Erosion and Sedimentation in the Pacific Rim (Proceedings of the Corva l l i s Symposium, August, 1987). IAHS Pub l . no. 165.

Modelling bed-load transport in steep mountain streams

J.G. WHITTAKER Ministry of Works and Development, Central Laboratories, Lower Hutt, New Zealand

ABSTRACT Steep mountain streams often exhibit a characteristic step-pool structure, which itself gives rise to the distinctive tumbling flow pattern. Bed-load transport processes in such streams are strongly linked to the bed configuration. Typically derived from limited input sites, bed load tends to move through the stream system as waves. Step-pool streams were modelled and clear water scour and sediment transport processes studied. Results were used to develop an equation for predicting sediment transport rates in step-pool streams. The applicability of these idealised model results to prototype step-pool streams is examined, showing that work is required on adapting the model to more correctly reflect field conditions.

NOTATION

d sediment size (m) h head driving the scour process (m) 1, distance from vertical through upstream step to maximum depth

of scour (m) 1 9 distance from vertical through upstream step to end of scour

c hole (m) n Mannings n q specific discharge (m3/s) q specific discharge at incipient motion conditions (m2/s) q, specific sediment bed-load transport rate (m2/s) B dummy variable D depth of scour below plane through top of steps, measured at

right angles to this plane (m) J slope J critical slope at incipient motion conditions L c r distance between steps (m) S vertical depth of scour hole (m) s vertical depth of flow in scour hole (m) a0 loss coefficient

319

320 J.G.Whittaker

8 angle of bed between end of scour hole and next step downstream, relative to horizontal (°)

Y angle of channel, relative to horizontal (°)

INTRODUCTION

Mountain streams with slopes greater than about 7%, particularly those with relatively low throughput rates of sediment, often exhibit a characteristic step-pool structure. Steps are formed from large bed elements, the size of which can be of the same order as the depth of flow, or even comparable to the width of the channel. The stream bed is typically armoured, and consequently particularly stable. The step-pool structure dictates the flow behaviour which accordingly has been designated tumbling flow. A significant amount of energy dissipation occurs in the pools due to turbulent mixing.

Sediment transport in step-pool streams has several characteristics. First, sediment for transport is derived from limited sites within the catchment. Second, once in the stream system, this sediment is stored in the pools when transport ceases. Once the bulk of this stored material has moved through a channel reach, the transport rate drops considerably until there is another input from one of the sediment production sites. Consequently, observed bed-load transport rates vary both spatially and temporally.

A variety of methods for predicting transport rates in steep mountain streams have been tried to date (see Whittaker, 1985), but with a marked lack of success. In fact, correct prediction of transport rates requires an understanding of the field flow and transport processes. Consequently, an investigation into the behaviour of step-pool streams was undertaken in which the steps were modelled by a succession of weirs. The intervening spaces were filled with sediment. Clear water scour and sediment transport processes were studied in this idealised system. The clear water scour results were used to develop a scour time development model which predicts sediment transport through a step-pool system for a given set of flow and sediment properties, and for given scour dimensions. The model was checked against the sediment transport results, and agreement found to be reasonably good. The model was then used to derive a loss coefficient, which used in conjunction with the Smart/Jaeggi sediment transport formula gives a generalised equation for predicting sediment transport in step-pool streams by allowing for energy lost to the transport process through turbulent mixing.

Despite the fact that the generalised model was developed with a step-pool model there is an immediate difficulty in applying the model to real mountain streams. First, prototype geomorphology is considerably more random than that of the weir-pool model. Second, armouring effects were not accounted for in the scour time development, yet these clearly limit scour depths in the pools of mountain streams. The time development model requires a reasonable estimate of not only the scour depths in the pools of a reach under consideration, but also of the maximum clear water

Modelling bed-load transport 321

scour depth for the particular flow rate. The aim of this paper is to describe briefly the development of the sediment transport model and to examine the difficulties in applying it to the field situation.

PHYSICAL MODELLING OF STEP-POOL STREAMS

The irregularity of step-pool streams can be simulated by a succession of discrete weirs (Rouse, 1965). The author performed a number of investigations with such models, three of which are reported here. The first two of these test series (series A and B) were performed in the Department of Agricultural Engineering, Lincoln College, New Zealand. The third series (C) was undertaken at the Laboratory of Hydraulics, Hydrology and Glaciology (VAW), ETH, Zurich, Switzerland.

