Comparison of bed load transport in torrents

Transcription

1 WATER RESOURCES RESEARCH, VOL. 37, NO. 12, PAGES , DECEMBER 2001 Comparison of bed load transport in torrents and gravel bed streams Dieter Rickenmann Department of Water, Soil and Rock Movements, Swiss Federal Research Institute WSL Birmensdorf, Switzerland Abstract. Data are compiled on bed load transport and discharge in gravel bed rivers and torrents with bed slopes up to The transport characteristics of 19 streams are compared with simple bed load transport equations and with measurements from flume experiments. In simple form the bed load transport is a function of an "effective" discharge times a bed slope factor. The efficiency of the streams in transporting bed load is defined as the deviation from the transport function. It varies over several orders of magnitude, particularly for smaller and steeper streams, whereas flume experiments and larger streams are in better agreement with the simple bed load transport relation. The large variation and the strong decrease in efficiency in smaller streams appears to be associated with substantial form losses for relative flow depths (defined by h/d9o ) smaller than Introduction critical conditions for beginning of motion [e.g., Komar and Shih, 1992; Wathen et at., 1995]. Many bed load transport formulas for gravel bed rivers were developed or calibrated on the basis of experiments in laboratory flumes. Measured transport rates in natural streams may differ by several orders of magnitude from values predicted with such formulas [e.g., Bathurst et at., 1987; Gomez and Church, 1989]. Also, measured transport rates in gravel bed rivers and boulder bed streams may vary by several orders of magnitude at similar (mean) flow conditions [e.g., Bathurst et at., 1987; Gomez, 1987; Reid and Laronne, 1995; Hegg and Instantaneous bed load transport rates generally do not vary continuously with changing flow rate but show a pulsing behavior; furthermore, different transport rates are typically observed between the rising and falling limb of a flood hydrograph [Bathurst, 1987; Gomez, 1987; Rickenmann et at., 1998]. Bed load discharge usually makes up less than ---25% of total sediment load in lowland (sand bed) streams, whereas this proportion is highly variable in upland or boulder bed mountain streams and may be higher than 75% [Bathurst, 1987; Rickenmann, 1999]. Gomez, 1987]. In very steep streams and at high transport Gravel bed and boulder bed streams (in the sense of intensities (e.g., "immature" debris flow conditions) the dis- Bathurst [1987]) are characterized by a wide range of sediment tinction between bed load and suspended load transport besizes and temporally and spatially variable sediment sources. comes blurred [Takahashi, 1987; Rickenmann, 1990]. Reasons for the large variability of observed bed load trans- These two features clearly distinguish them from sand bed streams. Channels with coarse bed materials are often characport rates in coarse gravel bed rivers can be summarized as follows [Bathurst, 1987]: (1) Breakup of an "armor" layer near terized by morphological units such as pools, riffles, rapids, the time of peak flow can lead to a hysteresis loop during a cascades, and isolated steps [Grant et at., 1990; Woht, 2000]. flood hydrograph; (2) seasonal and interstorm variations can The occurrence of these units appears to be slope-dependent result from depletion of the sediment supply, for example, [Grant et at., 1990]. In Europe the term "torrent" is used for during summer; (3) even at a constant water discharge, unhigh-gradient streams steeper than steadiness of bed load transport may be enhanced by sporadic It appears that in coarse gravel bed streams the size distri- breakup of clusters of sediment particles or bed forms; and (4) bution of the bed load approaches that of the bed material only in-channel storage volumes may delay or suppress sediment for higher flow intensities [Bathurst, 1987; Liste, 1995; Rickenmann eta!., 1998; Lenzi et at., 1999]. In the Rio Cordon, a steep Italian stream with an average bed slope of 0.15, the mean grain size of the transported material approaches that of the bed material at about 3 times the critical discharge for beginning of bed load motion [Lenzi et at., 1999]. Similarly, Kuhnte [1992] found from measurements in Goodwin Creek that equal mobility is approached only when the shear stress is waves due to external inputs, particularly where storage volumes are a significant proportion of average annual sediment output. For boulder bed channels, which are typical for torrents and mountain streams steeper than a few percent, a two-phase model as