that, for the plane strain cell converted by Ind-raratna et al. [12] method, we assumed the ratio of thesmear zone permeability to the undisturbed zone perme-ability, ksp/khp, equal to 1/5. On the other hand, the planestrain unit cell of our proposed method has no plane strain

smear zone; therefore, such a kind of ratio is not requiredin our plane strain unit cell.

Regarding the assumption of vertical permeability, ofboth the two clay layers, the vertical permeability in thesmear and undisturbed zones of both the axisymmetric celland Indraratna et al. [12] plane strain cell is assumed to bezero in the case of VTD-ES. Similarly, the vertical perme-ability in the equivalent zone of our proposed plane straincell is also assumed to be zero.

In the case of VS-FS, the vertical permeability of theplane strain cells was assumed as follows:

(1) For the plane strain cells converted by Indraratnaet al. [12] method, for the upper and lower clay layers,both the vertical permeability of the smear zone, kvsp,and of the undisturbed zone, kvp, are equal to the ver-tical permeability of the undisturbed zone, kva, of thecorresponding layer in the axisymmetric cell. Thismeans that kvsp = kvp = kva for each of the upperand lower clay layers.

(2) For the plane strain cells converted by our proposedmethod, in each of the two clay layers, the verticalpermeability of the equivalent zone, kvp, is the sameas that of the undisturbed zone in the axisymmetriccell. It should be noted here that kvsp is not needed

in our proposed plane strain cell.

The case of two clay layers

The case of one homogeneous clay layer

k = 11

k = 11

k = 0.51

1k = 0.5

o0.5p

op

opp

op

o

op

op

VTD-ES VS-FS

0.5po

oo

o0.5p 0.5p

o

ppoo

op

op

pp

VS-FSVTD-ES

VS-FSVTD-ESVS-FSVTD-ES

Fig. 6. Illustration of computed cases.



9/18

Table 2Input parameters of the axisymmetric and plane strain unit cells for the case of one clay layer

Unit cell parameter Symbol Computed cases

VTD-ES VTD-ES VS-FS VS-FS

The axisymmetric unit cell

Radius of the axisymmetric cell (m) R 0.45 0.45 0.45 0.45

Smear zone radius (m) rs 0.180 0.180 0.180 0.180Drain radius (m) rw 0.025 0.025 0.025 0.025Discharge factor Fd 20 0.1 20 0.1

Elastic modulus (kN/m2) and Poissonsratio of the drain

Eand m E= 1000 and m = 0

Horizontal permeability of the undisturbedzone (m/day)

kha 4.30E04 4.30E04 4.30E04 4.30E04

Horizontal permeability of the smear zone(m/day)

kha (1/5)kha (1/5)kha (1/5)kha (1/5)kha

Vertical permeability of the undisturbed andsmear zones (m/day)

kva andkvsa

0.0 0.0 4.30E04 4.30E04

Elastic modulus (kN/m2) and Poissonsratio of the undisturbed and smear zones


Computed cases

VTD-ES VTD-ES VTD-ES VTD-ES VS-FS VS-FS VS-FS VS-FS

The plane strain unit cell converted by our proposed method

Half-width of the plane strain cell (m) B 0.45 0.75 0.45 0.75 0.45 0.75 0.45 0.75Half-width of the drain-wall (m) bw 0.025 0.025 0.025 0.025 0.025 0.025 0.025 0.025Elastic modulus (kN/m2) and Poissons

ratio of the drain-wallEand m E= 1000 and m = 0

Horizontal permeability of the equivalentzone (m/day)

k#1hp 2.86E05 7.93E05 2.86E05 7.93E05 2.86E05 7.93E05 2.86E05 7.93E05

Vertical permeability of the equivalent zone(m/day)

kvp 0.0 0.0 0.0 0.0 4.30E04 4.30E04 4.30E04 4.30E04

Elastic modulus (kN/m2) and Poissonsratio of the equivalent zone


Note: The superscript #1 in the table means that this value was calculated based on Eq. (12).

