ÉC OL E PO LYT EC H NIQU EFÉ DÉRALE D E L AUSANNE

HYDRAULIC HEAD LOSSES IN AN UNLINED PRESSURE TUNNEL OF A HIGH HEAD POWER PLANT

THEORETICAL APPROACH AND COMPARISON WITH THE MEASURED VALUES

Case Study of 2x72 MW Chimay Hydropower Project, Peru

Postgraduate Diploma Project work by Manoj Kumar Garnayak LCH, EPFL 1999-2001

Cycle d'études postgrade en aménagements hydrauliques

1999 - 2001

1 Contents

1 INTRODUCTION 2

2 BASIC EQUATIONS FOR THE CALCULATION OF HEAD LOSSES IN THE WATER CONDUCTOR SYSTEM OF CHIMAY 2

2.1 Friction losses 2 2.2 Bend losses 3 2.3 Losses due to expansion 3 2.4 Losses due to contraction 3

3 LITERATURE REVIEW FOR THE CALCULATION OF FRICTION HEAD LOSSES IN AN UNLINED DRILL & BLAST TUNNEL 4

3.1 Rahm’s method 4 3.2 Colebrook’s method 4 3.3 Huval’s method 5 3.4 Priha’s method 5 3.5 Reinius’s method 5 3.6 Wright’s method 5 3.7 Johansen’s method 6 3.8 Solvik’s method 6 3.9 Ronn’s IBA method 6 3.10 Czarnota’s values from model study on a different project 7

4 STATISTICAL DISTRIBUTION AND CALCULATION OF FRICTION COEFFICIENT FOR CHIMAY DRILL & BLAST TUNNEL 7

5 HEAD LOSSES IN TBM TUNNEL 9

6 FRICTION AND SINGULAR HEAD LOSSES IN CHIMAY TUNNEL 10

6.1 Singular head losses 10

7 ESTIMATED HEAD LOSSES DUE TO FRICTION IN THE DRILL & BLAST TUNNEL 11

8 SUGGESTED DESIGN METHOD FOR D&B UNLINED TUNNEL 13

9 CONCLUSION 14

2 1 INTRODUCTION

Key words

Unlined drill & blast tunnel, TBM tunnel, friction losses, singular losses, statistical distribution, overbreak, measured head loss, roughness, lined invert, shotcrete,

Cross sections and shapes of an unlined drill & blast tunnel vary randomly from one section to another. The flow in it lacks longitudinal and lateral symmetry. Most convenient measurable dimensions of a tunnel at a construction site are the cross sections at 5 to 10 m intervals. The surface of the excavated rock has irregular projections due to blasting that offer high resistance to flow. This is commonly termed as micro-roughness. Variation in areas from one location to another causes continuous expansion, contraction and change of direction, called macro-roughness. Based on the statistical distribution of the measurements from site, methods to evaluate friction losses in an unlined tunnel, taking both these roughness together into account, are available in literature. Analytically, it is difficult to separate both the roughness. Addition of micro and macro roughness shall give equivalent hydraulic roughness of the unlined drill & blast tunnel.

2x72 MW Chimay hydropower project in central Peru has a pressurised water conductor system which consists of an unlined drill & blast (D&B) tunnel (D-shape, nominal base width 7.0 m, height 6.15 m) of 5’130 m long; a 5.7 m diameter TBM excavated tunnel of 4’021 m long; a 4.5 m diameter concrete lined vertical pressure shaft of 107 m height; a 3.8 m diameter steel lined high pressure horizontal tunnel of 220 m long; and two penstocks, 2.65 m diameter, of 63 m long each. Head losses in the water conductor system have been measured between the entry to the tunnel and at the end of the penstocks. The paper deals with the design discharge of 82 m3/s.

2 BASIC EQUATIONS FOR THE CALCULATION OF HEAD LOSSES IN THE WATER CONDUCTOR SYSTEM OF CHIMAY

Friction losses and singular losses in the pressurised water conductor system have been treated separately. Singular losses arise due to bends, expansion, contraction, transition, obstruction in rock traps, flow combination, flow division, and entry losses at the tunnel intake.

