Ellis,J. Pressure transients in water engineering, A guide to analysis and interpretation of behaviour

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of any backﬁll is ignored in unconstrained conditions. Where a

compacted backﬁll is involved, the properties of ﬁll have an important

inﬂuence on behaviour but the short-term effect of pipe ovalisation on

buckling pressure could also be included as outlined in the following

section.

462

210

200

190

180

170

160

150

140

130

120

110

100

Internal

ressure head

(

mWG

)

–10 0 10 20 30 40 50

Acoustic wavespeed a (m/s)

HDPE pipes

SDR = 26

= 6

Di = 0.04

SDR = 26

= 6

Di = 0.02

SDR = 17.6

= 6

Di = 0.04

SDR = 17.6

= 6

Di = 0.02

SDR = 26

= 2

Di = 0.02

Fig. 22.4. Wave speed as a function of deformation

Pressure transients in water engineering

22.8 Short-term elastic buckling under hydraulic transient

effects

22.8.1 Unconstrained conditions

If the pipe is installed in air, water or in very soft soils such as wet clay

which provide little support, critical pressure becomes:

¼ 24:0 0:7E

ð0Þ

=ð12D

Þ=ð1  

¼ 2ð0:7E

ð0Þ

Þðs=D

=ð1  

Þð22:31Þ

22.8.2 Constrained conditions

If ﬂexible pipes are exposed to a sudden negative pressure, the risk of

buckling can be calculated according to the following formula, provided

that the pipes are surrounded by granular soil which can give elastic

support to the pipe.

In studies of GRP pipe, Carlstrom (1981) introduced a reduction

factor  for the stabilising effect of the soil modulus into the equation

for critical pressure, giving:

¼ð1 "

=sÞ

1:5

f32E

t0:7E

=ð12D

Þg ð22:32Þ

where  ¼ 1 p=ðp þp

exð50Þ

Þ, p being the transient vacuum pressure

within the pipe and p

exð50Þ

the long-term vertical pressure on the

pipe. Long-term pipe stiffness ¼ 0:7E

=ð12D

Þ has been adopted as

the pipe may be more prone to buckle in the longer term when subject

to the same pressure transient. E

is the initial ﬂexural circumferential

modulus.

exð50Þ

¼ Cg

where 

is the weight of soil/m

and Table 22.5 provides some typical

values of 

. Cg may be obtained from Fig. 22.5.

Carlstrom (1981) considered friction between the soil prism and the

surrounding ﬁll as one component  and friction between the native

material forming the trench wall and the neighbouring ﬁll as a second

463

Table 22.5. 

v. soil type

Soil type Granular/

non-cohesive

Stony/

sandy

Wet—

loamy

Clay/heavy

loam

Water-

saturated clay



=

water

1.7 1.9 2.0 2.1 2.2

Flexible pipe

component 

, so that the vertical weight of overburden was modiﬁed

by a factor 

. He also considered that over time the vertical pressure

would increase and so he proposed use of the factor ð1 þ Þ=

2ð1 þ 

Þ=2 in compensation for long-term effects so that vertical

pressure was given by:

exð50Þ

¼ð1 þÞ=2ð1 þ

Þ=2ðp

þ p

 p

where p

is the pressure from overburden on the pipe, p

is pressure due

to trafﬁc loading, p

is uplift pressure due to groundwater, and p

internal water pressure.

464

2.5

1.5

0.1 0.15 0.2 0.3 0.4 0.5 0.6 0.8 1.0 1.5

1 1.5 2 3 4

H/B

1.5

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.25

0.2

0.15

0.10

Fig. 22.5. C

as a function of depth of cover/trench width

Pressure transients in water engineering

Much of the foregoing is based upon the assumption of a generally

elliptical deformation with y=h ¼ 0:91.

A number of studies have been carried out of ﬂexible pipe defor-

mation behaviour using ﬁnite-element analysis of the soil structure.

The possibility exists of coupling hydraulic transient analysis with the

numerical analysis of soil behaviour to provide a more detailed picture

of deformations.

22.9 Application of a ﬂexible pipe model

Some examples have been included to illustrate the effects of pipe

ﬂexibility upon the propagation of pressure transients. Different types

of simple pipeline conﬁguration have been considered.

22.9.1 Long horizontal pipeline

The ﬁrst case considers a long horizontal pumping main. Overall

length of the pipeline is 12.5 km and an overburden or external

loading equivalent to 2.5 mWG has been included.