(a) Axisymmetric unit cell, R = 0.45 m

w

Impermeable boundary

Vertical r oller boundaryVertical rol ler boundary

Centreline of the drain


Fixed, impermeable boundary

Horizontal roller boundary

Half drain, r = 2.5 cm

Smear zone, 15.5 cm


Undisturbed zone, 27 cm


Periphery of the cell

50 kPa

C

Undrained rigid plate

Top of the cell

Periphery of the drain

Vertical r oller boundary

(b) Plane strain unit cell, B = 0.45 m

Equivalent zone, 42.5 cm

Half drain wall, b = 2.5 cm



Impermeable boundaryw

Negative 50 kPa excess pwpTop of the drain

Top of the cell


50 kPa


Impermeable boundaryVertical rol ler boundary



Impermeable boundaryw

Fixed, impermeable boundaryEquivalent zone, 72.5 cm

Top of the cell


CL50 kPa


Impermeable boundaryVertical rol ler boundary

Vertical ro ller boundary





H=10m




Vertical rol ler boundary


Top of the drainNegative 50 kPa excess pwp


(c) Plane strain unit cell, B = 0.75 m

L CL

Fig. 7. Boundary conditions and meshes of the axisymmetric and plane strain cells used in VTD-ES cases of one homogeneous clay layer.



10/18

In summary, all computed cases for this two-clay-layercase are illustrated in Fig. 6, and all input parameters usedfor these plane strain cells are shown in Tables 3 and 4.

In this two-clay-layer case, when Indraratna et al. [12]conversion expression was used for the lower clay layer,then a question arose as to whether we should choose thelength of the drainage path of the drain (l) of the axisym-metric unit cell in the lower clay layer equal to the totalthickness of both two clay layers (l= 10 m) or equal tothe thickness of the lower clay layer only (l= 5.05 m). Allpermeability for the lower clay layer of plane strain cellsconverted by Indraratna et al. [12] conversion expression,in accordance with both l= 10 and 5.05 m, are listed inTable 4.

In Table 4, in the case that the plane strain cell hasB= 0.75 m together with qwp = 0.0043 m

3/day (i.e. corre-sponding to Fd = 0.1 of the axisymmetric cell), the con-verted permeability of the lower clay layer based onIndraratna et al. [12] method becomes negative, eventhough using either l= 10 or 5.05 m. Therefore, we could

not model this case by Indraratna et al. [12] method. Forthis reason, the comparison between our method andIndraratna et al. [12] method was not conducted for thiscase.

Also in Table 4, when the plane strain cell hasB= 0.45 m together with qwp = 0.0043 m

3/day, the con-verted permeability of the lower clay layer based on Ind-raratna et al. [12] method also becomes negative if usingl= 10 m (i.e. equal to the total thickness of both two claylayers). Hence, for the lower clay layer, we did not choosel= 10 m, but chose l= 5.05 m (i.e. equal to the thickness ofthis layer) to input into the conversion expression of Ind-raratna et al. [12].

4. Results and discussion

4.1. In the case of one homogeneous clay layer

FE results of the degree of consolidation of the axisym-metric cell, and of our proposed plane strain cell are shownin Figs. 10 and 11, in which all curves of degree of consol-idation were calculated based on the surface settlement ofthe soil layer.


Vertical roller boundaryImpermeable boundary


Vertical roller boundary


w





Equivalent zone, 42.5 cm

(b) Plane strain unit cell, B = 0.45 m

wHalf drain, r = 2.5 cmImpermeable boundary




Smear zone, 15.5 cm

Undisturbed zone, 27 cm

(a) Axisymmetric unit cell, R = 0.45 m

H

=4.9

5m

Negative 50 kPa excess pwpTop of the drain

50 kPa

Vertical roller boundary



Vertical roller boundaryImpermeable boundary


Top of the cell


Impermeable boundaryVertical roller boundary

H=5.0

5m

Upperclaylayer

Lowerclaylayer

Top of the cell


C50 kPa


Impermeable boundaryVertical roller boundary

LCL

Fig. 9. Boundary conditions and meshes of the axisymmetric and plane strain cells used in VTD-ES cases of two clay layers.

Integration point

Pore pressure unknown

Displacement unknown

Fig. 8. The element type used for FE analyses.



11/18

The proposed conversion method was validated in thecase of VTD-ES with k1 = 1. As shown in Fig. 10a, itcan be seen that a good agreement in the degree ofconsolidation, between the proposed plane strain cell andthe axisymmetric cell, was obtained not only under no-

well-resistance condition, but also under high wellresistance condition. After that, the proposed methodwas also examined in the case of VS-FS with k1 = 1; asshown in Fig. 10b, the same good agreement as that inFig. 10a was obtained under both no-well resistance andhigh well resistance conditions.