2.1 Friction losses

Darcy-Weisbach equation for friction head losses in a pressure pipe is expressed as:

DL

gv

fh f ⋅⋅

⋅=2

2

(2.1)

The friction coefficient f is determined from Colebrook-White modified friction formula given by equation (2.2):

3

2

9.0Re74.5

7.3log

25.0

+

=

Dk

f (2.2)

2.2 Bend losses

The bend loss coefficients in the layout are calculated from Idel’cik (equation 2.3)

mb ςβας ⋅⋅= (2.3)

zyxm ⋅⋅=ς (2.4)

Where

( )θFx = , deflection angle of the bend

=

DR

Fy 0 , ratio of radius of curvature of the bend and the equivalent hydraulic

diameter of the tunnel

=

wh

Fz , ratio of height and width of the cross section of the tunnel

2.3 Losses due to expansion

The expansion loss coefficients are calculated from Borda-Carnot’s relation

2

2

11

−⋅Φ=

AA

eeς , 0≤ζe≤1 (2.5)

Where

φe= Coefficient of correction =

+⋅ θ

πθ

α 2sin2

for θ≤π/6

=

−⋅

πθ

α24

5 for π/6≤θ≤π/2

α= shape factor of the conduit = 1 for circular, 0.75 for open channels

A1, and A2 are the areas upstream and downstream of the expanded flow.

2.4 Losses due to contraction

The contraction loss coefficients are Calculated by Gardel’s equation

4

21

1

−=

µς c (2.6)

Where

( ) ( ) ( )b

bbaba⋅−

−⋅⋅⋅+⋅⋅−−=

03.003.1495.138.1032.111 49.07.048.1

µ

a = ratio of downstream area to upstream area=1

2

AA

°

°

=180θ

b

Other local losses like entry, combination, division, obstructions are referred from standard literatures.

3 LITERATURE REVIEW FOR THE CALCULATION OF FRICTION HEAD LOSSES IN AN UNLINED DRILL & BLAST TUNNEL

Methods to define equivalent absolute hydraulic roughness k or the friction coefficient f of an unlined D&B tunnel are reviewed from the literature and are presented in this section.

3.1 Rahm’s method

This method makes a statistical distribution of the measured areas to find out hypothetical cross section areas with 99% and 1% cumulative frequency to determine relative overbreak δ and is calculated as per equation (3.1).

100%1

%1%99 ⋅−

=A

AAδ % (3.1)

The relative equivalent roughness in the rock tunnel can be expressed through

Rk /15

log105.05.0 ⋅=−δ (3.2)

The relationship between f and δ is empirically given as

δ⋅⋅= −31075.2f (3.3)

3.2 Colebrook’s method

In this method the normal overbreak tm of an unlined D&B tunnel is defined as the half of the difference between the mean hydraulic diameter and the hydraulic diameter of the area with 1% cumulative frequency. The normal overbreak tm is equal to the absolute

5

roughness k of the surface. The value of the friction factor f is calculated from equation (3.4).

5.2

5.1

)(55.0

m

m

tRRt

f+⋅

⋅= (3.4)

3.3 Huval’s method

The overbreak is measured as the difference between the mean hydraulic diameter and the nominal hydraulic diameter of the drill & blast cross section. The overbreak is equal to the equivalent hydraulic roughness k.

)(4

nmnm AADDk −⋅=−=π

(3.5)

3.4 Priha’s method

5.0

%1

5.0%13

)9(1030.3

+⋅⋅⋅= −

AA

f δ (3.6)

3.5 Reinius’s method

The method suggests that if drill & blast tunnel is constructed in the direction of flow, there would be more head losses than if the construction is in opposite direction. Different friction coefficients are empirically given for normal, slow and rapid progress of work.

For a normal excavation the friction factor f is described by

δ⋅+= 0016.002.0f (3.7)

For a careful excavation the friction factor is

δ⋅+= 00085.003.0f (3.8)

And, for a rapid excavation, the friction factor is

δ⋅+= 0027.001.0f (3.9)

3.6 Wright’s method

Natural overbreak tn of a drill & blast tunnel is described according to equation (3.10).

)(5.0 %1%50

%1%50

PPAA

tn +⋅−

= (3.10)

The relative overbreak is calculated as per equation (3.11).

