IPENZ Transactions, Vol. 25, No. 1/CE, 1998

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Hydraulic roughness of bored tunnelsMark Stuart Pennington1, BScEng, MScEng, TM.IPENZ

Physical roughness data from various bored tunnels in Southern Africa werecollected. An in-depth investigation into energy losses in conduits and themechanisms associated therewith was made. The roughness data were analysedmathematically and linked to these energy loss mechanisms, to predict theexpected hydraulic friction effects. Similar investigation was made on the innersurface of spun concrete pipes, and the results predicted using the methodsdeveloped accurately matched those obtained by physical testing of the pipes,giving confidence in the accuracy of the methods employed. Expected hydraulicloss coefficients for various surface types encountered in the tunnels sampledwere calculated, and these resembled closely those values predicted in other suchtunnels. A method for accurate determination of hydraulic friction losscoefficients from measurements of physical roughness data was developed.

Keywords: water tunnels tunnel design unlined bored tunnels hydraulicresistance

1Montgomery Watson, PO Box 9463, Hamilton

This paper, first received on 15 September 1997, was received in revised form on6 April 1998.

1. IntroductionThis paper presents the results obtained from a research project carried out at the University of Natal,Durban, South Africa during 1994 and 1995. The topic researched was the hydraulic roughness orresistance of unlined, bored water tunnels, and the research was sponsored by the Water ResearchCommission of South Africa.

The expected hydraulic resistance is one of many factors which need to be known in the design ofwater tunnels. Knowing the resistance, the necessary diameter and/or difference in head required for agiven design flow rate may be calculated. This is commonly done using the Darcy-Weisbach head lossequation

h flvgdf

=2

2.................................................(1)

where hf = head lost due to friction

f = Darcy-Weisbach friction factor

l = length of conduit

v = mean velocity of fluid

g = acceleration due to gravity

d = diameter of conduit

In terms of friction factor, f, the Colebrook-White equation (given in (2) below) has been shown toyield results consistent with those obtained experimentally.

12

371252

210f

kd f

= - +

log

..

Re.....................( )

where Re is the pipe Reynolds number and k is the equivalent sand grain diameter fromNikuradses (1933) experiments on roughness in pipe flow.

For hydraulically rough or fully-developed flow at high Re (usually encountered in tunnels),equation (2) reduces to

IPENZ Transactions, Vol. 25, No. 1/CE, 1998

2

12

371310f

dk

=

log

....................................( )

3As an alternative to equation (1) with (2) or (3), Mannings formula is often used as an hydraulicresistance equation. The advantage offered by this is that Mannings n is independent of diameter andeffectively depends only on surface roughness. For a closed conduit flowing full at high Re,Mannings n and Darcy-Weisbach f are related by

nd f

g=

4 8

4

1

6

..........................................( )

In order to evaluate friction factor f from equation (3), the Nikuradse equivalent sand graindiameter k needs to be known (or estimated). The problem is that k is an equivalent dimension anddoes not relate specifically to any measurable surface roughness dimension other than the diameter ofsingle-sized sand grains glued to the inside of a pipe (Nikuradse, 1933).

2. Measurements of surface roughnessThe first step towards establishing this link was to obtain a large amount of sufficiently detailed rawdata. For this purpose, a tunnel roughness measuring device, consisting of a movable laser scannermounted on a track, was developed at the University of Natal. By picking up the reflection of anemitted laser beam from the tunnel wall in consecutive, discrete intervals, the wall roughness profilewas digitised. Distance readings from the scanners one metre long track to the tunnel wall were takenevery 0.5mm to an accuracy of 0.1mm, giving 2000 points per metre. In this way a two-dimensionalprofile of the roughness sampled at various locations along a tunnel wall could be plotted. A three-dimensional picture of the surface roughness was obtained by taking parallel scans each a set distanceapart, dependent on the degree of accuracy required. Examples of the two-dimensional plots obtained(to an exaggerated vertical scale) are shown in Figure 1. A more detailed description of the apparatus,its operation and its accuracy is given by Pennington (1995).