The laboratory channel used for series A and B is shown in Whittaker (1982, 1985). This 10-m-long, tilting, recirculating flume was able to be adjusted to slopes of up to 0.248. Steps dimensioned 0.285 m high by 0.132 m wide by 0.033 m thick were placed at 0.5-m intervals to represent the steps in a step-pool system. The fields between the steps were filled with gravel (dg0 = 0.0049 m) whose size distribution is shown in Fig. 1.

In test series A clear water scour was investigated for various combinations of slope and flow rate. A definition sketch of the variables measured is presented in Fig. 2. In the derivation and analysis of the results presented here, interest was limited to those tests in which the scour hole did not develop to the extent of being strongly distorted by the step at the downstream end of the scour field. Where such strong distortion did occur, a flow instability was noted; these features of step-pool behaviour are described in Whittaker (1982, 1985, 1987).

For test series B, the scour dimensions defined in Fig. 2 were measured for various combinations of flow rate, slope, and sediment transport rate. Measurements were made when it had been determined that equilibrium transport conditions had been established.

The laboratory channel used for the series C tests is shown in Whittaker (1985) and Smart and Jaeggi (1983). This channel was 0.235 m deep, 0.2 m wide, and approximately 6 m long. Steps dimensioned 0.14 m by 0.2 m by 0.15 m were placed at 0.25-m intervals to simulate a step-pool system. Clear water scour tests were performed at slopes up to 0.254 for a number of flow rates, and with the two sediment mixtures described in Fig. 1. The scour dimensions defined in Fig. 2 were again measured. The results of test series A and C were analysed using multiple regression to evaluate a number of descriptive equations, viz:

S 1.4115 (q - q ) h

ç H _ (1) o , 0 7 2 3

d90

322 J.G.Whittaker

90 (7> C v, 60 V) O

° - 70

*; 60

50

to

30

20

10

V

/ II

tnl

| •« j

ri

il

f ,•

.'

f 1 /

i

1 !

1

1 f

f

Fig. 1 : Grain size distribution curves

12 16 20

Grain size d [mrn|

Fig. 2: Basic scour situation

Fig. 3: Loss coefficient a : Predicted (eqns 8 & 9) vs observed

" 0. 00 0. 20 0. 40 0. 60 0. B0

LOSS COEFFICIENT COBS)

Modelling bed-load transport 323

w h e r e q - [0.047 d (s - I) ] 1 ' 6 6 7 (2) where qcr , 16?

n J is the discharge at which material characterised by Mannings n begins to move at slope J, with

0.047 = critical Shields parameter

and n = 0.04168 d 0 - 1 6 7 (3)

(n n ^- 4 5 h°-59

S = 0.9121 (q " %r} n (4) . 0.27 d90

So/l2 = 0.3404 J0'7 (L/d 9 0)

0' 4 (5)

Equations 1, 4, and 5 were used to develop a general clear water scour model (Whittaker, 1986). The model is based on the assumption that the final scour state for a discharge q. has exactly the dimensions of an intermediate scour state for a larger flow rate q.. In other words, similarity of scour hole shape and position isJassumed for the time development of a scour hole (channel slope, and material size constant), even with a variable discharge.

The sediment transport rate from the scour hole can be calculated from the Smart/Jaeggi equation (specifically developed for steep slopes, see Smart, 1984; and Smart and Jaeggi, 1985). This equation is

H °-2 „ _ _ 4 / 9 0 \ „ ,0.6 n , > (6) % -TiTTj(d^) q J (J " V

in which , _/ [0.049 d (s-1)] 1' 6 6 7 ) ' (7) cr ^ nq

If the situation shown in Fig. 2 v5rresponds to the final scour shape for a specific discharge q., then q, = 0 because J = tan$ = J . However, if the1situation in Fig. 2 corresponds to the intermediate scour development state for a larger specific discharge q., then the transport out of the scour hole can be calculated nfrom equations 6 and 7, with 0 = tanB > J . This sediment transport rate is obviously less than that obtained by using J = tany, i.e., that occurring if the pools were filled to the plane through the top of the steps. It is possible to add into equation 6 a coefficient a which reduces the sediment transport rate from the maximum possible with J = tany to that corresponding to the appropriate state of development of the scour hole. In this sense a is a loss coefficient, characterising the flow energy lost to the transport process through turbulent mixing in the pools. An expression for a was derived in Whittaker (1986) as

324 J.G.Whittaker

a = e"D (8)

