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Geotextiles and Geomembranes 27 (2009) 227–231

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Geotextiles and Geomembranes

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Technical note

Equivalent diameter of a prefabricated vertical drain

Hossam Abuel-Naga a,1, Abdelmalek Bouazza b,*

a Department of Civil and Environmental Engineering, University of Auckland, Private Bag 92019, Auckland Mail Centre, Auckland 1142, New Zealandb Department of Civil Engineering, Building 60, Monash University, Melbourne, Victoria 3800, Australia

a r t i c l e i n f o

Article history:Received 2 February 2008Received in revised form21 October 2008Accepted 8 November 2008Available online 18 January 2009

Keywords:ConsolidationPVDNumericalFlow rateWellEquivalent diameter

* Corresponding author. Tel.: þ61 3 9905 4956; faxE-mail addresses: h.naga@auckland.ac.nz (H. A

eng.monash.edu.au (A. Bouazza).1 Tel.: þ64 9 373 7599x89067; fax: þ64 9 373 746

0266-1144/$ – see front matter � 2008 Elsevier Ltd.doi:10.1016/j.geotexmem.2008.11.006

a b s t r a c t

The aim of this study is to evaluate numerically the equivalent diameter of a prefabricated vertical drain(PVD) using an equal flow rate approach. This approach requires that the equivalent circular well to beable to yield similar discharge as a PVD well when subjected to similar pressure-controlled pumpingcondition. Simple equivalent diameter equation was obtained from the numerical study. The proposedequation is discussed in light of the equivalent diameter equations and experimental results currentlyavailable in the literature. The outcomes of this study give more insight in this subject and improve themethod of calculating the equivalent diameter of a PVD well.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

A prefabricated vertical drain (PVD), also referred to as wick orstrip drain, is a composite geosynthetic system consisting ofa polymeric inner core with formed flow path grooves on both sidesand an outer nonwoven geotextile filter jacket. It is commonly usedto accelerate consolidation or to dewater fine-grained soils (Li andRowe, 2001; Bergado et al., 2002; Lorenzo et al., 2004; Shen et al.,2005; Rowe and Li, 2005; Abuel-Naga et al., 2006; Sinha et al.,2007; Rowe and Taechakumthorn, 2008; Chai et al., 2008). Acomprehensive overview about the use and geotechnical applica-tions of PVDs is given elsewhere (Bo et al., 2003; Chu et al., 2004).Recent works have also adopted PVDs for in situ remediation ofcontaminated soils below the groundwater table. The technology isan enhancement of the soil flushing technique (Gabr et al., 1996;Quaranta et al., 2000; Warren et al., 2006; Sharmin et al., 2008). Forthe case of remediation above the groundwater, i.e. in the vadosezone, Collazos et al. (2002, 2003) and Abuel-Naga et al. (2008)investigated the possibility of incorporating prefabricated verticaldrains into soil vapour extraction (SVE) systems to enhance theireffectiveness, to reduce the time required for clean-up, and to lower

: þ61 3 9905 4944.buel-Naga), malek.bouazza@

2.

All rights reserved.

the residual contaminant levels achieved at the end of thetreatment.

The design of prefabricated vertical drains varies according tothe specific application. One of the key parameters usuallyaddressed in the PVD design is the determination of the equivalentdiameter of band-shaped drains with radial drains which dictatesthe size of the inflow surface (the focus of this paper); otherparameters such as discharge capacity of the drain, drain spacing,well resistance, effect of smear disturbance and installationmethods are not addressed in the present paper. PVDs are oblong incross sectional and not round. In order to design a PVD, forgeotechnical or remediation purposes, an equivalent circular draindiameter must be used. There are many methods and proceduresfor determining the equivalent diameter of a PVD with an oblongcross section (Welker et al., 2000). Table 1 shows the mostcommonly used formulations to determine PVD equivalent diam-eter. These equations were developed using different methodolo-gies. The Hansbo (1979) equation is based on the surface area ofa PVD. Long and Covo (1994) used an electric analogue approach todevelop their equation. The Atkinson and Eldred (1981) equationtakes into account throttle that takes place close to the drain.Fellenius and Castonguay (1985) established their equationconsidering the cross-sectional area of a PVD. However, no defini-tive recommendation is available regarding the validity of theseequations.

