PREDICTION OF BED-LoAD TRANSPORT BY DESERT FLASH FLOODS

By Ian Reid/ D. Mark Powell/ and Jonathan B. Laronne3

ABSTRACT: A number of predictive bed-load sediment transport equations are rated against a unique set offield data collected by automatic slot samplers during flash floods in a desert wadi. The Meyer-Peter and MUllerequation is shown to perform well, providing a median ratio of calculated to observed (C/O) bed-load flux of1.18. The Bagnold equation is shown to underpredict considerably, with a median C/O of 0.44. The Parkerequation performs better, though it still underpredicts with a median C/O of 0.78. The apparent success of theMeyer-Peter and MUller formula is attributed to the ready supply of sediment to the channel system in desertand semidesert environments. This ensures that the channel bed remains unarmored, in contrast to supply-limited,armored, perennial rivers of humid zones. It also ensures that bed-load flux responds to changing hydraulicconditions in a comparatively simple fashion.

INTRODUCTION

Establishing field-based flux for bed-load transport is ex­tremely labor-intensive and is often logistically impossible.This has led to the development of a large number of predic­tive formulas, each of which is usually shown to perform rea­sonably well when rated against a particular transport data set[e.g., Parker et al. (1982)]. However, engineers and geomor­phologists remain sceptical about the general application ofsuch formulas (Carson and Griffiths 1987). Problems ariselargely because each predictive equation requires an adequatedescription of the channel-bed material, yet bed sediments arecomplex and variable in nature. As a result, there is often greatdifficulty in characterizing them appropriately for modelingpurposes. This problem is then compounded by the compli­cated and often unpredictable interaction between the flow andthe sediments that make up the channel bed (Maddock 1970).

Gomez and Church (1989) provided a valuable assessmentof a large number of bed-load formulas. Their analysis givesan indication of the relative merits of existing transport for­mulas, at least when they are applied to armored gravel-bedchannels with perennial or snowmelt runoff regimes. However,the stricture placed on the choice of the field data againstwhich each formula was rated inevitably means that severalquestions remain unanswered. In particular, there is consider­able interest in predicting bed load where flows are unsteady.This is the case in semiarid areas where problems associatedwith river sedimentation are often magnified (Vanoni 1975).Indeed, it is especially in this environmental setting that itwould be beneficial to know whether tolerable estimates oftransport rates could be derived.

The installation of the Birkbeck-type automatic samplers(Reid et al. 1980) on channels in the northern Negev Desert,Israel, has provided a unique data set that describes the bedload carried by flash floods in ephemeral gravel-bed streams(Laronne and Reid 1993), and has given an opportunity toassess the performance of a selection of predictive bed-loadequations in this environmental setting.

FIELD DATA

The Yatir is a small fourth-order tributary of the Nahal Be­sor and drains part of the southern flanks of the Hebron Moun-

'Prof.• Dept. of Geography. Loughborough Univ. of Techno\.. Lough­borough, LE11 3TU, U.K.

'Res. Ofcr.• Dept. of Geography, Loughborough Univ. of Techno\..Loughborough, LE11 3TU, U.K.

'Assoc. Prof.• Dept. of Geography and Envir. Deve\., Ben Gurion Univ.of the Negev, Beer Sheva, 84105, Israe\.

Note. Discussion open until August I, 1996. To extend the closingdate one month. a written request must be filed with the ASCE Managerof Journals. The manuscript of this technical note was submitted for re­view and possible publication on May 23, 1994. This technical note ispart of the Journal of Hydraulic Engineering. Vo\. 122, No.3, March,1996. ©ASCE, ISSN 0733-9429/96/0003-0170-0173/$4.00 + $.50 perpage. Technical Note No. 8503.

170/ JOURNAL OF HYDRAULIC ENGINEERING / MARCH 1996

tains. Annual rainfall of the region averages between 220 and280 mm and it can be expected to produce about five or sixflash floods whose time of rise is typically less than 10 min,and whose total duration is typically only a few hours. Thechannel is roughly rectangular in cross section, with an av­erage bed width of 3.5 m and banks that are 0.9 m high. Thebed is alluvial and consists of planar "flats," with are inter­rupted by slightly steeper and coarser bars. The average lon­gitudinal bed slope is 0.0087. There is no armor-layer devel­opment in the bed material, in strong contrast with mostperennial gravel-bed rivers (Laronne et al. 1994). This reflectsthe abundant supply of sediment that is contributed by thesparsely vegetated water catchment. It is a characteristic thatis shared with other channels where sediment is readily movedinto the stream network from adjacent hillslopes (Buffingtonet al. 1992; Dietrich et al. 1989; Lisle and Madej 1992). Be­cause of this abundant supply of sediment, the surface bedmaterial of the Yatir is comparatively fine. In the channel flats,the median grain size (Dso ) of the surface layer is 6 mm. whilein the channel bars it is 20 Mm.