0 A0 ,0.24 ,0.11 0.22 ,n/n ,1.16 ,_, P _ 2.42 J L g (D/Dmax) (9)

„0.11 A 0.44 g d9o

[Note, J is the channel slope, i.e. tany] in which Dm,„ is given by max 3

D = [S + (h - 1, J L)] COSY (10)

max L v i /J i \ i in which lj = 0.452 12 (11) A comparison of equation 8, with the results of test series B

(Fig. 3) shows that the general model described the sediment transport phenomenon reasonably well, although a better fit with the observed data was given by

a . = 1.0065 a , + 0.0368 (12) obs pred v '

This affects smaller transport rates most, as is to be expected from equation 9: equation 9 tends to a finite limit of a with D = D , instead of a = 0.

Wiïft the equations listed above it should be possible to predict sediment transport in a prototype step-pool stream. The channel slope J must be measured, as must the specific discharge q, bed material size d, step spacing L, and scour depth D. D can be calculated. But considerable difficulties stand in tWix

way of using such a procedure to predict prototype mountain stream bed-load transport rates.

APPLICABILITY OF IDEALISED SCOUR/SEDIMENT TRANSPORT MODEL

The generalised predictive model described above is based on results obtained from an idealised representation of step-pool streams. Naturally, the applicability of the model for predicting field bed-load transport rates will depend inversely on the extent to which the prototype situation deviates from the idealisation.

The point to be considered is the difference of the actual geomorphology of step-pool streams to the idealised weir-pool model. Hayward (1978, 1980) notes that there are three distinct step types (Figs. 4 and 5):

(a) Boulder steps consist of a group of boulders arranged in a straight or curved line across the channel.

(b) Riffle steps are a collection of larger than average sized sediments that steepen the channel. Riffle steps may incorporate boulder steps (and so step types are in a sense hierarchical) and may occur at slopes of less than 0.05, thus falling in the range given by Bathurst et al.(1981) as being definitive of boulder-bed channels.

Modelling bed-load transport 325

Fig. 4: Schematization of step-pool morphology (after Hayward, 1980)

rock step

• • . • V > ï : ' ; - "

!" *>:<

• ^ • ¥ . • • " ; •

. ^ . ' • V --^'N,y: -Î"' • • • • • • • • / . . .

•.vjsv w

4*.\#£k

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Fig. 5(a): Boulder step

326 J.G.Whittaker

iM:M;*v $*^*m*w

Fig. 5(b): R i f f l e step

"Sh,

. * • ,

/, .v,:^v:r;k|-"

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Fig . 5(c) : Rock step

Modelling bed-load transport 327

(c) Rock steps are found where the channel is confined by bed-rock boundaries.

[Note: the presentation here does not cover steps and pools created by instream vegetative debris, although they function in the same manner.]

Given the range of physical characteristics of steps, an immediate difficulty is encountered in evaluating quantities such as step length L, stream width W (required to calculate the specific discharge q ) , and the channel slope J. There can be a significant effect of observer dependence in such an evaluation. For example, the author used the survey plans and field notes from Hayward's investigation of the Torlesse Stream in New Zealand (he was also familiar with the stream, having done a limited amount of field observation there subsequent to the completion of Hayward's work) to evaluate average step lengths versus channel slope for specified sub-reaches. The results (presented in Whittaker, 1985) differed from those of Hayward (1980, Fig. 86). In fact, while some observers have no difficulty in, discerning a step-pool pattern in a mountain stream, others see very little evidence of ordered structures (Mosley, 1986). It is recognised that step structure becomes better defined and more regular at steeper slopes. Mosley (1986) is correct in pointing out that well defined steps and pools appear to be correlated with low long-term sediment yield streams. The section of the Torlesse Stream experiencing the greatest continual inundation of sediment also exhibits the most poorly defined and least stable step-pool structures.

Further difficulties arise when the field scour hole shape is compared to that observed in the laboratory. As noted in the introduction, boulder and riffle steps are comprised of large bed elements. Although the bed material in the pools is finer, the whole bed structure tends to be extremely imbricated. 'Minor' step structures (e.g. boulder steps within a riffle step) may be rearranged by relatively frequent floods, but floods with return periods in excess of 50 years are required to restructure 'major' step structures. Whittaker and Jaeggi (1982) showed that steps and pools form under low return period events. While armouring limits the scour depth in the pools associated with these events, the large discharges create quite a flat slope between the end of the scour hole and the next step (see Fig. 6). These features become relics that are unchanged by normal flow conditions.-