Further, we tested the proposed method in the case ofVTD-ES with k1 = 0.5. The results in Fig. 11a indicatedthat, under both no well resistance and high well resistanceconditions, the proposed method produced good matchingresults. Finally, we tested the proposed method in the caseof VS-FS with k1 = 0.5. As can be seen in Fig. 11b, thesame good matching results as those in the case of VTD-

ES (k1 = 0.5) were also obtained.

4.2. In the case of two clay layers

The FE results of the degree of consolidation of theaxisymmetric cell, of the Indraratna et al. [12] planestrain cell, and of our proposed plane strain cell are pre-sented in Fig. 12, in which all curves of degree of consol-idation were calculated based on the surface settlementof soil layers.

Besides, the difference in the degree of consolidation(Ua Up) between the axisymmetric unit cell (Ua) and

the plane train unit cell (Up) of Indraratna et al. [12]

method, and the plane strain cell (Up) of our proposedmethod is illustrated in Fig. 13.

As shown in Fig. 12a and b, the results of degree of con-solidation revealed that both methods are very good underno well resistance condition (Fd = 20).

Fig. 13a shows that, under the condition ofFd = 20, themaximum difference in the degree of consolidation(Ua Up), of the whole two clay layers is 2% for Ind-raratna et al. [12] method, and 4% for the proposedmethod. This means that our method produced matchingresults which are almost as good as those of Indraratnaet al. [12] method (under condition of Fd = 20). Besides,also under condition of Fd = 20, Fig. 13b showed thatthe maximum difference in the degree of consolidation ofthe lower clay layer is 1% for Indraratna et al. [12] method,and 2.5% for the proposed method.

Under high well resistance condition, Figs. 12a and13a revealed that, of the whole two clay layers, the max-

imum difference (Ua Up) of Indraratna et al. [12]method is 9%, which is considered to be fairly high,whereas the maximum difference of the proposed methodis 2%. Figs. 12b and 13b show that, of the lower claylayer, the maximum difference (Ua Up) under high wellresistance condition of Indraratna et al. [12] method evenreach 14%, whereas the difference of the proposedmethod is less than 1%.

Further, we examined both methods in the case of VS-FS with k1 = 0.5 as shown in Fig. 12c, d and 13c, d. Ascan be seen in these figures, almost the same results as thosein the case of VTD-ES (k1 = 0.5) are obtained. However, in

this VS-FS case, the maximum difference in the degree of

Table 3Input parameters of the plane strain unit cells converted by our method for the case of two clay layers

Parameters of the proposed plane strain unitcell

Symbol Computed cases


Half-width of the plane strain cell (m) B 0.45 0.75 0.45 0.75 0.45 0.75 0.45 0.75Half-width of the drain-wall (m) bw 0.025

Discharge capacity of the drain-wall (m

3

/day) qwp 1.222

1

1.222

1

0.0061

2

0.0061

2

1.222

1

1.222

1

0.0061

2

0.0061

2

Elastic modulus (kN/m2) and Poissonsratio of the drain-wall


The upper clay layer


k#1hp 2.86E05 7.93E05 2.86E05 7.93E05 2.86E05 7.93E05 2.86E05 7.93E05

Vertical permeability of the equivalent zone(m/day)

kvp 0.0 0.0 0.0 0.0 4.30E04 4.30E04 4.30E04 4.30E04



The lower clay layer


k#1hp 1.43E04 3.97E04 1.43E04 3.97E04 1.43E04 3.97E04 1.43E04 3.97E04

Vertical permeability of the equivalent zone(m/day) kvp 0.0 0.0 0.0 0.0 2.15E03 2.15E03 2.15E03 2.15E03



Note: The superscript #1 means that this value was calculated based on Eq. (12). The superscript 1 means that this value was calculated based on Eq.(11) and is corresponding to Fd = 20. The superscript 2 means that this value was calculated based on Eq. (11) and is corresponding to Fd = 0.1.