Pipe bedding was included which offered some lateral support to a

deforming pipe. Initial ovality was set at 2% and gas release at low

pressure was included. The suction well level was set at þ0.0 mWG

and discharge elevation was þ1.0 mWG. Figure 22.6 shows predicted

465

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10 000 11 000 12 000

Chaina

e (m)

Series 1

Series 2

Series 3

Horizontal flexible pipeline. con = 1.0, del/D = 2%, h

= 12.3, gas included

Elevation (mAD)

–10

Fig. 22.6. Envelope curves for horizontal pipe and 2% deﬂection ratio

Flexible pipe

minimum head following a pumping failure. Maximum head is that of

steady pumping. In Fig. 22.7 changing head at the pumping station

and approximately halfway along the main are shown. A signiﬁcant

amount of damping is evident in these results. If gas release is ignored

and the pipe has the same trench support as before, the minimum head

along the main after pump trip falls to a greater extent and buckling

occurs at chainage 11.5 km as depicted in Fig. 22.8. The model was

466

Time (s)

Horizontal flexible pipeline. con = 1.0, del/D = 2%, h

= 12.3

Head (mAD)

–10

2.46

19.68

36.90

54.12

71.34

88.56

105.78

123.00

140.22

157.44

174.66

191.88

209.10

226.32

243.54

260.76

277.98

295.20

312.42

329.64

346.86

364.08

381.30

398.52

415.74

432.96

450.18

467.40

484.62

Series 1

Series 2

Fig. 22.7. Head variation for horizontal pipe after pump failure

Chaina

e (m)

Series 1

Series 2

Series 3

Horizontal flexible pipeline. con = 0.9, del/D = 2%, h

= 12.3, no gas

Elevation (mAD)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10 000 11 000 12 000

–10

Fig. 22.8. Envelope curves showing prediction of pipe buckling

Pressure transients in water engineering

set to terminate computations in the event of a pipe collapse. Where

the buckling limit is below the gas release level then the formation of

gas- or vapour-ﬁlled cavities will inhibit any further fall in head and

so internal pressure should not reach the level at which buckling occurs.

22.9.2 Descending outfall

The second example considers an idealised outfall similar to one of the

case studied by Larsen (1976). The pipeline descends uniformly over a

distance of 3 km. Pumping is used to overcome resistance with the

suction well level set at þ0.0 mAD. The pipeline is at 30.0 mAD at

its discharge point. An initial uniform eccentricity of 2% was adopted.

Gas release was represented and the pipeline was without lateral

support. After pump trip the minimum head shown in Fig. 22.9 was

predicted. The damping effect shown is essentially due to the pipe ﬂex-

ibility as positive internal pressures preclude gas release. Appreciable

damping of the pressure wave is also evident in this ﬁgure.

If the pipeline is supported by stiffeners, ﬂexibility is reduced. After

trip, the minimum head along the main falls to a greater extent than

before, as shown in Fig. 22.10. Greater amplitude of head variation

also occurs than without supports, again indicating a smaller amount

of damping in this stiffer pipeline.

Chaina

e (m)

Series 1

Series 2

Series 3

Descending flexible pipeline. con = 6, del/D = 2%, h

= 12.3, gas c = 0.000 01

Elevation (mAD)

100

200

300

400

500

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

1700

1800

1900

2000

2100

2200

2300

2400

2500

2600

2700

2800

2900

3000

–10

–20

–30

Fig. 22.9. Envelope curves for descending outfall after pump failure and 2%

deﬂection ratio

Flexible pipe

22.9.3 Uniformly rising main

A further example studied a uniformly rising pipeline of length 3 km.