described by Beschta [1987] and Bathurst [1987] can provide a conceptual framework for the description of sediment transport behavior. In "phase 1 transport," there is batimes the critical shear stress for initiation of motion. Other sically no destruction of the coarse surface layer, and the transfield observations in gravel bed streams suggesthat equal ported sediment is generally made up of the finer fractions mobility is reached at discharges which are 2-5 times the (sand and fine gravel). Transport rates largely depend on fine material available in pools and behind channel obstructions or Copyright 2001 by the American Geophysical Union. Paper number 2001WR /01/2001WR on external sediment supply which can vary by orders of magnitude. In "phase 2 transport" the surface or armor layer is broken up, and there is essentially equal mobility of all size 3295

2 3296 RICKENMANN: BED LOAD TRANSPORT IN GRAVEL BED STREAMS fractions; that is, the size distribution of the bed load is similar dimensionless critical shear stress at initiation of bed load to that of the subsurface material. Several studies have applied different bed load transport formulas to field measurements in gravel bed rivers and have tried to assess their performance. For example, Gomez and Church [1989] recommended the BagnoM [1980, 1986] equation to estimate the magnitude of bed load transport when only limited hydraulic information is available. They also found that Schoklitsch's [1934] formula gave reasonable predictions. Bathurst et al. [1987] compared six bed load transport formulas with field and flume data at bed slopes up to 0.09 and found that the best predictions were given by Schoklitsch's [1962] equation. The intent of this study is to examine how transported bed load volumes vary for streams with a wide range of channel gradients, including particularly steep torrents with bed slopes above For many of these streams, only limited data exist on bed load and flow characteristics. In steep streams with coarse bed material and an irregular channel geometry it is difficult to define an appropriate flow depth which could be transport, h is the flow depth, S is the channel bed (or water surface) slope, and Fr is the Froude number equal to v/(17h) ø'5, where v is the mean fluid velocity. In terms of predicted versus measured qt, values the correlation coefficient is r = 0.98 and standard error of estimate is Se = 35 %. In the analysis leading to (1), the measured flow depths were corrected for sidewall influence. In the steep flume experiments, sedimentransport intensities were high, and flow depths used in development of (1) refer to the flowing mixture of water and transported particles. This mixture flow depth was up to 100% greater than the calculated (fictitious) clear water depth for the fluid (water) only. With regard to the Meyer-Peter and MMler [1948] experiments, total bed shear stress (including both grain and bed form resistance) was used in the development of (1). To compare (1) with other formulas, the exponent of the Froude number is set to 1.0 instead of 1.1, which results in a deviation of <10% for the range 0.35 < Fr < 2.6. For a uniform sediment mixture the factor (d9o/d3o) ø'2 may be approximated by 1.05 [Smart and Jaeggi, 1983]. The factor (s - used in shear stress-based bed load transport calculations. 1)-ø'5 has an exponent different from other similar formulas Furthermore, flow depths may be available for an artificial channel reach at the observation site only; they are seldom measured in natural channel reaches. Therefore, in this study, only discharge and bed slope are used as parameters to characterize flow hydraulics. Observations on bed load transport in steep experimental flumes are taken as a reference condition, but is in agreement with another study varying the solid to fluid density ratio s [Low, 1989], as well as with the coal and barite grain experiments of Meyer-Peter and MMler [1948] and the steep flume experiments with a varying fluid density [Rickenmann, 1991]. Here s = 2.68 is assumed for quartz grains in water. With these simplifications, (1) is rewritten as which defines maximum transport rates ("transport capacity") for the idealized case of rather uniform bed material, essen- (Do = 2.50ø's(0 - Oc) Fr. (2) tially no morphological features, and hence no significant form Using the definitions of (Dt, and 0 and the continuity equation roughness effects. For such idealized conditions a dischargeq - vh, (2) can be transformed into based bed load transport equation is shown to be equivalent to a shear stress-based approach. Given the scarce field data on qb = 1.5(q - qc)s 's, (3) controlling