6

%100)

21(

12

2

%50

%50

⋅

⋅−

⋅⋅=

RtR

t

n

nδ (3.11)

After knowing the values of the relative overbreak δ, friction coefficients are read out from the graphs provided by the author for an exposed drill & blast rock surface, or with a concrete lined invert of a drill & blast unlined tunnel.

3.7 Johansen’s method

The absolute roughness for a cross section is defined in equation (3.12).

i

ii

A

Ak

∆⋅+= βα (3.12)

For m measured cross sections along a drill & blast tunnel stretch, the absolute roughness is calculated according to equation (3.13).

i

im

A

Am

k∆

Σ⋅⋅+=1

1βα (3.13)

Where, α and β are experimentally determined constants having the values 0.15 m and 0.37, respectively.

3.8 Solvik’s method

Relative area difference between adjacent cross sections defines the roughness ki arising due to the variation of cross section areas. To it wall roughness kw of 0.15 m is added to obtain the overall roughness k of the drill & blast tunnel.

i

iii P

AAk 1−−

= (3.14)

∑=

=m

iiavg k

mk

1

1 (3.15)

Total roughness of the tunnel, k = wall roughness (kw) + area roughness (kavg)

667.0

449.0

⋅⋅=

avgRk

f (3.16)

3.9 Ronn’s IBA method

The root mean square (rms) value of r measurements on any of the lines parallel to a reference line along the tunnel is calculated as

7

r

xxrms

r

ii

j

∑=

−= 1

2)( (3.17)

For m numbers of such lines (at least 3 numbers of lines on the blasted rock surface parallel to a reference line are required for which distance measurements are taken at 0.25 m to 0.50 m intervals), the wall roughness is expressed in equation (3.18).

m

rmsrms

m

jj

wall

∑== 1

2)( (3.18)

The area roughness for any stretch (measured at 0.5 m to 1.0 m intervals) is calculated as

r

AArms

r

jj

Ai

∑=

−⋅= 1

5.05.0 )(53.0 (3.19)

For m number of stretches, the area roughness is

m

rmsrms

m

iA

A

i∑== 1

2)( (3.20)

Total equivalent roughness of the tunnel is expressed as:

Awall rmsrmsk += (3.21)

3.10 Czarnota’s values from model study on a different project

The representative friction coefficients from the model test for a different project were:

• The friction coefficient of the drill & blast rock f = 0.0730

• Average friction coefficient with the invert lined f = 0.0623

• Average friction coefficient with unlined invert and shotcrete on wall and roof f = 0.0519

• Average friction coefficient with lined invert and shotcreted on wall and roof f = 0.0411

4 STATISTICAL DISTRIBUTION AND CALCULATION OF FRICTION COEFFICIENT FOR CHIMAY DRILL & BLAST TUNNEL

The drill & blast section of the tunnel is divided into 5 stretches, named km 1, km 2, km 3, km 4, and km 5. Cross sections have been recorded at 5 m or 10 m intervals

8

covering around 92% length of the drill & blast tunnel. The statistical distribution of the surveyed areas and the calculated perimeter are shown in figure 4.1 and 4.2, respectively. Out of 903 measured cross sections, 888 number of cross sections have been chosen (204 in km 1, 190 in km 2, 126 in km 3, 199 in km 4, and 169 in km 5). Rest of the cross sections are disregarded due to incompatibility.

0

10

20

30

40

50

60

70

80

90

100

110

-5 0 5 10 15 20 25 30 35 40 45

variation of area over the nominal area %

Num

ber

of s

ampl

es

km 1

km 2

km 3

km 4

km 5

all

Figure 4.1: Chimay, Statistical distribution of area with respect to nominal area of 37.792 m2.

0

10

20

30

40

50

60

70

80

90

100

110

-2 2 6 10 14 18 22

variation of perimeter %

Num

ber o

f sam

ples

km1km2km3km4km5all

Figure 4.2: Chimay, Statistical distribution of perimeter with respect to nominal perimeter of 23.296 m.

9

Absolute roughness k and friction coefficient f have been calculated directly for each km by applying the suggested methods in section 3. The values are presented graphically in figures 4.3 and 4.4, respectively. In the figure, the values are joined by representative smooth curves, but it is not to be understood that k values and f values keep on changing from one point to another. They remain same for a particular km. The k and f values are for the exposed rock in a drill & blast tunnel. These values shall be corrected when either the invert is lined or the surface is shotcreted or both.