This was also one of the conﬁgurations examined by Larsen. If the pipe-

line is supported along its length then following a pump trip, minimum

head is able to fall to a considerable extent as shown in Fig. 22.11. If the

468

Chaina

e (m)

Series 1

Series 2

Series 3

Descending flexible pipeline. con = 1.0, del/D = 2%, h

= 12.3, gas c = 0.000 01

Elevation (mAD)

100

200

300

400

500

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

1700

1800

1900

2000

2100

2200

2300

2400

2500

2600

2700

2800

2900

3000

–10

–20

–30

Fig. 22.10. Enve lope curves for descending outfall after pump failure and with

stiffeners

Chaina

e (m)

Series 1

Series 2

Series 3

Rising profile. Flexible pipeline. con = 1.0, del/D = 2%, h

= 12.3

Elevation (mAD)

–5

100

200

300

400

500

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

1700

1800

1900

2000

2100

2200

2300

2400

2500

2600

2700

2800

2900

3000

Fig. 22.11. Rising main, envelope curves with pipe stiffeners

Pressure transients in water engineering

pipeline is unsupported, damping due to shape change is greater and

minimum head after trip does not fall to the same extent as before,

as illustrated in Fig. 22.12. If a rigid pipe is assumed, the minimum

head after pump failure is determined by vapour cavity formation, as

illustrated in Fig. 22.13, while if gas release and vapour cavity formation

469

Chaina

e (m)

Series 1

Series 2

Series 3

Rising profile. Flexible pipeline. con = 6.0, del/D = 2%, h

= 12.3

Elevation (mAD)

–5

100

200

300

400

500

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

1700

1800

1900

2000

2100

2200

2300

2400

2500

2600

2700

2800

2900

3000

Fig. 22.12. Rising main, envelope curves for unsupported pipe

Chaina

e (m)

Series 1

Series 2

Series 3

Rising profile. Rigid pipeline. con = 0.0, del/D = 2%, h

= 12.3

Elevation (mAD)

–5

–10

100

200

300

400

500

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

1700

1800

1900

2000

2100

2200

2300

2400

2500

2600

2700

2800

2900

3000

Fig. 22.13. Rising main, envelope curves for rigid pipe and gas release

Flexible pipe

are both ignored, the minimum head falls to a much greater extent, as

depicted in Fig. 22.14.

The previous two examples considered produced predictions closely

in line with Larsen’s results when similar pipe properties were used. This

ﬁnding would indicate that it is quite feasible to utilise a ﬁxed grid

computational scheme for such studies as well as a variable wave

speed scheme.

22.9.4 Pipeline of differing properties

Many pipeline networks are made up of a set of pipelines, each of which

has different properties. If ﬂexibility of a pipe is considered then even if a

pipe of uniform characteristics is considered — that is, constant diameter,

material and wall thickness — a radical change in the pipe support

arrangement may be sufﬁcient to produce a different response from

individual sections of a main. An outfall system may comprise a landward

section of pipeline which is laid in a trench with well-compacted backﬁll

and a seaward section which is essentially unsupported. The landward

stretch of main is unlikely to deform to the same extent as the seaward

element under changing pressure.

As an example, consider a pumped outfall having a horizontal land-

ward pipeline section of length 3.0 km and invert level þ1.0 mAD,

470

Chaina

e (m)

Series 1

Series 2

Series 3

Rising profile. Rigid pipeline. No air content. con = 0.0

Elevation (mAD)

–5

–10

–15

–20

100

200

300

400

500

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

1700

1800

1900

2000

2100

2200

2300

2400

2500

2600

2700

2800

2900

3000

Fig. 22.14. Rising main, envelope curves for rigid pipe and no gas release

Pressure transients in water engineering

leading to an embankment or seawall at level þ2.5 mAD. A seaward

section extends for a further 3.0 km, initially falling at a gradient

1:200 and then more steeply at 1:100 to a ﬁnal discharge level of

14.0 mAD. Pumping station suction well level was at 3.0 mAD.

After pump trip, head falls steeply at the pumping station and for an 8 s

interval Fig. 22.15 shows rarefaction waves propagating along the

landward section with little damping evident, producing a minimum

pressure of around 8.0 mAD. Once over the seawall and into the

descending outfall section, noticeable damping occurs. Envelope curves

for the 6 km main show most clearly the effects of the different responses

of the two sections of main. The relatively stiff landward section allows a

substantial fall in head after trip. Within the seaward section, damping

inhibits the development of low pressures to an appreciable extent

(Fig. 22.16).

The cases considered were primarily used to demonstrate the

feasibility of carrying out computations for deforming pipelines under

changing transient pressure. Including what appears to be a signiﬁcant

effect within day-to-day investigations, as a matter of routine is con-

sidered worthwhile. Obtaining the necessary information to allow

471

Time

(

)

d/s PS

Seawall

Outfall

Head (mAD)

–5

–10

Fig. 22.15. Composite system, head variations after pump trip

Flexible pipe