parameters, the study focuses on an order of magnitude comparison of average bed load transport volumes tween gravel bed streams and torrents reflecting a wide range of bed material compositions, channel gradients, and discharges. The potential effect of relative smoothness (used partly as a surrogate for other factors such as channel and bed where q is the unit discharge and qc is the critical discharge at initiation of bed load transport. Note that (3) is not strictly valid for the conditions of those steep flume experiments in which bed load concentrations were nonnegligible (e.g., bed load volume concentrations above -10%); in this case the form roughness) on bed load transport efficiencies is examined transformation of mixture flow depth used in (2) into water by means of an empirical (regime type) equation to estimate flow depths as a function of discharge, bed slope, and a characteristic grain size. discharge is more complex, and (3) is not equivalento (2). Introducing the bed load transport rate by immersed weight, it, = qt,(ps - P)17, and the unit stream power, 6o = pt7sq, (3) is rewritten as 2. Bed Load Transport Formulas for Gravel Bed Streams At Eidgen6ssische Technische Hochschule (ETH) in Zurich, experiments on bed load transport in gravel bed channels were performed both for bed slopes of [Meyer- Peter and Miiller, 1948] and for bed slopes of [Smart and Jaeggi, 1983; Rickenmann, 1990]. For the entire slope range, including a total of 252 experiments, the following dimensionless bed load transport equation was developed [Rickenmann, 1991]: (D o -- 3.](d90/d30)ø'20ø'5(O- Oc)Fr ' (s- 1) -ø's, (1) where the dimensionless bed load transport rate (Do = qt,/ [(s - 1)#d3m] ø-5, qt, is the bed load transport rate per unit of channel width, s = Ps/P is the ratio of solid to fluid density,!7 is gravitational acceleration, d m is the mean grain size, 0 is the dimensionless shear stress equal to hs/[(s - 1)dm], 0 c is the 2.5(6o- 6oc)S ø's, (4) where 6oc is the critical stream power at initiation of bed load transport. For comparison, the bed load transport equation developed by Meyer-Peter and MMler [1948] may be written 8(0'- Oc).5 (5) where 0' = (no/n,) 0 is the dimensionless shear stress due only to grain resistance, i.e., excluding resistance due to bed forms, with n o = Manning's n owing to grain roughness, and /'l t = Manning n as measured for the total bed resistance. An equation similar to (3) has been developed by Schoklitsch [1962]: Simplifying for s = 2.68 gives qo = (2.5/s)(q - qc)s 's. (6) qo = 0.93(q - qc)s 1'5. (7)

3 RICKENMANN: BED LOAD TRANSPORT IN GRAVEL BED STREAMS 3297 Equation (7) is similar to (3), except for a coefficient of 0.93 instead of 1.5. Equations similar to those presented above have been proposed in other studies [e.g., Bagnold, 1956, 1980; Yation of (3), (7), or (9) over the channel width and averaging over the time of the flood event yield the following general equation: lin, 1977]. In fact, Yalin [1977] showed theoretically that t, should depend on 0, Fr, and s if viscosity effects are neglected. Qt, = AX (Qrn - Qc), (10) Considering equations of the type t, versus 0 as in (5), incluwhere Qt, is the average bed load transport rate, Qm is the sion of the Froude number has been proposed for flows chang- average flow rate, Qc is the critical discharge at initiation of ing from the lower and upper stage plane bed regime [Daido, bed load motion, A is an empirical coefficient, and the expo- 1983; Bridge and Dominic, 1984]. nent/3 varies between 1.5 and 2.0 in many bed load transport For the steep slope range, i.e., < S -< 0.20, a regresequations for gravel bed rivers and steeper streams. Integrasion analysis performed with the steep flume data on bed load tion of (10) over the time of a flood event results in transport obtained at ETH Zurich resulted in the equation [Rickenmann, 1990] Gœ = ASt3Vre, (11) qb = 12.6(dvo/d3o)ø'2(q- qc)s2'ø(s- 1) -1'6. (8) For simplification, setting (d9o/d3o) ø'2 = 1.05 and s = 2.68 yields qb- 5.8(q - qc)s 2'ø. (9) where G is the total bed load volume per flood event and the effective runoff volume, V,, is the integral of the discharge above the critical discharge at initiation of bed load motion (Q - Q c). In a more generalized form of (11) the bed load transport data can also be compared using the relationship Equation (9) predicts the steep flume data better than (3) 5;Ge = AS13 Vre, (12) since for conditions with nonnegligible sedimentransport the where 5; refers to observation periods including several flood transformation from (2) to (3) is invalid. Other bed load transand sediment events. The parameter Ks = AS t3 represents an port