The scour hole shape described above (and illustrated in Fig. 6) will of course be modified by sediment deposited in the pool during transport through the step-pool system. This bed-load material is usually considerably finer than the imbricated bed material, and is typically derived from active but isolated input sites. The input from such sites may be reasonably continuous as when the stream is continually under-cutting an unstable side slope. However, the input can be massive, and of very short duration (Ackroyd and Blakely, 1984). In some situations, sediment can even be supplied to the stream via avalanche thaw

328 J.G.Whittaker

Fig. 6: Typical field scour hole shape

(b)

Fig. 7: Possible sediment transport states

Modelling bed-load transport 329

deposits (Ackroyd, 1986). The bed-load material, once in the stream, is transported as a series of waves. These waves are only sometimes due to the step nature of input to the stream. The mechanics of flow and sediment transport through a step-pool system can themselves encourage the development of sediment waves, even with a constant sediment input rate (Whittaker and Davies, 1982; Whittaker, 1985), although the processes are not yet fully understood. Once the flow drops below the level required to move the stream's bed-load material, this is stored within the pools until re-activated by a freshet.

Noting the comments made in the two preceding paragraphs, a number of situations could be encountered when measuring D for the predictive model. (Note: D is calculated for the bed-load material, which is easily sampled). These situations are shown in Fig. 7.

(a) In Fig. 7a, D > D (q ). For this discharge, no transport will occur. max u

(b) In Fig. 7b, the flow has been able to move the bed material. If D = D (q), then there is no transport through the scour hole, m however, D < D (q), then there is transport from upstream into the scour hole, and the same transport rate from the scour hole into the next pool downstream. In this latter case, the predictive model will be able to be used. (For the former case, the structure of Eq. 9 will result in a finite predicted transport rate, rather than a zero value as required).

(c) In Fig. 7c, the D measured is that to the bed material. This could be < D (q»qu) for a range of q, values from zero to some finite limit q.-, . In this case, the measured D value will result in an incorrect finite value of transport rate, unless (D = D (q,q,,)).

As well as these difficulties in using the measured value of D in the predictive computation, D is a difficult quantity to measure in the field. Obviously field scour shapes can differ strongly from those observed in the laboratory. Only for high transport rates (relative to the discharge, as in Fig. 5b) is the field situation accurately simulated in the laboratory.

The next difficulty is still linked to the field step-pool geomorphology. Because of the structure of the steps, the flow plunges over a rather complicated boundary. This complicates the determination not only of q, but also of the head h. Further, scour in the pools will be reduced from two-dimensional values if the pool is wider (at the point at which the jet is actively scouring) than the step width. Moreover, before q can be determined, Q must be measured. There is of course no problem if a calibrated weir site is to hand. Practically, however, we must assume that such a facility is not available. Gaugings introduce their own errors, and the use of portable Parshall flumes, for example, may not be possible because of access problems.

330 J.G.Whittaker

It is clear that the actual geomorphology of step-pool streams deviates considerably from the weir pool laboratory model, and this presents difficulties in evaluating the physical quantities required for the sediment transport model.

Now, as opposed to the differences in the geomorphology of step-pool streams and laboratory models, non-steady flow does not pose a problem. The laboratory tests were undertaken with steady inputs of both water and sediment (the latter for test series B only). The prediction of the time development of scour with a hydrograph has been shown to be possible from such a laboratory data base using the steady-state assumption (Whittaker, 1985b). The generalised model described above supports the same hypothesis. Further, Whittaker (1985) showed that this approach yields realistic sediment wave motion through a step-pool reach, for a steady discharge, following an instantaneous bulk input of sediment to the system. The author has done no calculations involving unsteady flow and sediment conditions, but is confident that this could also be done from the steady-state assumption.