0

0.2

0.4

0.6

0.8

0 1 2 3 4 5 6

km

k (m

)

Rahm

Colebrook

Huval

Wright

Johansen

Solvik

IBA

Figure 4.3: Chimay; Calculated k values from different methods

0

0.02

0.04

0.06

0.08

0.1

0.12

0 1 2 3 4 5 6

km

f [-

]

Rahm

Colebrook

Huval

Priha

Reinius

Wright

Johansen

Solvik

IBA

Figure 4.4: Chimay; Calculated f values from different methods

5 HEAD LOSSES IN TBM TUNNEL

Nominal diameter of the TBM tunnel is 5.7 m. Cross section areas of the TBM excavated tunnel vary along its length due to placement of supports in form of shotcrete and steel ribs within the excavated diameter (figure 5.1). In addition to the friction losses, there shall be losses due to flow expansion and contraction in the tunnel.

Area variation in the TBM tunnel is not erratic, but it remains constant over a stretch. Each stretch shall have its flow velocity and effective hydraulic radius, therefore, a different friction coefficient. The roughness of the TBM exposed rock and the shotcrete layer is of the same magnitude. The roughness of the shotcrete and the rock is varied from 4 mm to 10 mm. Average friction head losses in the TBM is of the order of 8.23 m. The friction coefficient f of the TBM tunnel varies from 0.0181 to 0.0233.

10

Figure 5.1: Chimay; Cross section area variation in TBM tunnel

Area variation in a TBM bored tunnel is not abrupt. The difference in cross section area is gradual by applying shotcrete between two adjacent stretches. The transition angles vary from 20° to 45°. Average head losses due to expansion and contraction due to area variation in the TBM tunnel are of the order of 0.28 m and 0.19 m, respectively.

On average, without considering the effect of the expansion and contraction, the TBM tunnel has a friction coefficient of f = 0.02, k = 6.03 mm, and K=59.24 m1/3s-1. Taking into account the effect of local expansion and contractions, TBM tunnel has an average f = 0.0211, k = 7.48 mm, and K=57.67 m1/3s-1.

6 FRICTION AND SINGULAR HEAD LOSSES IN CHIMAY TUNNEL

A smaller part of Chimay water conductor system has concrete lining, and the penstocks are steel lined. The absolute roughness of the concrete in the lined part of the tunnel is varied with 0.50 mm, 0.75 mm and 1.0 mm. An average, calculated head losses due to friction are 0.62 m. Similarly, varying the absolute roughness of the steel lined part with 0.025, 0.050, 0.075, and 0.1 mm, an average friction head loss of 2.05 m is expected.

The cross section area of the rock trap goes on varying. Therefore, taking an average roughness of the wall surfaces, friction head losses of 0.14 m is calculated. Similarly in the transition between the D&B tunnel and the TBM tunnel a friction head loss of 0.02 m is expected.

6.1 Singular head losses

There are 15 bends in the water conductor system of Chimay. The cross section areas of the bends in the D&B stretch keep on changing. Therefore, average area of the cross sections at a bend is taken to define the bend geometry. Head losses in the bend with lower R0/D ratios are much more in comparison to those with higher bend ratios. For practical reasons, bends with R0/D > 50, and v < 5 m/s can be neglected during preliminary design. Bends placed within 30 times the hydraulic diameter (with respect to the upstream bend) between them interfere. In Chimay head losses due to bends can come down by 9% on account of the bend interference. A head loss of 1.5 m is estimated in the bends.

Effective flow areas in TBM excavated portion of the tunnel

02468

101214161820222426

5100 5600 6100 6600 7100 7600 8100 8600 9100

chainage,m

Area

, m2

11

Losses due to entry at the tunnel, expansion, contraction and transition in the defined geometry of the tunnel are estimated at 0.32 m.

Similarly head losses due to flow division, combination and bifurcation in the system works out to another 0.32 m.

Head losses due to obstruction of flow in the rock trap caused by the beams and the concrete wall is of the order of 0.1 m.