12/18

Table 4Input parameters of the plane strain unit cells converted by Indraratna et al. [12] method for the case of two clay layers

Parameters of the plane strain unitcell of Indraratna et al. [12]

Symbol Computed cases


Half-width of plane strain cell (m) B 0.45 0.75 0.45 0.75 0.45 0.75 0.45 0.75Half-width of drain-wall (m) bw 0.025

Discharge capacity of the drain-wall(m3/day) qwp 0.864

1

0.864

1

0.0043

2

0.0043

2

0.864

1

0.864

1

0.0043

2

0.0043

2

Elastic modulus (kN/m2) andPoissons ratio of the drain-wall


The upper clay layer; we chose l = 4.95 m (i.e. equal to the thickness of this layer)Horizontal permeability of the

undisturbed zone (m/day)k#2hp 1.01E04 2.27E04 2.27E04 6.55E04 1.01E04 2.27E04 2.27E04 6.55E04

Horizontal permeability of the smearzone (m/day)

ksp (1/5)khp

Vertical permeability of theundisturbed and smear zones(m/day)

kvp andkvsp

0.0 0.0 0.0 0.0 4.30E04 4.30E04 4.30E04 4.30E04

Elastic modulus (kN/m2) andPoissons ratio of the undisturbedand smear zones


The lower clay layer; if choosing l = 5.05 m (i.e. equal to the thickness of this layer)Horizontal permeability of the



ksp (1/5)khp

Vertical permeability of theundisturbed and smear zones(m/day)

kvp and

kvsp

0.0 0.0 0.0 0.0 2.15E03 2.15E03 2.15E03 2.15E03

Elastic modulus (kN/m2) andPoissons ratio of the undisturbedand smear zones


The lower clay layer; if choosing l = 10 m (i.e. equal to the total thickness of the two clay layers)Horizontal permeability of the



ksp (1/5)khp

Note: The superscript #2 means that this value was calculated based on Eq. (*) in Table 1. The superscript 1 means that this value was assumed to beequal to qwa, which is corresponding to Fd = 20. The superscript 2 means that this value was assumed to be equal to qwa, but, in this case, qwa iscorresponding to Fd = 0.1.

(b) VS-FS with k1=1

0

20

40

60

80

100

0.1 1 10 100

Time factor Th

DegreeofconsolidationUh

(%)

(a) VTD-ES with k1=1

0

20

40

60

80

100

0.1 1 10 100

Time factor Th


(%)

B0.45: Proposed

R0.45: Axisymmetric

B0.75: Proposed

B0.45: Proposed

R0.45: Axisymmetric

B0.75: Proposed

R0.45: Axisymmetric

B0.45: Proposed

B0.75: Proposed

R0.45: Axisymmetric

B0.45: Proposed

B0.75: Proposed

o

op

po

op

p

i

h

i

: = :

i

i

: :

Fig. 10. Comparison of FEM results of the axisymmetric unit cell having R = 0.45 m (R0.45: Axisymmetric) with that of the plane strain unit cell having

B= 0.45 m (B0.45: Proposed) and with that of the plane strain unit cell having B= 0.75 m (B0.75: Proposed); these graphs corresponding to the case of

one clay layer and k1 = 1.



13/18

0.1 1 10 100

(a) VTD-ES with k1=0.5

0.1 1 10 100

Time factor Th

= 0.1>

B0.45: Proposed

R0.45: Axisymmetric

B0.75: Proposed

B0.45: Proposed

R0.45: Axisymmetric

B0.75: Proposed

B0.45: Proposed

B0.75: Proposed

R0.45: Axisymmetric

B0.45:

B0.75:

o

o0.5p

po

0.5po

p

1

f

= 0.1>

.7

:

. 5:

(b) VS-FS with k1=0.51

0

20

40

60

80

100

Time factor Thf

Degreeofconsolidatio

nUh

(%)

0

20

40

60

80

100

Degreeofconsolidatio

nUh

(%)

R0.45: Axisymmetric

Proposed

:

5: Proposed

o

Fig. 11. Comparison of FEM results of the axisymmetric unit cell having R = 0.45 m (R0.45: Axisymmetric) with that of the plane strain unit cell having

B= 0.45 m (B0.45: Proposed) and with that of the plane strain unit cell having B= 0.75 m (B0.75: Proposed); these graphs corresponding to the case ofone clay layer and k1 = 0.5.