SANDSTONEChainage 3500 Ngoajane South

-10

-5

0

5

10

0 100 200 300 400 500 600 700 800 900 1000

mm

mm

GRANITEChainage 3200 Emolweni

-10

-5

0

5

10

0 100 200 300 400 500 600 700 800 900 1000

mm

mm

SHOTCRETEChainage 1600 Emolweni

-10

-5

0

5

10

0 100 200 300 400 500 600 700 800 900 1000

mm

mm

FIGURE 1: Physical roughness plots obtained from scanner.

4Field trips to four different tunnels bored by Tunnel Boring Machine (TBM) were made. All of thetunnels were sampled, with samples being taken at 100m intervals. Within these tunnels, fourdifferent surface types were encountered and sampled. A summary of the sampling done is given inTable 1.

TABLE 1: Summary of sampling - numbers of samples

In-situ Unlined Unlined ShotcreteTOTALTunnel Name concr lininggranite sandstoneEmolweni, Inanda-Wiggins Aqueduct 5 27 0 18 50Clermont, Inanda-Wiggins Aqueduct 0 0 36 30 66Ngoajane North Drive, Lesotho Highlands 0 0 108 0 108Ngoajane South Drive, Lesotho Highlands 0 0 37 0 37TOTAL 5 27 181 48 261

Differences in texture obtained from the various surface types are evident in Figure 1. In thisfigure any trends in the data have been removed using a linear least squares best fit, and are measuredrelative to the mean value which corresponds to the zero of the ordinate.

3. Statistical analysis of dataHaving collected all these data, the challenge was then to extract something meaningful from them. Inattempting to evaluate f, as given by equation (3), the phenomenon of flow at the boundary must bestudied.

Energy lost in boundary shear in a fluid flowing in fully developed turbulent flow past a roughboundary is due to the formation and subsequent dissipation of eddies. The amount of energy lostdepends on at least two factors, these being the size of the eddies generated and the frequency atwhich they are shed. The size of eddy is directly dependent on the height of the roughness projectionfrom which the eddy is spawned, and the frequency of eddy generation is dependent on thelongitudinal spacing between consecutive roughness projections. Other factors such as shape andspatial distribution of the roughness elements will also have an effect on the amount of energy lost.Due to the random nature of the roughness pattern produced by TBM on rock, these other factors areassumed to have a common effect, with the major differences being due to the roughness height andspacing. It should, however, be noted that the spacing is a means of quantifying the number of eddiesspawned (per unit length) and it is this (ie the number of eddies) which is directly linked to theamount of energy lost due to friction at the boundary.

After identifying height and spacing as the two roughness characteristics having the dominantinfluence on energy loss due to boundary roughness in tunnels, meaningful analysis could be carriedout on the physical roughness data collected.

In this study, two different roughness height measurements were compared against each other.They were (i) the standard deviation, hs , and (ii) the mean range, hl. The spacing was measured bythe mean wavelengths of the surface, obtained via the power spectrum of the data. It is also possible torepresent physical roughness with a sinusoid of equivalent dimensions. These roughness descriptorsare defined and described in detail in Appendix 1.

4. Linking physical roughness to hydraulic resistance

For flow at high values of Reynolds Number, Re, the flow essentially skims over the roughness crests,causing the formation of cylindrical eddies in the space between the tops of the roughness elementsand the bed, with sizes varying according to the space available. In considering the energy lost tofriction at the boundary, it may be seen that it is the net volume (or area in a two-dimensionalrepresentation) of eddies generated which is the causing mechanism. It is evident that this eddyvolume is directly proportional to the heights of the roughness projections causing the turbulence.However, it would appear that the spacing of such projections has no bearing on the net eddy volume,provided Re is sufficiently high to cause wake interference flow (Morris, 1955). It is proposed thatfor surfaces of random roughness, the overall eddy generation is equivalent, by definition, to that overa sinusoid of equivalent dimensions. Note that hydraulic resistance in the context of wakeinterference flow is primarily by the mechanism of eddies being spawned in the hollows between

5protrusions and being transported into the flow where their energy is dissipated through mixing andviscosity. Consider the sinusoidal boundary shown in Figure 2.

6FIGURE 2: Flow at sinusoidal boundary.