The monitoring station that was installed on the Nahal Yatirhas been described fully elsewhere (Laronne et al. 1992).Briefly, it consisted of three Birkbeck-type slot samplers thatwere set side by side across the stream. Each sampler actedindependently. Each automatically and continuously weighedthe bed load that fell through its horizontal slot into an un­derlying container. The water stage was measured simultane­ously at two locations in order to provide a continuous mea­sure of the water-surface slope in the station approach reach.This varying measure of water-surface slope has been used inconjunction with values of the hydraulic radius and fluid den­sity that take account of high suspended sediment concentra­tions to rate bed-load transport against contemporary channel­average shear stress. No sidewall corrections have beenapplied. Grain-size distributions of the bed load were derivedafter layer-sampling the sediment that filled each sampler. Be­cause a Birkbeck-type sampler continuously monitors the ac­cumulation of the bed load, each of these layers can be as­signed to a narrow time slice, and contemporary hydraulicconditions can be described.

Bed-load discharge on the Yatir can be shown to respondto changes in shear exerted by the flow in a comparativelyuncomplicated manner, reflecting the ready availability of bedmaterial and the lack of armoring (Laronne and Reid 1993).This is in strong contrast to the behavior of most perennialcounterparts, where sediment is usually supply-limited; finer­grained material is selectively entrained, thus leaving the bedsurface armored; and the armor layer is usually studded withmicroforms such as pebble clusters that increase both flowresistance and bed strength, thereby complicating the relationsbetween sediment transport and hydraulics (Laronne andCarson 1976; Reid et al. 1992; Reid and Laronne 1995).

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10(c)

10

(b)(a)

10

'eIfBet:

~ MPM ! 0

~ 0.1 D5O _!i (.-MPMD.0.01 '--~'--L..1""""'~-,----~~

1 10 100Shear stress, N m-2

FIG. 1. (a) Bed-Load Flux Observed In Ephemeral Gravel­Bedded Nahal Yatlr, Israel, as Function of Contemporary BedShear Stress and Selected Bed-Load Functions; (b) Yatlr Bed­Load Flux for Material >2 mm and Bed-Load Function of Parker(1990); (c) Yatlr Bed-Load Flux and Bed-Load Function of Meyer­Peter and Muller (1948), with OlIO =6 mm and Om =8 mm

,'"'e

OIl..:.:

~8-'"g 0.1 0.1

]:a MPM~ Dm

O.ol 0.01 '--~'--'--' .........~~'--'--' ............1 10 100 1 10 100

Shear stress, N m-2 Shear stress, N m-2

bears some resemblance to the homogenized materials thatformed the beds of the laboratory flumes.

In the light of the positive conclusions of Gomez andChurch (1989) about Bagnold's (1980) equation, its perfor­mance is disappointing when rated against the Yatir data.However, as discussed elsewhere (Laronne and Reid 1993;Reid and Laronne 1995), this northern Negev ephemeralstream appears to be more efficient at moving bed load thanthe upper limit that was intuitively ascribed to streams byBagnold.

The serious underprediction by the Parker et al. (1982)equation [Fig. l(a)] is more easily accounted for. The size­distribution of bed load in Oak Creek, a small, well-armored,perennial stream in Oregon that provided the data againstwhich the equation was tested, had suggested strongly that thesubarmor layer was the chief source of bed load, either be­cause of a partial or complete breaching of the armor or be­cause of an active exchange between the armor and subarmorlayers once entrainment of the bed material had commenced.Because of this, their equation uses the subarmor-layer sizedistribution to characterize the bed material. However, in theYatir, as in other streams of the Negev, there is a lack ofvertical layering. Indeed, in the channel flats of the Yatir, thesubsurface material is marginally coarser (Dso = 10 mm). Inusing the Parker et a1. (1982) formula, the predicted bed-loadflux rates are, therefore, based on this coarser component ofthe bed material and, as a result, they are underestimates. In­deed, a comparison of bed-load and bed-material size distri­butions in the Yatir points positively to the surface material asthe source of mobilized sediment (Laronne et al. 1994). It canbe shown that the D so of the bed load hovers narrowly between