investigations for steep slope conditions [Mizuyama, 1981; average value for the total observation period in (12) or for the Ward, 1986] also suggest an exponent of 2 rather than 1.5 as in time of the flood event in (11). (3), which is valid for the entire investigated bed slope range For the data of Table 1 used in this study, G r in the follow _< S -< For the case of intense bed load transing diagrams refers to the volume of bed load sediment. In the port or so-called "immature" debris flow conditions a relationcases with a sediment retention basin (denoted SRB in Table ship with the proportionality t, " 02'5 has been proposed 1) the bed load volume is estimated to be half of the total [Hanes and Bowen, 1985; Takahashi, 1987]. deposited sediment volume. This ratio of 50% bed load is based on a grain size analysis at different locations in the 3. Analysis of Field Data An overview of the data on bed load transport used in this study is given in Table 1. To emphasize the slope range of steep streams, 13 cases refer to bed load data measured in channel reaches with bed slopes equal to 0.05 or greater. For comparison, another eight cases are included with bed slopes as low as ; in these cases the mean diameter of the transported material is also in the gravel size range. In a number of Swiss torrents (cases 1, 2, 4, 5, 9, 11, and 12 in Table 1), sediment measurements have been performed over several decades, using either sediment retention basins or sediment retention basin of the Erlenbach stream, where 50% of the deposit consists of grains larger than ---1 cm [Rickenmann, 1997]. This percentage may vary considerably depending on stream characteristics and hydraulic conditions in the sediment retention basin (for the cases of this study the fraction of bed load deposits may be assumed within a probable range of 20-90%). However, the main intent of this study is a gross comparison of the coefficienta in (10), (11), and (12) for streams with a wide range of channel gradients. Since A is found to vary over several orders of magnitude, the above assumption is acceptable. As a characteristic value of the channel gradient, an average bed slope value over a reach of sediment traps. In general, two consecutive sediment surveys m length upstream of the measuring site has been used. cover a time period of 1 year which includes several flood events. More frequent sediment surveys were carried out for the Erlenbach stream where special sensors give more detailed information on bed load transport intensities every minute An overall average value of bed load transport intensity for a given stream is obtained by calculating the ratio K s -- 5;G r/ 5; Vr according to (12). In Figure 1, K s is shown as a function of the bed slope at each measuring site (for all streams in Table during flood events [Hegg and Rickenmann, 1999; Rickenmann 1 except case 14). The data points are determined from meaet al., 1997]. For Bas Arolla (case 7), bed load data were available on a yearly basis. For a number of other streams (cases 3, 10, 13, 15, 16, and 17), bed load transport data were averaged over the flood event. For the rest of the streams (cases 6, 8, 14, 18, and 19) a bed load sampler was used in most cases, with measuring intervals ranging from minutes to hours. The values of the critical discharge for the beginning of bed sured values and compared with theoretical predictions of K s by (3), (7), and (9). A similar representation can be made using mean bed load transport rates and mean flow rates per flood event in the form of (10), instead of cumulated values over the whole observation period. Since for most of the Swiss torrents (case 2, 4, 5, 7, 9, 11, and 12) no data with sufficientemporal resolution are available, they are not included in this repreload transport (Table 1) are either those given in the respective sentation in Figure 2. For comparison, the laboratory data publications, or they were determined by back extrapolating from ETH Zurich are also shown. discharges with small bed load transport rates to a probable discharge with zero bed load transport. For many streams of Table 1, no within-event sediment measurements are available, which requires the use of a timeintegrated bed load transport equation in the analysis. Integra- From Figures 1 and 2 it may be inferred that the exponent/3 in (10) and (11) is not constant over the examined slope range (ifa and Qc are assumed not to vary substantially). There is a trend for /3 to increase with increasing bed slope. For bed slopes greater than a value of/3 = 2 appears to be

4 3298 RICKENMANN: BED LOAD TRANSPORT IN GRAVEL BED STREAMS ß l/ II II