0 0

60

80

100

0.1 1 10

Time factor T

Degreeofconso

lidationUh

(%)

o0.5p

o

po

0.5po

of lower clay layer

Degree of consolidation

o0.5p

op p

o

0.5po


of lower clay layer

B0.45:

R0.45:

B0.45: In

B0.45: Proposed

R0.45: Axisymmetric

B0.45: Indraratna

B0.45: Proposed

R0.45: Axisymmetric

B0.45: Indraratna

B0.45: Proposed

R0.45: Axisymmetric

B0.45: Indraratna

B0.45: Proposed

R0.45: Axisymmetric

B0.45: Indraratna

B0.45: Proposed

R0.45: Axisymmetric

B0.45: Indraratna

VTD-ES with k1=0.5; Uh for two clay layers

20

40

100

0.1 1 10 100

Time factor T

0.1 1 10

Time factor T

100

0.1 1 10 100

Time factor T

0

60

80

100

Degreeofconsolidation

Uh

(%)

20

40

0

60

80

100

Degreeofconsolidation

Uh

(%)

20

40

60

80

100


(%)

20

40

(a) VTD-ES with k1=0.5; Uh for lower clay layer(b)

VS- FS with k1=0.5; Uh for lower clay layer(d)VS-FS with k1=0.5; Uh for two clay layers(c)

p

p

of lower clay layer


o0.5p

op p

o

0.5po


of lower clay layer

Proposed

Axisymmetric

draratna

B0.45:

R0.45:

B0.45: In

Proposed

Axisymmetric

draratna

B0.45:

R0.45:

B0.45: In

Proposed

Axisymmetric

draratna

B0.45: Proposed

R0.45: Axisymmetric

B0.45: Indraratna

B0.45: Proposed

R0.45: Axisymmetric

B0.45: Indraratna

B0.45: Proposed

R0.45: Axisymmetric

B0.45: Indraratna

B0.45: Proposed

R0.45: Axisymmetric

B0.45: Indraratna

B0.45: Proposed

R0.45: Axisymmetric

B0.45: Indraratna

Fig. 12. Comparison of FEM results of the axisymmetric unit cell having R = 0.45 m (R0.45: axisymmetric) with that of the plane strain unit cellconverted by Indraratna et al. [12] method (B0.45: Indraratna) and with that of the plane strain unit cell converted by the proposed method (B0.45:

Proposed); these graphs corresponding to the case of the two clay layers and k1 = 0.5.



14/18

consolidation in the lower clay layer of Indraratna et al.[12] method is 12%, i.e. little smaller than the value 14%in the case of VTD-ES, but this value 12% is still be consid-ered to be significant.

5. Conclusions

By modifying Indraratna et al. [12] method, which wasdeveloped for cases of single layer subsoil, a new conver-sion method has been proposed in this study to convertan axisymmetric unit cell to an equivalent plane strain cellunder vacuum-surcharge preloading condition.

In the proposed method, the widths and the permeabil-ity of the smear and undisturbed zones in the axisymmetriccell are converted theoretically into an equivalent perme-ability of the equivalent zone in the proposed plane straincell. And, as the proposed cell has no plane strain smearzone, a full-scale plane strain simulation of soft groundimproved by vertical drains under vacuum-surcharge pre-

loading can be conveniently made.

Besides, in the proposed method, the determination ofthe length of the drainage path of the drain (l) in each soillayer (of a multi-layer subsoil) is not required in determin-ing the equivalent permeability for each soil layer.

The proposed method was validated via analyzing con-solidation of the axisymmetric and plane strain unit cells intwo cases, one is a homogeneous clay layer, and the otherone is a two-clay-layer case. The vertical drains analyzed inthese cases are under two conditions, no well resistance(Fd = 20), and high well resistance (Fd = 0.1). The analyzedresults showed that the proposed method produced verygood agreements in all the cases. In addition, the resultsalso indicated that the proposed method can be used well,not only under equal strain condition, but also under freestrain condition.

Acknowledgements

The financial support from JICA (Japan International

Cooperation Agency) for this study through AUN/

(a) VTD-ES with k1=0.5; (Ua - Up) for two clay layers

0.1 1 10 100

Time factor T

Ua

Up

(%)

(b) VTD-ES with k1=0.5; (Ua - Up) for lower clay layer

Ua

Up

(%)

(c) VS-FS with k1=0.5; (U

a- U

p) for two clay layers

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

14

Ua

Up

(%)

(d) VS-FS with k1=0.5; (Ua - Up) for lower clay layer

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

14

Ua

Up

(%)

Indraratna

Proposed

Indraratna

Proposed

Indraratna

Proposed

Indraratna

Proposed

Indraratna

Proposed

Indraratna

Proposed

Indraratna

Proposed

Indraratna

Proposed

(a) VTD-ES with k1=0.5; (Ua - Up) for two clay layers

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

14

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

10 100

Th

0.1 1 10 100

Time factor T10 100

Th0.1 1 10 100

Time factor T10 100

Th

0.1 1 10 100

Time factor T10 100

Th

(b) VTD-ES with k1=0.5; (Ua - Up) for lower clay layer

Ua

Up

(%)