Assume that the area between the bed and the tops of the roughness crests comprises eddies. Fortwo different random surfaces whose equivalent sinusoids have the same amplitude, it is seen that thenet eddy generation per unit length in the direction of flow is the same. Therefore it is argued thatspacing has no effect on the amount of energy consumed in the formation and subsequent dissipationof eddies formed at the bed. If this is true, then the equivalent sand grain diameter, k, may be taken tobe a function of the average or representative height of the surface roughness elements only. Otherfactors such as shape are disregarded due to the random nature of the tunnel roughness. Note that theabove only applies to flow at sufficiently high values of Reynolds Number.

5. Relationship between k and hLeCocq and Marin (1976) showed that sufficient accuracy is obtained if k is equated with averageroughness height at the boundary, h. (No indication of how this h was calculated or measured wasgiven in their paper). This, however, can only be true for high Re at which the flow exhibits wakeinterference flow, as outlined above.

Using the values of hs and hl calculated for surfaces of known hydraulic resistance, this postulatewas tested.

The first surface used for this test was that occurring within spun concrete pipes. From theliterature it was found that a representative value for Mannings n of such pipes is n = 0.010(Pennington, 1995). It is, however, suspected that this is a lower bound for the roughness and thatmany spun concrete pipes would exhibit slightly higher n-values.

The laser scanner was used to sample the physical roughnesses of such pipes, and from thesemeasurements values of hs and hl were calculated. Using these, values of friction factor, f , werecalculated using equation (2) with k = hs or hl, and compared with expected values. Since thediameters of pipes tested varied, it is more convenient to compare the corresponding calculated valuesfor Mannings n obtained in this way. A summary of these values is given in Table 2.

Table 2: Mean estimated Mannings n-values for concrete pipes.

method Mannings nA : Colebrook-White equation, k=hs 0.0117B : Colebrook-White equation, k=hl 0.0113Expected value from literature 0.0100

From Table 2 it is evident that use of either hs or hl for k in the Colebrook-White equation (2)yields comparable values for n, and these are similar to the expected value, found in the literature.

Heerman (1968) attempted to establish the link between physical roughness and hydraulicresistance. His methods were found to be inaccurate for the types of flow found in tunnels(Pennington, 1995), but some valuable information was available in his thesis. He included thephysical roughness dimensions of the pipes of varying roughness configurations which he tested,providing a source of roughness data for which the corresponding hydraulic resistances have beenmeasured.

7The different roughnesses were made by casting plaster in the form of sinusoids of varyingdimensions within 150mm outside diameter pipes.

For each roughness, Heerman made head loss measurements in flowing air at various values of Refrom which the friction factor, f, could be calculated. By analysis of generated sinusoidal roughnessdata corresponding exactly to the roughnesses tested, theoretical values for friction coefficients couldbe calculated (by the methods given above), and these were compared to the experimental results.Very good correlation of these was observed. In selecting the best k/h relationship, the sums of squaresof the deviations of theoretical from measured values were calculated.

From this it was found that using k= hl yields very similar results to those obtained using k= hs,with the former being marginally more accurate. It was also found that both of these yielded results ofan acceptable degree of accuracy (Pennington, 1995), and therefore the postulate of k=h can beapplied with confidence to physical roughness data taken from bored tunnels, when attempting tocalculate the corresponding hydraulic resistance parameters.

6. Application to tunnel roughness dataUsing both the standard deviation and the mean range as measures of roughness height, and by lettingthese heights be equivalent to k in equation (2), values of f and hence n for all sets of data taken in alltunnels sampled were calculated. These resulting n values are summarised in Figure 3 in whichmaxima and minima, together with the ranges one standard deviation either side of the mean valuesare shown.