JOURNAL OF HYDRAULIC ENGINEERING / MARCH 1996/171

RESULTS AND ANALYSIS

Five bed-load equations have been rated against theNahal Yatir field data that were collected during the winter of1990-91. They have been chosen either because they are incommon use by practicing engineers and geomorphologists orbecause they have played a recent and important part in thecontinuing controversy that surrounds the nature and predic­tion of bed-load transport in gravel-bed streams. Full detailsof the derivation and application of the selected bed-load func­tions can be found in the original references and in subsequentreviews [e.g., Shulits and Hill (1968), Yalin (1972), Whiteet a1. (1973), Gomez and Church (1989)], although the readershould be on guard against inadvertent errors.

Except in the case of Parker (1990), the bed material grain­size distributions that have been used to compute the bed-loadfunctions are truncated at 1 mm. These size distributions arethose of the channel flats. This follows from an analysis thatindicates the flats to be the chief source of material, at leastover the range of shear stress for which there is bed-load data(Laronne et a1. 1994). However, there is some evidence thatelements of the coarser-grained channel bars are mobilized atand (by implication) beyond the highest measured values ofshear stress. As a consequence, any conclusions drawn fromthis present comparison can relate only to flows that fall withinthe measured range.

Fig. 1 indicates the Yatir's channel-average bed-load re­sponse to changing channel-average shear stress. Transportrates are given as the mass of sediment passing through unitchannel width in unit time. The values are 1.646 times higherthan those given in Laronne and Reid (1993) and Reid andLaronne (1995), where the effects of high concentrations ofsuspended sediment on fluid density are incorporated in abuoyancy factor that has been applied to the data. Superim­posed on the scattergrams are the bed-load functions of Meyer­Peter and MUller (1948), Parker (1979, 1990), Bagnold (1980),and Parker et a1. (1982). Parker's (1990) equation is givenseparately because it requires a truncation of the bed materialand bed-load size distributions at 2 mm. The curves of thebed-load functions have been extrapolated downward to em­phasise the abrupt rise in the bed-load flux that is anticipatedfollowing general entrainment. However, this should not en­courage a belief that the empirical field data would lie anylower. Indeed, as pointed out elsewhere (Reid and Laronne1995), bed-load transport rates in the Yatir start at compara­tively high values, values that are much higher than in armoredperennial rivers, where the initial phase of transport may in­volve only small amounts of relatively fine material movingin and around an intact armor layer (Emmett 1976; Bathurst1987).

Meyer-Peter and MUller's (1948) equation fits best withinthe limited scatter of the field data [Fig. l(a)], in completecontrast to the pattern established by Gomez and Church(1989), whose test data came from coarser-grained, armored,perennial rivers. The curve representing Parker's (1979) equa­tion also passes through the Yatir data points, although not asconvincingly as that of Meyer-Peter and MUller (1948). Boththese equations were founded on laboratory flume data, whichwere derived using sediments that had calibers not so far re­moved from that of the main source of bed load in the Yatiri.e., the channel flat immediately upstream of the slot samplers(Laronne et a1. 1994). Although the bed of the Yatir is not"underloose" [sensu Church (1978)]-as would have beenthe case with the flume studies-and may, therefore, have aslightly higher entrainment threshold, the similarity of grainsize may be one reason for the apparent success of the equa­tions. However, perhaps more important is that the nonar­mored nature of the Yatir channel bed means that its fabric

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Key nold (1980) and Parker (1990), among others. Part of this suc­cess undoubtedly stems from the fact that sediment is not sup­ply-limited by channel-bed armor development, in contrast tomost perennial rivers, ultimately reflecting the ease with whichmaterial is transferred to the channel system by hillslope pro­cesses in desert settings. Bed-load transport responds fairlysimply to changes in the hydraulic environment, again in con­trast to armored perennial rivers. As a consequence, predictedtransport rates can be shown to follow observed rates quiteclosely, even at measuring intervals as small as 1 min duringflash floods in which hydraulic conditions change very rapidly.