(c) VS-FS with k1=0.5; (U

a- U

p) for two clay layers

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

14

Ua

Up

(%)

(d) VS-FS with k1=0.5; (Ua - Up) for lower clay layer

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

14

Ua

Up

(%)

Indraratna

Proposed

Indraratna

Proposed

Indraratna

Proposed

Indraratna

Proposed

Indraratna

Proposed

Indraratna

Proposed

Indraratna

Proposed

Indraratna

Proposed

Indraratna

Proposed

Indraratna

Proposed

Indraratna

Proposed

Indraratna

Proposed

Indraratna

Proposed

Indraratna

Proposed

Fig. 13. The difference in the degree of consolidation between the axisymmetric unit cell and the plane strain unit cell converted by Indraratna et al. [12]method, and the plane strain unit cell converted by the proposed method; these graphs corresponding to the case of two clay layers with k1 = 0.5 that isshown in Fig. 12.



15/18

SEED-Net Project (ASEAN University Network/South-east Asia Engineering Education Development NetworkProject) is greatly appreciated. Besides, the authors wouldlike to thank Associate Professor, Dr. Hiroyuki Tanakaof Hokkaido University very much for his various com-ments and suggestions on this study.

Appendix A. Formulation of the analytical solution for the

average excess pore pressure of the axisymmetric unit cell

Consider a horizontal cross-sectional slice with a thick-ness dz from the radius r to the outer radius R of the axi-symmetric unit cell (see Fig. A1a).

The centripetal flow rate (into the drain) in the slice atradius r can be defined by

oq

ot

kh

cw

ou

or2prdz A1

where q, u, cw, and kh are horizontal flow of water in the

soil mass, excess pore water pressure, unit weight of water,and horizontal permeability coefficient of soil, respectively.

The rate of the soil volume change of the slice, from theradius r to the outer radius R, in the vertical direction canbe determined by

oV

ot

oe

otpR2 r2dz A2

where V and e are volume of the soil mass and verticalstrain, respectively.

Assuming that water is incompressible, therefore Eq.(A1) = Eq. (A2).

For undisturbed zone, rs6 r6 R we can obtain

ou

or

cw2

oe

ot

1

kha

R2 r2

r

A3a

where u and kha are excess pore water pressure and hori-zontal permeability coefficient of the soil in the undisturbedzone, respectively.

For smear zone, rw 6 r 6 rs we obtain:

ous

or

cw2

oe

ot

1

ksa

R2 r2

r

A3b

where us and ksa are excess pore water pressure and hori-zontal permeability coefficient of soil in the smear zone,

respectively.Considering a horizontal cross-sectional slice with athickness dz of the circular cylindrical drain with radiusrw (Fig. A1b), the change of vertical flow of water withinthe drain, in the z direction, from the entrance face to theexit face of the slice can be calculated by

dqz qwacw

o2u

oz2dzdt A4

where qwa is the discharge capacity of the drain.The horizontal inflow of water, in the radial direction,

from the outer face of the slice of the circular cylindricaldrain is determined by

dqr 2prwksa

cw

ou

ordzdt for r rw A5

where rw is the equivalent radius of the drain.For continuity of flow, the following equation must be

satisfied

dqz dqr A6

therefore

ou

or

rw

qwa

2prwksa

o2u

oz2

rw

0 for r rw A7

Substituting Eq. (A3b) into Eq. (A7), we obtain

o2uw

oz2

cwpr2w

qwan2a 1 oe

otA8

where na is the ratio R/rw; uw is the excess pore water pres-sure at rw.

rw

sr

R

zq

zzq + dq

dz

rdqdq

r

Smear zone

Undisturbed zoneUndisturbed zone

Undisturbed zone

Smear zone

q q

dz

r

R

rs

wr

Fig. A1. A horizontal cross-sectional slice of the axisymmetric unit cell.



16/18

By using the following boundary conditions:

At z 0 : uw p0.At z l : ouw

oz p0

1k1l

.

The solution of Eq. (A8) can be given by

uw p0 1 1 k1zl

h icwpr

2w

qwan2a 1 oe

otlz

z2

2

A9

After that, integrating Eq. (A3b) in the r direction with theboundary condition that at r = rw, us = uw (i.e. Eq. (A9)).