5 RICKENMANN: BED LOAD TRANSPORT IN GRAVEL BED STREAMS II eq. field (3), data Rickenmann (no. 1-13, (1990) 15-19) eq. (7), Schoklitsch (1962).(9), Rickenmann (1990o), B B I I I I I I I I I S 1 Figure 1. Relative bed load transport intensity expressed as mean value of the total observation period for each stream (Ks = GE/ Vre), shown in relation to the bed slope (S) above the measuring site. more appropriate than/3 = 1.5, in agreement with results from laboratory experiments on bed load transport performed at steep slopes. However, factors other than bed slope and mean flow conditions must affect bed load transport intensity, as is evident when considering the large variation of the factor Ks or A for the field data in Figures 1 and 2. For example, in some over the entire observation period. The data in Figure 3 have been ordered according to increasing values of Vre (Swiss torrents) or Vre (other streams). In this way a more systematic comparison can be made since the plotting position is less influenced by the more infrequent and random occurrence of high-intensity events. Most lines show more or less a linear experiments Meyer-Peter and Mueller [1948] measured trans- dependence of G r on Vre, in particular for the higherport rates ---1 order of magnitude smaller than the average, which they attributed to substantial form drag in addition to skin friction. Double mass plots provide a useful comparison between bed load transport in the different streams. In Figure 3 the cumulative bed load, ZG r, is shown as a function of the cumulaintensity events, thus being in agreement with the linear Q, (Qm - Qc) dependence in (10). To account for the slope effect on bed load transport, the Vre values were multiplied by the factor S '5 (Figure 4). Equations (3) and (7) tend to overestimate bed load transport for most of the streams. The closest agreement is found, on tive effective runoff volume, ;Vre. 52 refers to observation average, for the data of Sperbelgraben, Rappengraben, Aare, periods of sediment surveys which normally include several flood events for the Swiss torrents (cases 1, 2, 4, 5, 7, 9, 11, and 12) and for shorter time periods for the other streams (see and Inn (cases 4, 5, 9, 18, and 19) in Figure 4. Considering all the streams, the scatter between the lines (representing the different streams) is reduced in Figure 4 compared with Figure Table 1 and footnote), while ; indicates values cumulated 3. This suggests that inclusion of a slope factor S.5 can explain o 0.1 o ii 0.01 o field data (no. 1,3,6,8,10,13-19) o lab data (no ) eq. (3), Rickenmann (1990) eq. (7), Schoklitsch (1962) -- -eq. (9), Rickenmann (1990) S Figure 2. Relative bed load transport intensity expressed as mean value per flood event or per single observation period for each stream [Ks = Qb/(Qm - Qc)], shown in relation to the bed slope (S) above the measuring site.

6 3300 RICKENMANN: BED LOAD TRANSPORT IN GRAVEL BED STREAMS 1E (.9 1E+1 1E E-3 1E+2 1E+4 ZZVre [m 3] 1E+6 1E+8 Figure 3. Double mass plot showing the cumulative bed load, E EG r, as a function of the cumulative effective runoff volume, E E Vre. The numbers representing the different streams are identified in Table 1. some of the variation of bed load transport intensities among 12) or intervals of a few hours (cases 6, 8, 18, and 19) were smaller and steeper streams. used. The results are plotted as bed load volumes against the A further analysis was made in terms of bed load volumes flow parameter Vre S '5 (Figure 5) and average unit bed load (GE) and effective runoff volume (Vre). For about a third of transport rates q, as a function of the discharge parameter the streams, information on these values is available for the (q,- qc)s '5(Figure 6). From a hydraulic perspective the time of the flood events (cases 1, 3, 10, 13, 15, 16, and 17 in representation of the data in Figure 6 is more useful than that Table 1). In other streams these values have been determined in Figure 5 because bed shear stress depends on channel crossfor the observation period of sediment transport: Irregular sectional form for a given discharge. In Figures 5 and 6, 11 data intervals including several flood events (cases 2, 4, 5, 9, 11, and sets are common, and the general pattern is similar in both 1E+5 eq. (;3)ø eq. (7) I,..; 6 ;. 5. 1E+1., - ß /. 1E-1 1E-3 1E+O 1E+2 Z ;VreSl.S [m 3] 1E+4 1E+6 Figure 4. Double mass plot showing the cumulative bed load, EEG r, as a function of the cumulative effective runoff volume multiplied by the bed slope factor, E E V,.eS.5. Also shown are two bed load transport equations transformed into the form of (12). The numbers representing the different streams are identified in Table 1.