APPENDIX I. REFERENCES

The Negev Bedload Sediment Monitoring Programme has been sup­ported by funds from The Israel Academy of Sciences, The Israel Hy­drological Service, The Natural Environment Research Council of theUK, The British Council, The Central Research Fund of the Universityof London and the British Geomorphological Research Group. We areindebted to Yitshak Yitshak and Sureen Leizerovitch for assistance in thefield. We thank Lev Meerovich for reading an early draft. We also thankfive anonymous referees and the editor, Jacob Odgaard, for their helpfulcomments.

ACKNOWLEDGMENTS

Maximum

Minimum

Upper quartile

Median

Lower quartile

4

+- - - - - _:- - -- -~- --$-----~---OL----- --"'=--_L--....L-

3

1

8 2

6 and 7 mm as the shear stress ranges from 10 to 40 N· m-2,

and this conforms with the channel flat surface bed materialD,o of6 mm.

The gross underestimation by the Parker et al. (1982) equa­tion is partially corrected by the reformulation in Parker(1990). This acknowledges the significance of the surface sizedistribution in controlling the bed load. However, the resultingcurve still lies along the right-hand edge of the field data formost of its length and appears to be too steep [Fig. l(b»).

It would seem that the Meyer-Peter and MUller (1948) equa­tion provides the best prediction of bed load when ratedagainst observed values in the Nahal Yatir (Fig. 2). The me­dian value of the ratio between calculated (C) and observed(0) flux is 1.18. This compares with a value of 1.47 for theempirical function of Parker (1979), 0.78 for Parker (1990),0.44 for Bagnold (1980), and 0.02 for Parker et al. (1982).However, Meyer-Peter and MUller's equation is extremely sen­sitive to the particle size chosen to represent the bed material,as are the other bed-load functions. Meyer-Peter and MUllerdefined an appropriate bed-material size-distribution parame­ter, Dm = ~7.1 /;D j , in which /; is the proportion of ith-sizefraction in the surface grain-size distribution, and D; is themean grain size of the ith fraction. But, data are rarely givenin a form that allows the computation of Dm , while D50 isusually always given or it is easy to derive. Because of this,it is often substituted for Dm • Without entering into a full sen­sitivity analysis, it might be useful to see the effect of choosingD 50 rather than Dm when applying the equation in the Yatir.Fig. l(c) shows the expected leftward shift in the Meyer-Peterand MUller (1948) curve that results from the adoption of D so=6 mm as opposed to Dm =D68 =8 mm. The median valueof C/O shifts adversely to 1.47. Self-evidently, there is somebenefit to the use of Dm , which produces a median value ofC/O = 1.18 that is much more acceptable, especially if engi­neering design criteria are under consideration.

SUMMARY

A comparison of predicted bed-load sediment transport withfield data collected by automatic slot samplers in an ephemeralgravel-bed channel indicates that the Meyer-Peter and MUller(1948) equation performs well, and better than those of Bag-

172/ JOURNAL OF HYDRAULIC ENGINEERING / MARCH 1996

Bagnold, R. A. (1980). "An empirical correlation of bedload transportrates in flumes and natural rivers." Proc.• Royal Soc. London, SeriesA, London, England, Vol. 372, 453-473.

Bathurst, J. C. (1987). "Modelling and measuring sediment transport inchannels with coarse bed material." River channels: environment andprocess, K. S. Richards, ed., Blackwell Scientific Publ. Ltd., Oxford,England,272-294.

Buffington, J. M., Dietrich, W. E., and Kirchner, J. W. (1992). "Frictionangle measurements on a naturally formed gravel streambed: implica­tions for critical boundary shear stress." Water Resour. Res., Vol. 28,411-425.

Carson, M. A., and Griffiths, G. A. (1987). "Bedload transport in gravel­bed channels." J. Hydro., Wellington, New Zealand, Vol. 26, 1-151.

Church, M. (1978). "Palaeohydrological reconstructions from a Holocenevalley fill." Fluvial sedimentology. Can. Soc. of Pet. Geologists Mem­oir; No.5, A. D. Miall, ed., Can. Soc. of Pet. Geologists, Calgary,Alta.• Canada. 743-772.

Dietrich, W. E., Kirchner, J. W., Ikeda, H., and Iseya, F. (1989) "Sedi­ment supply and the development of the coarse surface layer in gravel­bedded rivers." Nature, Vol. 340,215-217.

Emmett, W. W. (1976). "Bedload transport in two large gravel-bed rivers,Idaho and Washington." Proc.• 3rd Federal Inter-Agency Sedimenta­tion Conf, 4.101-4.114.