7. Values of Mannings n from literatureAs part of the study, an extensive literature survey was conducted for information pertaining tohydraulic resistance in conduits, in particular bored water tunnels. From the limited amount ofliterature which was found on the resistance in bored tunnels, the following summary was compiled:

Unlined:

Stutsman (1988) : estimated from measurements n = 0.0159

Stutsman (1988) : calculated from head loss n = 0.0154

HDTC (1988) report n = 0.016

LeCocq and Marin (1976) : from measurements n = 0.016

Metcalf (in Pegram & Pennington, 1995) : from head losses n = 0.0161

Other:

HDTC (1988) report - shotcrete linings n = 0.016

Metcalf (in Pegram & Pennington, 1995) - shotcrete n = 0.015

HDTC (1988) report - cast in situ concrete n = 0.012

Metcalf (in Pegram & Pennington, 1995) - in situ concrete n = 0.012

From Figure 3 it is evident that, contrary to suggestions found in the literature, shotcrete isrougher than unlined rock. This ties in well with what has been observed from a physical inspection oftunnels and what may be evident from Figure 1. However, having said this, it is accepted that thenumber of shotcreted surfaces sampled is relatively small. Furthermore, shotcrete roughness may varyconsiderably with particular material constituents, and with the method of application.

8. Recommended values for Mannings n in bored tunnelsThe following emerge as the recommended values for Mannings n for use in the design of tunnelsbored by machine:

cast in situ concrete lining n = 0.0119 0.0009

unlined sandstone n = 0.0154 0.0010

unlined granite n = 0.0157 0.0008

8shotcrete n = 0.0161 0.0011

9Manning's n by Variancefor different surfaces

0.013

0.014

0.015

0.016

0.017

0.018

0.019

0.02

sandstone1 sandstone2 sandstone3 granite shotcrete1 shotcrete2

Surface

n

Manning's n by Mean Rangefor different surfaces

0.013

0.014

0.015

0.016

0.017

0.018

0.019

0.02

sandstone1 sandstone2 sandstone3 granite shotcrete1 shotcrete2

Surface

n

FIGURE 3: Values of Mannings n for tunnels.

The agreement between these values derived from physical measurements and those found in theliterature is encouraging. In consideration of the expected micro-roughness of bored tunnels, it issuggested that the above values for n be taken as representative. It should, however, be emphasisedhere that these n values only apply to the surface texture, and do not incorporate macro roughnesseffects (diameter / alignment changes, steps and other irregularities), as do commonly occur in boredtunnels (for example, when TBM cutters are changed).

In deciding on the applicability of the results of this investigation to shotcrete roughness, it issuggested that, at the least, visual comparison of the surfaces concerned is made. This is madepossible by the inclusion of photographs of every sampled section (shotcrete and unlined) beingincluded in the references, Pennington (1995) and Pegram & Pennington (1996). In addition, rawdata files (in ASCII format) of all physical roughness data sets used are included on diskette in thedocument by Pennington (1995).

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9. ReferencesColebrook, CF & White, CM (1937), Experiments with Fluid Friction in Roughened Pipes, Proceedings, Royal

Society of London, Vol.161, p.367-381.Heerman, DF (1968), Characterization of Hydraulic Roughness, Thesis submitted in partial fulfilment of

requirements for PhD, Colorado State University, Colorado.Highlands Delivery Tunnel Consultants (1988), Delivery Tunnel Design Contract TCTA- 01, Technical

Memorandum H2, Tunnel Roughness Report, Unpublished.LeCocq, R & Marin, G (1976), Evaluation des Pertes de Charges des Galleries DAmenee DEau Forees au

Tunnelier et Non-Revetues, Translation by J Capell (1994), Keeve Steyn Inc., Unpublished.Manning, R (1889), On the Flow of Water in Open Channels and Pipes, Transactions, Institution of Civil

Engineers of Ireland, Vol.20, p.161-207.Nikuradse, J (1933), Stromungsgesetze in Rauhen Rohren, Verein Deutscher Ingenieure, Forschungsheft,

Vol.361.Morris, HM (1955), Flow in Rough Conduits, Transactions, American Society of Civil Engineers, Vol.120,

p.373-410.Pegram, GGS & Pennington, MS (1996), A Method for Estimating the Hydraulic Roughness of Unlined Bored

Tunnels, Water Research Commission (WRC) Report No 579/1/96, Johannesburg, South Africa. ISBN No. 186845 219 0.

Pennington, MS (1995), Hydraulic Roughness of Bored Tunnels, Thesis submitted in fulfilment of requirementsfor MScEng, Department of Civil Engineering, University of Natal, Durban, South Africa.