Welker et al. (2000) and Welker and Herdin (2003) investigatedexperimentally the validity of two different approaches that can be

Table 1PVD equivalent diameter equations

References Equations

Hansbo (1979) deq ¼ 2ðwþ tÞ=P

Long and Covo (1994) deq ¼ 0:5wþ 0:7tAtkinson and Eldred (1981) deq ¼ ðwþ tÞ=2Fellenius and Castonguay (1985) deq ¼ ð4ðwtÞ=PÞ0:5

w is the prefabricated vertical drain (PVD) width and t is the PVD thickness.

re

X

X

Y

Y

re

w/2

t/2

deq/2

a

b

Fig. 1. Typical finite element configuration. (a) PVD well. (b) Equivalent circular cross-section well.

H. Abuel-Naga, A. Bouazza / Geotextiles and Geomembranes 27 (2009) 227–231228

used to evaluate the equivalent diameter of PVD namely; equalequipotentials and equal flow rate approach, respectively. The firstapproach requires that both the oblong well and the equivalentcircular well to generate equal equipotentials behaviour in the flowzone when subjected to similar flow-controlled pumping/injectioncondition. The second approach requires that the equivalentcircular well to be able to yield similar discharge as the oblong wellwhen subjected to similar pressure-controlled pumping condition.Welker et al. (2000) indicated that evaluation of PVD equivalentdiameter should not be based on equal equipotential approachsince the change in the well shape from oblong to circular impactsthe head values to such an extent that the equipotential lines willnever be similar. Welker and Herdin (2003) investigated experi-mentally the PVD equivalent diameter equation based on the equalflow rate approach. However, no firm conclusions were reached onthe equivalency calculation. The objective of this paper is todetermine numerically the equivalent diameter of a PVD-wellbased on the equal flow rate approach and to improve its method ofcalculation.

2. Numerical experiment program

Determining the equivalent diameter of PVD numerically offersthe possibility of eluding the following uncertainties that mightaffect the outcomes of the investigation if a laboratory experi-mental approach is used:

� Homogeneity and repeatability of tested soil samples.� Level of saturation of tested soil samples.� Disturbance in the flow regime due to piezometer installation

around a PVD well.� Difference in well resistance between a PVD and an equivalent

circular section well which might lead to different pressuregradients within the tested soil sample for each well type atequal flow rate condition.� Efficiency of top cover sealing during the extraction/pumping

process.

Moreover, as similar PVDs cross-section dimension can havedifferent well resistance (Quaranta and Gabr, 2000), the wellresistance change can also affect the general applicability of theacquired experimentally relation.

In this study numerical pumping tests were conducted oncylindrical unit cell soil specimen (1.0 m in diameter and height)that includes a central vertical drain. Pressure-controlled conditionwas employed. It involves applying different steady uniform pres-sures along the intake section of the well (PVD/circular crosssection) and determining the flow rate to the well. Considering thevalidity of Darcy’s law and the homogeneity of the tested soil, thetransport behaviour of fully saturated porous media in twodimensions can be expressed by the following governing equation:

v

vx

�k

vPvx

�þ v

vy

�k

vPvy

�¼ 0 (1)

where k is the hydraulic conductivity and P the fluid pressure head.A finite element solver, FlexPDE, with an automatic adaptive mesh

approach was used to solve Eq. (1). Taking the advantage of thesymmetrical behaviour of the problem, only one quarter of the areawas used in the model as shown in Fig. 1. The boundary conditionsof the numerical experiment pumping test conducted are describedas follows:

� Steady uniform pressure head, Pr, at the perimeter of the testedcell.� No flow boundary is applied along the two axis of symmetry (X,

Y).� Steady uniform pressure head, Pw, is applied along the PVD or

equivalent circular cross-section circumference.

The mode of pumping (extraction/injection) can be controlledby the pressure values (Pr and Pw). However, it should bementioned that the mode of pumping should not theoreticallyinfluence the relation between the oblong cross-section well and itsequivalent circular cross-section well. The size of PVD used in thenumerical modelling was varied from t/w¼ 0.0333–0.0875 where tand w are the thickness and width of the PVD, respectively, to covermost PVD sizes available commercially.