Therefore, for rw 6 r 6 rs, the following equation isobtained

us p0 1 1 k1z

l

h icw2

oe

ot

1

ksaR2 ln

r

rw

r2 r2w

2 ksa

pr2w

qwan2

a

12lz z2 !A10

or

us pz cw2

oe

ot

1

ksaR2 ln

r

rw

r2 r2w2

ksaqz

!A11

where

pz p0 1 1 k1z

l

h iand qz

pr2w

qwan2a 1

2lz z2

Then, integrating Eq. (A3a) along the r direction with theboundary condition that at r = rs, u = us (i.e. Eq. (A11)).

Hence, for rs 6 r 6 R, we can obtain the followingequation

u pz cw2

oe

ot

1

ksaR2 ln

r

rs

r2 r2s2

kha

ksaR2 ln

rs

rw

r2s r2w

2

khaqz

!A12

Let u be the average excess pore water pressure throughouta horizontal cross-section at depth z and for a given time, t

u

Rrsrw

2pusrdrRRrs

2purdr

p R2 r2w A13

After substituting Eqs. (A11) and (A12) into Eq. (A13),

and integrating it, finally Eq. (A13) becomes

u p0 1 1 k1z

l

h icw2

oe

ot

R2

khaln

na

sa

kha

ksalnsa

3

4 pz2l z

kha

qwa

!A14

We assume that

oe

ot mv

or0

ot mv

ou

otA15

where mv is the coefficient of volume compressibility forone dimensional compression.

Substituting Eq. (A15) into Eq. (A14) then integrating itwith time, t, and introducing the boundary condition thatat t = 0, u r1, then the following expression can beobtained

u r1 p0 1 1 k1z

l

h i exp

8Thalza

p0 1 1 k1z

l

h iA16

where

lza lnna

sa

kha

ksalnsa

3

4 pz2l z

kha

qwaA17

Tha Chat4R2

khamvcw

t4R2

A18

Appendix B. Formulation of the analytical solution for the

average excess pore pressure of the plane strain unit cell

Let us consider a horizontal cross-sectional slice with athickness dz from the vertical cross-section x to the widthB of the plane strain unit cell (see Fig. B1a).

In the slice, the horizontal flow rate into the drain, atcross-section x, can be defined by

B

dz

qzdqxx

dq

wb

q + dqz z

q q

x

B

dz

wb

Equivalent zone Equivalent zone

dy= 1

dy =1

Fig. B1. A horizontal cross-sectional slice of the plane strain cell.



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oq

ot

khp

cw

ou

oxdzdy

khp

cw

ou

oxdz 1

khp

cw

ou

oxdz B1

where q is the horizontal flow of water in the soil mass; khpis the equivalent horizontal permeability coefficient of thesoil in the plane strain cell.

The rate of the soil volume change of the slice in the ver-

tical direction, within the space from width x to the widthB, can be determined by

oV

ot

oe

otB xdz B2

where V, e, and B are volume of the soil mass, verticalstrain, and half-width of the equivalent plane strain cell,respectively.

Assume that water is incompressible, therefore Eq.(B1) = Eq. (B2), then the following equation can beobtained

ou

ox

cw

khp

oe

otB x B3

where u is excess pore water pressure in the equivalent zoneof the plane strain cell; khp is the equivalent horizontal per-meability coefficient of the equivalent zone of the planestrain cell.

The change of vertical water flow within the drain-wall,in the z direction, from the entrance face to the exit face ofthe slice (see Fig. B1b) can be calculated by

dqz qwp

cw

o2u

oz2dzdt B4

where qwp is the discharge capacity of the drain-wall.

The horizontal inflow of water, from the outer face ofthe drain-wall slice, flows into the drain-wall, can be deter-mined by

dqx khp

cw

ou

oxdzdt for x bw B5

where bw is the half-width of the drain-wall.For continuity of flow, the following equation must be

satisfied

dqz 2dqx B6

therefore, it leads to

o2u

oz2

2khpqwp

ou

ox

bw

for x bw B7

Substituting Eq. (B3) into Eq. (B7), we obtain

o2u

oz2

2cwqwp

oe

otB bw at x bw B8

By using the following boundary conditions:

At z = 0 and x = bw: u = p0.At z = l and x bw:

ouoz

p01k1l .

The solution of Eq. (B8) for uw at x = bw can be given by

uw p0 1 1 k1z

l

h i

2cwqwp

B bw oe

otlz

z2

2

B9

Integrating Eq. (B3) with the following boundarycondition.