7 ESTIMATED HEAD LOSSES DUE TO FRICTION IN THE DRILL & BLAST TUNNEL

The invert of the drill & blast tunnel is concrete lined from ch.1+035 m onwards, covering 80% of the total length. First 16.5 m of the D&B tunnel has a defined shape, and losses in this are already included in the lined part. Therefore, ch.0+016.5 m to ch.1+035 m of the D&B tunnel has an unlined invert. From the geological mapping drawings, it was calculated that around 45% of the excavated surface (walls and crown) of the D&B tunnel has been covered with a layer of shotcrete. The lined invert, a shotcrete layer and the exposed rock bring in the concept of composite roughness. Very often the shotcrete is applied only to a part of the exposed rock without full coverage of the walls and the crown. Therefore, the D&B tunnel length is linearly proportioned in equivalent lengths consisting of completely exposed rock, completely covered shotcrete, completely lined invert and unlined invert which can be seen in the table 7.1. Length of the rock trap (108 m in km 5) is excluded from the table for which friction losses have been computed separately and included in section 6.

Type of surface

All [m]

km 1 [m]

km 2 [m]

km 3 [m]

km 4 [m]

km 5 [m]

Exposed rock

2751.86 (55%)

574.46 451.08 578.20 394.88 753.24

Shotcrete 25 cm

3.50 (0%)

3.50

Shotcrete 10/12 cm

139.70 (3%)

73.00 35.00 31.70

Shotcrete 8 cm

846.94 (17%)

238.93 192.04 131.05 204.92 80.00

Shotcrete 5 cm

1167.5 (23%)

201.61 227.88 290.75 290.2 157.06

Steel ribs with

shotcrete

96.00 (2%)

21.00 75.00

Table 7.1: Chimay; covering over the excavated surface of D&B tunnel

From Czarnota’s laboratory model test on a D&B tunnel in Sweden, it is indirectly interpreted that when one layer of shotcrete, 5 cm to 10 cm thick, is applied the roughness of the natural D&B rock surface reduces by around 11 cm.

Absolute roughness of the exposed rock surface is calculated as per the existing methods briefed in section 3. Where shotcrte is applied, the surface roughness of the blasted rock is, therefore, reduced by 11 cm.

12

An absolute roughness of 3 mm is assumed for the concrete invert, wherever applicable. The friction factor of a composite section is given by euation (7.1).

PfPfPfP

f ssccrr ⋅+⋅+= ⋅ , and (7.1)

scr PPPP ++= (7.2)

After taking into account the influence of invert lining and shotcrete, as applicable, theoretically, the expected friction head losses for every km and for the total tunnel is furnished in the table 7.2.

Method 1st km [mwc]

2nd km [mwc]

3rd km [mwc]

4th km [mwc]

5th km [mwc]

Total [mwc]

Rahm 2.13 1.38 1.72 1.89 1.79 8.91

Colebrook 1.34 0.88 1.18 1.13 1.10 5.63

Huval 2.23 1.53 1.48 1.51 1.66 8.41

Priha 2.31 1.50 1.87 2.06 1.92 9.66

Reinius 1.78 1.22 1.51 1.59 1.52 7.62

Wright 1.96 1.35 1.66 1.73 1.66 8.36

Johansen 1.41 1.12 1.39 1.24 1.25 6.41

Solvik 1.04 0.83 1.09 0.92 0.99 4.87

IBA 1.87 1.37 1.77 1.53 1.55 8.09

Czarnota* 1.55 1.30 1.60 1.41 1.54 7.40

Table 7.2: Chimay; friction head losses in the D&B tunnel calculated by various methods (*only recommended friction factors considered).

Figure 7.1: Chimay; Comparison of friction loss in unlined tunnel

0 1 2 3 4 5 6 7 8 9 10

RAHMHUVALPRIHA

REINIUSWRIGHT

IBACZARNOTAMEASURED

COLEBROOKJOHANSEN

SOLVIK

Head loss (mwc)

13

Head losses for the entire water conductor system, amounting to 21.8 m, has been measured at the design discharge of 82 m3/s. Deducting the losses already calculated in sections 5 and 6, balance losses are accounted towards the measured head losses in the unlined D&B tunnel. Theoretically calculated head losses and the measured one are compared graphically in figure 7.1.