Gomez, B., and Church, M. (1989). "An assessment of bed load sedimenttransport formulae for gravel bed rivers." Water Resour. Res., Vol. 25,1161-1186.

Laronne, J. B., and Carson, M. A. (1976). "Interrelationships betweenbed morphology and bed material transport for a small gravel-bedchannel." Sedimentology, Vol. 23, 67-85.

Laronne, J. B., and Reid, I. (1993). "Very high rates of bedload sedimenttransport by ephemeral desert rivers," Nature, Vol. 366, 148-150.

Laronne, J. B., Reid, I., Yitshak, Y., and Frostick, L. E. (1992). "Re­cording bedload discharge in a semiarid channel, Nahal Yatir, Israel."Erosion and sediment tTransport monitoring programmes in river ba­sins; Publ. N. 210, 1. Bogen, D. E. Walling, and T. J. Day, eds., Int.Assoc. of Hydrol. Sci. (IA), Oslo, Norway. 79-86.

Laronne, J. B., Reid, I., Yitshak, Y., and Frostick, L. E. (1994). "Thenon-layering of gravel streambeds under ephemeral flood regimes." J.Hydro., Amsterdam, The Netherlands, Vol. 159, 353-363.

Lisle, T. E., and Madej, M. A. (1992). "Spatial variation in armouringin a channel with high sediment supply." Dynamics of gravel-bed riv­ers, P. Billi, R. D. Hey, C. R. Thorne, and P. Tacconi, eds.• John Wiley& Sons, Ltd., Chichester. England, 277 -291.

Maddock, T. (1970). "Indeterminate hydraulics of alluvial channels." J.Hydr. Div., ASCE, Vol. 96. 2309-2323.

Meyer-Peter, E., and MUller, R. (1948). "Formulas for bedload trans­port." Proc.• Int. Assoc. of Hydr. Struct. Res., 39-64.

Parker, G. (1979).• 'Hydraulic geometry of active gravel rivers." J. Hydr.Div., ASCE, Vol. 105,1185-1201.

Parker, G. (1990). "Surface-based bedload transport relation for gravelrivers." J. Hydr. Res., Vol. 28, 417-436.

Parker. G., Klingeman, P. C., and McLean, D. G. (1982). "Bedload and

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LE

A U

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opyr

ight

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E. F

or p

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ved.

size distribution in paved gravel-bed streams." J. Hydr. Div., ASCE,Vol. 108, 544 57 I.

Reid, I., and Laronne, J. B. (1995). "Bedload sediment transport in anephemeral stream and a comparison with seasonal and perennial coun­terparts." Water Resour. Res., Vol. 31, 773-78I.

Reid, I., Layman, J. T., and Frostick, L. E. (1980). "The continuousmeasurement of bedload discharge." J. Hydr. Res., Vol. 18,243-249.

Reid, I., Frostick, L. E., and Brayshaw, A. C. (1992). "Microform rough­ness elements and the selective entrainment and entrapment of particlesin gravel-bed rivers." Dynamics of gravel-bed rivers, P. BiIli, R. D.Hey, C. R. Thome, and P. Tacconi, eds., John Wiley & Sons, Ltd.,Chichester, England, 253-266.

Shulits, S., and Hill, R. D. (1968). "Bedload formulas." Rep. ARS-SCW­I, Agric. Res. Service, U.S. Dept. of Agr., Washington, D.C.

Vanoni, V. A. (1975). Sedimentation engineering. American Society ofCivil Engineers manual on sedimentation. ASCE, New York, N.Y.

White, W. R., MiIIi, H., and Crabbe, A. D. (1973). "Sediment transport;and appraisal of available methods." Rep. 119, U.K. Hydr. Res. Sta­tion, Wallingford, England.

Yalin, M. S. (1972). Mechanics of sediment transport. Pergamon Press,Inc., Elmsford, N.Y.

APPENDIX II. NOTATION

The following symbols are used in this paper:

c/o = ratio of calculated (C) to observed (0) unit bed­load flux;

V{ = mean grain size of ith-size fraction (mm);V m = characteristic grain size of the surface bed material

(mm);V x = grain size at the xth percentile of a size distribution

(mm);it = proportion of the ith-size fraction in the surface

bed-material grain-size distribution;MPM V m =Meyer-Peter and Muller (1948) bed-load function

deploying characteristic surface bed-material grainsize; and

MPM V~o = Meyer-Peter and Muller (1948) bed-load functiondeploying surface bed-material median grain size.

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