At x = bw, we have u = uw being Eq. (B9).Therefore, for bw 6 x 6 B, the following equation is

obtained

u pz cwoe

ot

1

2kp2Bx x2

B bwqwp

2lz z2

"

1

2kp2Bbw b

2w

#B10

where

pz p0 1 1 k1z

lh i

Let u be the average excess pore water pressure through-out a horizontal cross-section at a given depth z and for agiven time, t

u

RBbwudx

B bwB11

After substituting Eq. (B10) into Eq. (B11), and integratingit, finally Eq. (B11) becomes

u p0 1 1 k1z

lh i B bw

2

khp

cw2

oe

ot

2

3

2khpB bwqwp

2lz z2

" #B12

Substituting Eq. (A15) (in Appendix A) into Eq. (B12),then integrating it with boundary condition that at t = 0,u r1, finally the following expression can be obtained

u r1 p0 1 1 k1z

l

h i exp

8Thplzp

p0 1 1 k1z

lh iB13

where

lzp 2

3

2khpBqwp

2lz z2 B14

Thp Chpt

4B2

t

4B2khp

mvcwB15

References

[1] Biot MA. General theory of three-dimensional consolidation. J ApplPhys 1941;12:15564.

[2] Barron RA. Consolidation of fine-grained soils by drain wells. ASCETrans 1948;113:71854.

http://-/?-http://-/?-


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[3] Chai JC, Miura N, Sakajo S, Bergado DT. Behavior of vertical drainimproved subsoil under embankment loading. Soil Found1995;35(4):4961.

[4] Chai JC, Shen SL, Miura N, Bergado DT. Simple method ofmodeling PVD improved subsoil. J Geotech Geoenviron Eng, ASCE2001;127(11):96572.

[5] Chai JC, Carter JP, Hayashi S. Vacuum consolidation and itscombination with embankment loading. Can Geotech J 2006;43:98596.

[6] Hansbo S. Consolidation of fine-grained soils by prefabricateddrains. In: Proceedings of 10th international conference onsoil mechanics and foundation engineering, vol. 3; 1981. p.67782.

[7] Hansbo S. Design aspects of vertical drains and lime columninstallations. In: Proceedings of the ninth southeast Asian geotech-nical conference, vol. 2; 1987. p. 812.

[8] Hird CC, Pyrah IC, Russel D. Finite element modeling of verticaldrains beneath embankments on soft ground. Geotechnique1992;42(3):499511.

[9] Indraratna B, Redana IW. Plane strain modeling of smear effectsassociated with vertical drains. J Geotech Geoenviron Eng, ASCE1997;123(5):4748.

[10] Indraratna B, Redana IW. Numerical modeling of vertical drainswith smear and well resistance installed in soft clay. Can Geotech J2000;37:13245.

[11] Indraratna B, Bamunawita C, Khabbaz H. Numerical modeling ofvacuum preloading and field applications. Can Geotech J 2004;41:1098110.

[12] Indraratna B, Rujikiatkamjorn C, Sathananthan I. Analytical andnumerical solutions for a single vertical drain including the effects ofvacuum preloading. Can Geotech J 2005;42:9941014.

[13] Jamiolkowski M, Lancellotta R. Consolidation by vertical drains-uncertainties involved in prediction of settlement rates, Panel discus-sion. In: Proceedings of the 10th international conference on soilmechanics and foundation engineering; 1981.

[14] Kjellman W. Consolidation of fine-grained soils by drain wells. TransASCE 1948;113:74851 [Contribution to the discussion].

[15] Kobayashi M, Minami J, Tsuchida T. Determination method ofhorizontal consolidation coefficient of clay. Tech Rep Res CenterHarbor Eng Transport Ministry 1990;29(2):6383.

[16] Mesri G, Lo DOK. Field performance of prefabricated verticaldrains. Proceedings of the international conference on geotechnicalengineering for coastal development theory and practice on softground Yokohama, vol. 1. Japan: Coastal Development Institute ofTechnology; 1991. p. 2316.

[17] Rixner JJ, Kraemer SR, Smith AD. Prefabricated vertical drains.Engineering Guidelines vol. 1, Report No. FHWA-RD-86/168.Federal Highway Administration, Washington, DC; 1986.

[18] SAGE Engineering Ltd. SAGE CRISP users manual. Bath, UnitedKingdom: SAGE Engineering Ltd.; 1999.


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