7 _ RICKENMANN: BED LOAD TRANSPORT IN GRAVEL BED STREAMS IOO0 IO0 E '-' :-- ß S>0.1 (no. 1-5) o 0.01<S<0.1(no. 6-13),,.-:, S < 0.01 (no ) ß,,, '½,,, o ---eq. (3) : eq. (7) ß ß,,0 ß ß A-v. AA,,,,, o --- o ø oo o :. o- o ø ø ' o o o _,,, > xx o. ß o _= z o > o _--,,;,,, - '- %o - 0 O.O01 I i! i i i ii I I I I IIII I I I i i i II I I I I I I II I I I I I III I I I, I I II O O Vre sl's [m 3] Figure 5. Bed load volumes, GE, shown as a function of the hydraulic load, i.e., the effective runoff volume multiplied by the slope factor, Vre S '5. Data points refer either to flood events or to periods of sediment surveys. Also shown are two bed load transport equations. Figures 5 and 6, suggesting that the use of integral stream flow resistance in small and steep streams increasesignifiparameters is reasonable in this study and allowing inclusion of cantly if the relative flow depth (relative smoothness) h/d9o is many of the steep torrents in the analysis. below [Rickenmann, 1994, 1996], where h is flow depth and d9o is the grain size for which 90% of the surface bed material is finer by weight. 4. Effect of Relative Smoothness on Bed Load For most of the bed load transport data in Table 1, no Transport Efficiency information is available on flow depth. Therefore an approxi- Owing to coarse bed material and irregular channel mor- mate value of the flow depth is estimated empirically using phology it may be assumed that higher energy losses occur in relationships developed for torrents and gravel bed rivers for small and steep gravel or boulder bed streams than in channels which information is available (only) on mean flow velocity with a more uniform bed material. It has been observed that (v), discharge (Q), bed slope (S), characteristic grain size of OOOOO 1OOOO 1000 ß S>0.1(no. 1,3) o 0.01 =< S < 0.1 (no. 6,8,10,13,14) ' ß S < 0.01 (no ) ß lab data (no ) ---eq.(3) x, ß e ß 0 e..:. o 1 I i i ill i i i IIIII I i i i iiii O O. 001 O. 01 O. 1 (qm'qc)sl's [m 31slm ] Figure 6. Average bed load transport rates per flood events, q t,, shown as a function of the hydraulic load, i.e., the effective runoff multiplied by the slope factor, (q,- qc)s '5. Also shown are two bed load transport equations.

8 3302 RICKENMANN: BED LOAD TRANSPORT IN GRAVEL BED STREAMS. loo =- ß >. 0 o 1 o o ii Aeee ß i - o... ß -, ß ' '... ' :. h ' 'r' ' '.; '... ' - o A ß '- _ A t, ß ß,A A ß ß 't j ' " o 0.01=<S<0.1(no. 6,8-14) - ['o ß S>0.01(no ), ß lab data (no ) o... eq. (3) - o o eq. (7) I I I i I I ii I I I I I I I I I I I I I I I I i i i I I i ii O h/d9o 1 O Figure 7. Back-calculated values of the coefficient A in the bed load transport equation (2), shown in relation to the relative flow depth, h/d9o. the surface bed material (d9o), and mean flow width (W) [Rickenmann, 1994, 1996]' v = 0.37gø'33Qø'34Sø'2ø/d b 35 S -> 0.008, (13) U = 0.96g' 0 36,r- ' 029c, o' 035/./0.23 /a9o S < 0.008, (14) W = 5.01Qø'32d 2 /( g0.]650.25). (]5) In terms of predicted versus measured velocities or flow widths, the following statistical parameters were determined: correlation coefficient r = 0.93 for both (13) and (14), r = 0.82 for (15), standard error of estimate s e = 34% for (13), S e = 24% for (14), and S e = 44% for (15). Using the continuity equation v = Q/(Wh) together with (13) and (15), the following expression is obtained to determine the flow depth, h, for bed slopes S -> 0.008: h Qø'34Sø'ø5 d / ] _0.17 S -> (16) All the relationships (13)-(16) are dimensionally correct. Equation (16) is used to estimate a representative flow depth for cases 1-6, 8-12, and in Table 1; in most cases the mean discharge used in (16), and for the Swiss torrents with longer observation periods (cases 2, 4, 5, 9, 11, and 12) the maximum discharge during that period is used. Using mean discharges results in an average value of h representing the bed load transport event. For the Swiss torrents the maximum discharge appears to be the best approximation characterizing the entire observation periods including several flood events. (If the maximum discharge is 10 times as high as the mean discharge, the corresponding h value determined by (16) is twice as high.) For the Jordan River data, h is given, and for the Inn River, h is determined from the known cross-sectional area and the flow width. For the Aare and the Rhein Rivers the flow width, W, is known, and h is determined together with (14). For all the bed