Stutsman, RD (1988), TBM Tunnel Friction Factors for the Kerckhoff 2 Project, Water Power 87, Brian WClowes ed., p1710-1725.

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10. Appendix 1: Details of statistical analysis of roughness data

10.1 Standard deviation

In order to obtain a number representative of the average height of roughness projection for each dataset, the variance of the data was calculated. The square root of this, the standard deviation s, is alinear measure of the average deviation of the data from the mean. This deviation may be linked toamplitude by considering a sinusoid, the variance of which is given by

( )[ ]( )[ ]

s p l

lp l

l

2

2

0

2

2

12

2

=

=

=

var .sin /

.sin /

a x

a x dx

a

That is, standard deviation and amplitude are linked by

s =a

2The crest-to-trough height of a sinusoid is equal to twice the amplitude, so the average height ofroughness elements obtained via the standard deviation, hs, may be given by

h as s= =2 283.This measure of roughness height, hs, is easily calculated from the data sets.

10.2 Mean wavelength

In order to extract from the physical roughness data some measure of average or representativewavelength associated with the roughness, frequency analysis was used.

The power spectrum of a set of data, obtained via the Fourier Transform, shows how the varianceof the data is distributed with frequency. For each of the data sets obtained, the sample spectrum, asgiven by equation (6) below, was calculated and plotted.

( )CN

xeXX tt N

Nj tf pf=

=-

--D D

12

2

6...........................( )

where CXX(f) = sample spectrum

N = number of data points

t = length of series

D = sampling interval

It is worth noting that the integral of the sample spectrum gives the variance of the data set. Inorder to find the longitudinal spacing representative of the roughness, we developed the idea of thecentroidal frequency, fC, which is found by calculating the centroid of the sample spectrum. This isdone using:

( )fs

f f fC i XX ii

N

C==1 72

1

. . ................................( )D

which yields the mean wavelength lC as:

lfC C

=1

The above treatment indicates how a set of tunnel roughness data consisting of 2000 points may berepresented by one (or both) of two parameters, these being the average roughness height and thecentroidal wavelength, or spacing. It should be stressed that the height measure, hs, obtained via thevariance of a data set is particularly susceptible to outliers or trends in the physical data, because of

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the squaring of the variation of the individual deviations from the mean. As an alternative to hs,another representative roughness projection height was sought.

13

10.3 Mean range

Having already found the representative interval between roughness projections of a rough surface(i.e. lC), it was possible to calculate the variation in height within intervals of this length. Thissampled variation, called the range ri, is given in a specific interval (xi, xi+lc ) by the relation

[ ] [ ]r x x x xi i i i c i i i c= -+ +max , min ,l l

with i ranging from 1 to N-lC.

The mean range hl obtained for the data set as a whole is then given by

hN

rc

ii

N c

l

l

l=

- =

-

11

This measure of average or representative roughness height is a far more robust estimator than hs.It requires more computation, but this is not a problem with modern computing power. In well-behaved data sets (i.e. no trends or outliers), the values of hs and hl are very similar, in which case nogreater accuracy is afforded by either one.

As would be expected of data of the sort being dealt with here, the value of lC has a markedinfluence on the resulting value of hl. This is shown in Figure A1. In this figure, various values forthe mean wavelength, lC , were arbitrarily chosen and plotted against the corresponding resultingvalues of hl.

Effect of Wavelength on Mean range

0

1

2

3

4

5

6

7

8

9

0 50 100 150 200 250 300 350 400 450 500

wavelength (mm)

mean r

ange (

mm

)

FIGURE A1: Effect of lC on hl.

Figure A1 shows the dependence of hl on lC , and emphasises the importance of maintaining aconsistent method for the computation of lC. Even though power spectra of certain data sets mayexhibit certain dominant frequencies at which the variance is concentrated, calculation of lC via themean range as outlined above will be, on the whole, representative of the particular variancedistribution. Also evident from this figure is that hl tends to be proportional to the square root of lC.

10.4 Equivalent sinusoid

Knowing both the representative wavelength (spacing) and height (twice the amplitude), a set ofroughness data may be represented by an equivalent sinusoid of wavelength lC and amplitude equalto either hs/2 or hl/2.