The last point to be considered is the generalised predictive model itself. Being based on a number of empirical equations, each possessing its own error, the model naturally has a built-in inaccuracy. Figure 3 shows that for low transport rates this can be of the order of 50%, although it is considerably less for higher transport rates. The combination of this error with those described above due to the non-similarity of the physical modelling of step-pool streams could result in predicted transport rates that are incorrect by an order of magnitude. Despite this, because realistic flow and transport processes are considered, calculated transport rates are probably considerably more accurate than those calculated by traditional techniques. For example, many researchers, noting the temporal and spatial variations in bed-load transport rates of orders of magnitude (even with constant flow conditions), have attempted to develop prediction models which have considered the watershed as a black box. Inputs to these models have been macroscopic factors such as climate and topography. Hayward (1980) pointed out the inappropriateness of some of these models, specifically those attempting to relate sediment transport to climate and relief. Another common but incorrect method for predicting bed-load transport rate is that of assuming it to be a certain percentage of the suspended sediment transport rate. Bed load can vary between a small percentage of the total transported sediment and up to ~90% (Whittaker, 1985). Griffiths (1980) developed a stochastic model for predicting bed-load yield from step-pool streams, but this requires calibration with flow records from a nearby instrumented catchment. While the method for predicting bed-load rates presented here requires a number of field measurements for application, it does not require calibration in the way that these alternative methods do. A further advantage is that instantaneous rates are predicted, rather than long term averages. The accuracy of the generalised scour model will be enhanced as work is conducted to reduce the difficulties described above, in par­ticular adapting the generalised model to reflect field conditions.

Modelling bed-load transport 331

CONCLUSIONS

An equation for predicting sediment transport in steep plane channels has been modified to predict sediment transport in steep step-pool streams. The modification is based on tests performed in a laboratory step-pool channel. Reasonable agreement was found between observed and predicted sediment transport rates.

Difficulties exist in applying the step-pool equation to prototype streams, large because of differences between the prototype stream's geornorphology and the structure of the laboratory idealisation. Despite the error that will result in field application of the equation, its field application offers distinct advantages, including greater accuracy than methods used to date.

REFERENCES

Ackroyd, P. (1986) Debris transport by Avalanche, Torlesse Range, New Zealand. Zeitschrift fur Géomorphologie N.F., Bd 30, Heft 1.

Ackroyd, P. & Blakely, R.J. (1984) En masse debris transport in a mountain stream. Earth Surface Processes and Landforms, Vol 9, 307-320.

Bathurst, J.C., Li, R.M. & Simons, D.B. (1981) Resistance equation for large scale roughness. J Hyd Div ASCE, Vol 107, HY 12, 1593-1613.

Griffiths, G.A. (1980) Stochastic estimation of bedload yield in pool and riffle mountain streams. Water Resources Research, Vol 16, No. 5, 931-937.

Hayward, J.A. (1978) Hydrology and stream sediments in a mountain catchment, PhD thesis (Vols 1-3), University of Canterbury, Christchurch, New Zealand.

Hayward, J.A. (1980) Hydrology and stream sediments from Torlesse Stream catchment. Tussock Grasslands and Mountain Lands Institute, Lincoln College, (NZ), Special Publication No. 17.

Mosley, M.P.(1986) Discussion of 'sediment transport in step-pool streams' by J.G. Whittaker, presented at 2nd International Gravel Rivers Workshop, Pingree Park, Colorado, 1985.

Rouse, H. (1965) Critical analysis of open channel resistance. J Hyd Div ASCE, Vol 91, No. HY4, 1-25.

Smart, G.M. (1984) Sediment transport formula for steep channels. J Hyd Engrg, Vol 110, No. 3, 267-276.

Smart, G.M. & Jaeggi, M.N.R. (1983) Sediment transport on steep slopes. Mitteilung Nr 64 der VAW, ETH, Zurich.

Whittaker, J.G. (1982) Flow and sediment movement in stepped channels. PhD thesis, University of Canterbury (Lincoln College), New Zealand.

Whittaker, J.G. (1985) Sediment transport in step-pool streams. In 2nd International Gravel Rivers Workshop, Pingree Park, Colorado.

Whittaker, J.G. (1985b) Local scour at steps in mountain rivers in floods. In 2nd International Conf on Floods and Flood Control, Cambridge.

332 J.G.Whittaker

Whittaker, J.G. (1986) An equation for predicting bed-load transport in steep mountain step-pool stream. In 9th Australasian Fluid Mechanics Conf., Auckland, NZ.

Whittaker, J.G. (1987) Design and behaviour of stepped channels. Mitteilung der VAW, ETH, Zurich, (in preparation).

Whittaker, J.G. and Davies, T.R.H. (1982) Erosion and sediment transport processes in step-pool torrents. In Recent Developments in the explanation and prediction of erosion and sediment yield, Proc Exeter IAHS Sym, IAHS Pub No 137.

Whittaker, J.G. & Jaeggi, M.N.R. (1982) Origin of step-pool systems in mountain streams, J Hyd Div ASCE, Vol 108, No. HY6, 758-773.

ACKNOWLEDGMENTS

This paper is published with permission of the Commissioner of Works.