load transport data of the streams and experimental studies in Table 1 a value for A is calculated using (10), (11), or (12) with/3 = 1.5. The parameter A represents a kind of bed load transport efficiency which is defined by the deviation of observed transport rates from those predicted by (3) or (7), i.e., the decrease (or increase) of an appropriate value for A. In Figure 7, A is shown in relation to the relative flow depth, h/d9o. Part of the decrease in bed load transport efficiency for the "smaller" and steeper streams is apparently associated with decreasing values of relative flow depth. All the lower efficiency streams occur when h/d9o is smaller than Discussion In the above analysis it is assumed that Q c is constant for a given stream. In reality, Qc may vary and depend on grain size, bed configuration (before breakup of the armor layer), and possibly on external sediment supply. Here back-extrapolated mean values of Q c were used. This seems acceptable because this study emphasizes an order of magnitude comparison of bed load transport. Another assumption is the linear dependence of bed load transport on effective discharge based on semiempirical evidence from flume experiments. By analyzing bed load data from 19 streams and considerin general trends using double mass plots, this study provides some confirmation for this assumption. Considering, in particular, the higher-flow and/or longer-duration events, Figures 3 and 4 indicate that for most of the data, there is roughly a linear dependence between bed load transport and "effective" flow rate, i.e., Qb (Qm - Qc) with a = 1, in agreement with (3), (7), and (9). Some of the lines representing one stream in Figures 3 and 4 show a considerable fluctuation around a mean "slope"(i.e., a fluctuating value of a), which is generally more pronounced on the lower left-hand side of the diagrams where flow conditions are more likely to be near critical conditions for initiation of bed load transport. For the higher-flow and/or longer-duration events (i.e., the upper right-hand side of the diagrams) the data of Bridge Creek (case 8) suggest a value a 2, while for the data of Rappengraben (cases 4 and 5), a 0.5 would be more appropriate. These extremes indicate an approximate range of a in this study. Hegg and Rickenmann [1999] compared observed bed load transport rates for the Erlenbach stream during minute inter-

9 RICKENMANN: BED LOAD TRANSPORT IN GRAVEL BED STREAMS 3303 <1: Spel belgrabe -' Rappengrabel 1 Bas Arol a 'A"" A re Bridge ß Rappen aben 2 Rotenl)ach! (Schw ndliba½ ) -Virginic (Melera) I Oak ß Turkey (Torle. sse) Rio"Cordon Edenbach Pitzbach! (Jordan) i i i i i i dso,b [cm] Figure 8. Back-calculated mean values of the coefficient A (average value for each stream) in the bed load transport equation (2), shown in relation to a characteristic grain size of the surface bed material, d9o,t,. Stream names in parentheses indicate estimated values of d9o,t,. vals with (9). They found that at higher flow intensities (-4-5 times the critical conditions) the measured values were, on average, about parallel to and about one third of the values predicted by (9). At lower flow intensities the same data show a large variability, a general trend of Qt, " Q4 or Qt, " Qs, can be inferred. If the Erlenbach data are averaged over the time of the flood events (analyzing a total of 142 events), the other torrents, where the transported material is generally finer than the bed material. Lower and variable bed load transport efficiencies for values of h/d9o smaller than (Figure 7) roughly correspond to the marked increase in flow resistance within a similar range of relative flow depths [Pitlick, 1992; Rickenmann, 1996]. Jarrett [1990] reported that flow resistance may be substantially larger corresponding value of a varies between 1 and 2 [Rickenmann, and more variable in steep boulder bed channels at smaller 1997]. The comparison of the measured data with (3) and (7) in Figure 6 or with the integrated form (11) with/3 = 1.5 in Figure 5 reveals that basically only the Inn, Sperbelgraben, and a part of the Rappengraben and Aare field data plot above the bed load transport equations, whereas the data of the other, primarily smaller streams exhibit a large variability. Most of the "smaller"(and steeper) streams tend to have a lower bed load transport efficiency than the "larger" streams. Bathurst et