It is seen that Rahm’s method overestimates the losses by 10.8%, Colebrook’s method underestimates it by 30%, Huval’s method overestimates by 4.6%, Prihas’s method overestimates by 20%, Reinius’s method underestimates by 5.2%, Wright’s method overestimates by 4%, Johansen’s method underestimates by 20%, Solvik’s method underestimates by 39%, and IBA method overestimates by 0.6%, than the measured values. The methods that estimate the friction losses of the D&B tunnel within an acceptable range are retained and the others are discarded to arrive at an average calculated value of the friction head losses. Therefore Rahm, Huval, Reinius, Wright and IBA methods are retained.

If the rock surface of the D&B tunnel were completely exposed, friction head losses of around 10.9 m would have been expected. Since 80% of the D&B tunnel length is invert lined and 45% of surface is covered by shotcrete, reduction of head loss has reduced by 2.86 m (26% of the losses). The invert lining contributes to 83% of this reduction, and balance 17% is from the shotcrete layer.

Equivalent average friction coefficient of the completely exposed rock surface would have been f = 0.0813, K = 28.41 m1/3s-1 and k = 454 mm. After 80% of the tunnel invert has been lined and 45% of the surface is shotcreted, the new parameters have been f = 0.0600, K = 33.06 m1/3s-1 and k= 273 mm.

8 SUGGESTED DESIGN METHOD FOR D&B UNLINED TUNNEL

The methods of Rahm, Reinius, Wright and IBA require enough measurement at site to have a statistical distribution of the areas and perimeter. However, the method of Huval depends on the mean and the nominal hydraulic diameter. It is seen that the average excavated area and the nominal area plot almost a linear graph. It means that, once knowing the nominal area, the average excavated area can be ascertained approximately for preliminary design. From the mean excavated area, a mean hydraulic diameter can be calculated. The roughness k will be calculated according to equation (3.5), and the friction coefficient can be known from equation (2.2).

However, as work progresses, rigorous site measurements of the excavated cross sections are required at closer intervals to apply other methods to verify the estimated head losses calculated during preliminary design stage.

The graph of nominal area versus mean excavated area is shown at figure 8.1.

14

Figure 8.1: Nominal area vrs. Average excavated area of a D&B tunnel

9 CONCLUSION

Head losses in an unlined D&B tunnel can be estimated as per some of the existing methods (Rahm, Huval, Reinius, Wright and IBA). Correction for invert lining and for shotcrete shall have to be applied in real site condition. Enough and systematic record of cross section data are required for better statistical distribution. Cross sections formed due to geological overbreak or due to intentional enlargement shall be excluded from the standard distribution.

If the invert of a D&B tunnel of Chimay size is lined, around 27% of the head losses, with respect to complete exposed D&B rock, can be reduced. If shotcrete is applied on the wall and the roof, around 10% of the head losses can be reduced.

Cross section of a TBM excavated tunnel may vary due to the placement of supports within the excavated diameter. In addition to the friction losses, there shall be losses due to expansion and contraction. The surface roughness of such a tunnel can be between 4 mm to 10 mm.

K value for an unlined D&B tunnel is around 28 m1/3s-1. This corresponds to a friction coefficient f = 0.0813.

Separation of macro and micro roughness is difficult in a D&B tunnel. Both are to be treated integrally as one roughness.