3. Test result and discussion

The PVD flow rate (Q) obtained from the numerical test fordifferent PVD sizes under constant soil transport properties(k¼ 5�10�7 m/s) and boundary conditions (Pw� Pr¼ 5.0 m) isplotted in the dimensionless plane Q/(kHS)� (t/w) to generalize theoutput of this study as shown in Fig. 2. The term S is the differencein pressure head (Pw� Pr), and H is the thickness of the soil layer(H¼ 1.0 m for the two-dimensional numerical analysis shown inFig. 1). The results show that for a constant PVD width, varying thethickness of a PVD has insignificant effect on the total PVD flow rate(Q) whereas varying the PVD width and keeping the PVD thicknessconstant affects the total PVD flow rate. The flow rate contributionsof the PVD width (Qw) and thickness (Qth) to the total PVD flow rate

Q/(

kHS)

[m]

1.5

1.75

2

2.25

2.5

0.03 0.05 0.07 0.09

t/w

w=0.08m

w = 0.09mw = 0.1m

w = 0.11m

w = 0.12mt = 0.007m

t = 0.006m

t = 0.005m

t = 0.004m

Fig. 2. Flow rate of PVD well for different PVD sizes (re¼ 0.5 m).

tw

H = 1.0 m

i

a

b Equipotentiallines

H. Abuel-Naga, A. Bouazza / Geotextiles and Geomembranes 27 (2009) 227–231 229

(Q¼QwþQth) were determined as shown in Fig. 3. For the casewhere the PVD thickness varied while keeping its width constant,the increase in Qth was cancelled by the decrease in Qw as illus-trated in Fig. 3a. On the other hand, if the PVD thickness is keptconstant and its width varied, Qw increases and minimal effect isobserved on Qth as shown in Fig. 3b.

Q/(

kHS)

[m

]Q

/(kH

S) [

m]

0.0

0.5

1.0

1.5

2.0

0.04 0.05 0.06 0.07

t/w

Qth Qw Q

0.0

0.5

1.0

1.5

2.0

2.5

2.5

0.04 0.045 0.05 0.055 0.06 0.065

t/w

Qth Qw Q

a

b

Fig. 3. Effect of PVD thickness and width change on PVD flow rate. (a) At constantwidth (w¼ 0.1 m). (b) At constant thickness (t¼ 0.005 m).

Fig. 4. Hydraulic gradient at PVD wall.

0

20

40

60

80

100

120

140

0 10 20 30 40 50 60

ii

(w+t)/2

t=3, w=100 mmt=6, w=100 mm

w/2 t/2

a

0

20

40

60

80

100

120

140

0 10 20 30 40 50 60

(w+t)/2

t=6, w=100 mm

t=6, w=80 mmt/2 w/2

b

Fig. 5. Evolution of hydraulic gradient at PVD wall as PVD cross-section dimensionschange.

0

0.5

1

1.5

2

2.5

3

3.5

0 0.02 0.04 0.06 0.08 0.1

deq (m)

Numerical

Analytical

Q/(

kHS)

[m

]

Fig. 6. Flow rate of equivalent circular cross-section well at different deq (re¼ 0.5 m).

deq = 0.4546w

R2 = 0.9983

0

0.02

0.04

0.06

0.08

0 0.04 0.08 0.12 0.16

w (m)

deq

(m)

Fig. 8. Relationship between deq and w.

H. Abuel-Naga, A. Bouazza / Geotextiles and Geomembranes 27 (2009) 227–231230

This behaviour can be explained using Darcy’s equation whichdetermines the PVD discharge (Q¼ kiA), where i is the hydraulicgradient at the PVD wall, and A is the PVD surface area(A¼ 2H[tþw]) as shown in Fig. 4a. The non-uniform distribution ofi along the PVD wall is attributed to the elliptical shape of theequipotential lines close to the PVD as shown in Fig. 4b. Darcy’sequation states that for a given soil, Q is function of both i and A.However, i and A are also interrelated as changing the PVD cross-section dimensions (t, w) will also change the distribution of i alongthe PVD wall. Therefore, the change in Q, as the PVD cross-sectiondimensions (t, w) vary, can be equal to zero if the product of i by Astays constant. To verify the above, the evolution of i along the PVDwall was determined numerically for different PVD cross-sectionsas shown in Fig. 5. The term iA, for each case, can be determined bycalculating the area delimited by the hydraulic gradient curve andx-axis. Fig. 5a shows that if w is constant and t increases then thisarea stays relatively constant. Thus Q does not change much.However, if t is constant and w increases then this area increases asshown in Fig. 5b, consequently Q increases.

3.1. PVD equivalent diameter

Fig. 6 shows the variation of the normalized flow rate Q/(kHS)against deq for a circular cross-section well tested under similar soil

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.03 0.05 0.07 0.09

deq

(m)

t/w

w = 0.08m

w = 0.09m

w = 0.10m

w = 0.11m

w = 0.12m

t = 0.007mt = 0.005m

t = 0.006mt = 0.004m

Fig. 7. Equivalent circular cross-section well diameter at different PVD sizes(re¼ 0.5 m).

transport properties and boundary conditions as in the PVD-wellcase. The relationship between Q/(kHS)� deq was establishednumerically and analytically to validate the numerical approach.

To evaluate the PVD equivalent diameter under equal flow ratecondition, the normalized flow rates Q/(kHS) given in Fig. 3 werereplaced by deq using Q/(kHS)� deq relationship given in Fig. 6.Thus, a relation between deq and t/w can be determined, at equalflow rate conditions, as shown in Fig. 7. This relation indicatesthat for the practical range of PVDs (0.0333� t/w� 0.0875) thethickness has insignificant effect on the calculated equivalentdiameter. Fig. 8 shows that the relation between deq and w, for0.0333� t/w� 0.0875, can be expressed as follows:

deq ¼ 0:4546w (2)

To evaluate the effect of re/w on the above relation (re is radius ofunit cell sample), numerical tests of a given PVD size (w¼ 0.1 m)and its deq according to Eq. (2) were conducted at different re/wratios and under similar soil transport properties and boundaryconditions. Fig. 9 shows that the total flow rates of the PVD (QPVD)and its equivalent circular cross-section ðQdeq

Þ are almost equal fordifferent re/w ratios. Consequently, it can be concluded that Eq. (2)is (re/w) independent over the practical range of PVD size andspacing (PVD with w¼ 100 mm and t¼ 5 mm, square pattern withspacing 1, 2, 3 m).

The experimental evaluation of deq of different drain typesconducted by Suits et al. (1986) and the four different equivalent

0.0

1.0

2.0

0 5 10 15 20re/w

Qde

q/Q

PV

D

w = 0.1m, t = 0.05m, deq = 0.04546m

Fig. 9. Effect of re/w ratio in Eq. (2).

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.035 0.045 0.055 0.065 0.075

t/w

d eq/

w

Suits et al. (1986)Long and Covo (1994)Atkinson and Eldred (1981)Hansbo (1979)Fellenius and Castonguay (1985)This study

Alidrain

Colbond CX 1000

Vinylex

Bando

Castleboard

Amer-Drain

Fig. 10. Comparison between different PVD equivalent diameter equations.

H. Abuel-Naga, A. Bouazza / Geotextiles and Geomembranes 27 (2009) 227–231 231

diameter equations given in Table 1 as well as Eq. (2) were plottedin deq/w� t/w plane as shown in Fig. 10. The equivalent diameterequations proposed by Hansbo (1979) and Fellenius and Cas-tonguay (1985) can be considered as the upper and lower limits,respectively, for the possible equivalent diameter zone in deq/w� t/w plane. It can be noticed that the four equivalent diameter equa-tions listed in Table 1 are function of t/w ratio which is not the caseof Eq. (2). Moreover, it can also be observed that the experimentalresults reported by Suits et al. (1986) are in acceptable agreementwith the equivalent diameter equations proposed by Atkinson andEldred (1981), Long and Covo (1994), and Eq. (2). This findingsuggests that the possible equivalent diameter zone in deq/w� t/wplane can be contracted to be between Eq. (2) and Long and Covo(1994) equation. The observed scatter in the experimental resultsreported by Suits et al. (1986) can be attributed to the uncertaintyin smear effect evaluation that might have affected the equivalentdiameter back-calculation process.

4. Conclusions

The numerical study conducted to establish an equivalentdiameter of a PVD well considering equal flow condition shows thatfor such oblong section (0.0333� t/w� 0.0875) the equivalentdiameter is only function of PVD width. The equivalent diameterequation presented in this study is close to the equations proposedby Atkinson and Eldred (1981) and Long and Covo (1994), and is inagreement with the experimental results reported in literature. It issuggested that the proposed equation together with Long and Covo(1994) equation can establish the boarders of possible equivalentdiameter zone in deq/w� t/w plane.

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