Nominal area versus excavated area of a drill and blast unlined tunnel

0

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

160

170

180

190

200

210

220

230

240

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200

Nominal area, m2

15 Symbols, abbreviations and subscripts

α [-] Coefficient β [-] Coefficient θ [-] Angle µ [-] Parameter for contraction loss δ [-] Relative overbreak (in decimal or percentage) ζ [-] Co-efficient for singular head losses ∆A [m2] Area variation ∆X [m] Length variation a [-] Ratio b [-] Ratio f [-] Friction factor / friction coefficient g [m/s2] Acceleration due to gravity (9.81 m/s2 hL [mwc] Head loss general hf [mwc] Head loss due to friction hb [mwc] Head loss due to bends and curves he [mwc] Head loss due to expansion hc [mwc] Head loss due to contraction hi [mwc] Head loss at intake k [m] Equivalent absolute roughness t [m] Overbreak v [m/s] Velocity x [-] Ratio xi [m] Measured distance x [m] Mean distance y [-] Ratio z [-] Ratio A [m2] Area A [m2] Mean area D [m] Diameter or hydraulic diameter F Functional relation K [m1/3s-1] Strickler’s coefficient L [m] Length P [m] Wetted Perimeter Pr [m] Wetted Perimeter of exposed rock Pc [m] Wetted Perimeter of concrete invert Ps [m] Wetted Perimeter of shotcreted wall and roof Q [m3/s] Discharge R [m] Hydraulic radius Re or Re [-] Reynolds number R0 [m] Radius of curvature Subscripts 1%, 50%, 99% cumulative frequency of 1%, 50%, 99% for statistical

distribution. avg average m mean, average n nominal, natural t theoretical

16 i,j,m,r integer variables Abbreviations ch. Chainage D&B Drill and Blast log Logarithm function to base 10 m asl Metre Above Mean Sea Level mean Arithmetic mean mwc Metre of water column rms Root mean square TBM Tunnel Boring Machine Bibliography Bollrich. G – Technische Hydromechanik 1, ISBN 3-345-00608-1 Borch E. – Unlined high pressure tunnels in areas of complex topography, Water Power & Dam Construction, November 1984. Carstens T. and Hansen S.E. – Rehabilitation in the unlined rock tunnels of Nedre Rossaga after 40 years, Hydropower’97. Cuesta L. – Unlined hydroelectric tunnels, Rock Mechanics and Power Plants, Pages 289-292, ISBN 90 61918 278. Czarnota Z. – Hydraulics of rock tunnels; Hydraulics Laboratory of The Royal Institute of Technology, Stockholm, 1986 Czarnota Z. – Rock Tunnels; Hydraulics Laboratory of The Royal Institute of Technology, Stockholm, 1980. Dann H.E.- Unlined tunnels of the Snowy Mountains Hydro-electric Authority, Australia, Journal of the Power Division, Proceedings of the American Society of Civil Engineers, October 1964. Dubois J.- Comportement hydraulique et modélisation des écoulement de surface. Communication 8 du Laboratoire de Construction Hydrauliques, EPFL, Laussane, 1998. Gulliver and Arndt – Hydropower Engineering Handbook, ISBN 0-07-025193-2. Hydraulic Design Criteria, Corps of Engineers, USA. Hydraulic considerations (Section 6) – describing method [Solvik, 1984] for unlined pressure tunnels, received from Kukule Ganga Hydropower Project, Sri Lanka. I.E. Idel’cik- coefficients de pertes de charge singulièrs et de pertes de charge par frottement, traduit par Mme. Meury, 1979. Lecocq R. et Marin G. – Evaluation des pertes de charge des galleries d’amnee d’eau forees au tunnelier et non revetus, 1986 (collected from Prof. Schleiss, EPFL, Laussane).

17 Metcalf J.R. and Jordaan J.M.- Hydraulic roughness change in the Orange-Fish Tunnel: 1975-1990, The Civil Engineer in South Africa, August 1991. Miller D. S.- Internal flow systems. BHR publication. ISBN0-947711-77-5. Nord G.- Drilling Accuracy in Underground Construction, World Tunnelling, December 2000. Pennington M.S- Hydraulic roughness of bored tunnel. Paper on internet, IPENZ Transactions, Vol.25, No. 1/CE,1998. Petrofsky A.M.- Contractor’s view on unlined tunnels, Journal of the Power Division, Proceedings of the American Society of Civil Engineers, October 1964. Polycopie du postgrade aménagement hydrauliques, Laussane, 2001-2003 – Pertes de charge singulièrs. Polycopie du postgrade aménagement hydrauliques, Laussane 1999-2001; Basic Hydraulics of Natural Water Courses Reinius E. – Head losses in unlined rock tunnels; Water Power July / August 1970. Ronn and Skog – New method for estimation of head loss in unlined water tunnels. Hydropower 1997. Solvik O. and Tesaker E.- Floor paving in unlined hydropower tunnels, Hydropower 1997. Spencer R.W.- Unlined tunnels of the Southern California Edison Company, Journals of the Power Division, Proceedings of the American Society of Civil Engineers, October 1964.