al. [1987] came to a similar conclusion on the basis of a study with values of relative flow depths, that is, when the ratio of the hydraulic radius to the medium grain diameter is less than Bathurst [1993] discussed the development of a roughly S- shaped velocity profile in boulder bed channels at relative flow depths h/d84 of Compared to a logarithmic velocity profile, the flow in the lower zone near the boulders is retarded, while there is a higher-velocity zone near the flow surface. Pitlick [1992] also observed a deviation of the velocity profile from the log law at relative flow depths h/ks (where ks is the roughness height) smaller than The additional drag a smaller number of streams. caused by the boulders in this range may be partly responsible Some data of the torrents with bed slopes steeper than plot within -- 1 order of magnitude above or below the relations (3) and (7) in Figures 5 and 6. DMgostino and Lenzi [1999] reported that the highest observed bed load transport rates approach values predicted by (7) for the largest flood event in the Rio Cordon in Transport efficiencies in a range similar to the Aare and Inn Rivers and the laboratory experiments are observed for some of the Swiss torrents, having bed load transport values higher than those predicted by (3) and (7) (Figures 5, 6, and 8). This is the case for the Sperbelgraben, the Rappengraben 1 and 2 (cases 4, 5, and 9), for the lower bed load transport "efficiency" as compared to flows with higher relative flow depths. Reid and Laronne [1995] compared the bed load transport in six gravel bed streams and found that the mean transport efficiencies tended to decrease with increasing median grain size of the surface bed material. Four of these streams (cases 10, 15, 16, and 17) are also included in this study. Considering all the data in the present study, the same trend cannot be detected. Plotting mean transport efficiencies against the d9o,t, of the surface bed material, there appears to be an upper envelope line indicating decreasing efficiencies with increasing and a few flood events in the Rotenbach (case 12). Considering d9o,t, values (Figure 8). A similar pattern of results is observed mean transport efficiencies over the entire observation period (Figure 8), predicted bed load volumes for Bas Arolla (case 7) are also of the same order of magnitude as observed ones. The bed material of both the Sperbelgraben and the Rappengraben is very well-rounded, and the grain size distribution is generally narrower than for the other torrents of this study. Thus bed load transport conditions are more similar to those of the experimental flows (cases 20, 21, and 22), in contrast to the when using d5o,t, instead of d9o,t,. However, the trend is weak in both cases, and there are only few data 13oints with large grain-size values. In a detailed analysis of flume experiments, Carson [1987] concluded that tractive stress acting on bed grains is an excellent predictor of bed load transport rates. It is probable that for many of the streams investigated in this study, there is sub-' stantial form roughness. In this case the transformation of (2)

10 3304 RICKENMANN: BED LOAD TRANSPORT IN GRAVEL BED STREAMS into (3) is not strictly valid, or, alternatively, a reduced dis- A possible explanation of lower transport efficiencies is subcharge value should be introduced. Owing to insufficient in- stantial form roughness in some of the smaller and steeper formation these losses could not be quantified in this study. streams. Other reasons for lower transport efficiencies may be However, it is possible that low values of relative smoothness substantially varying sediment supply conditions or flows near are an indicator of significant form roughness, and in this critical conditions for the beginning of transport, which prerespect the present results support the importance of Carson's vailed for many events analyzed in this study. conclusion. If an effective discharge-based bed load transport equation is to be successfully applied in these cases, an appro- Acknowledgments. I would like to thank all colleagues who helped priate discharge reduction might be introduced analogous to with field measurements and data processing, in particular Bruno correcting the tractive force for form roughness losses when Fritschi and Geri R6thlisberger. The study was partly supported by the using a shear stress-based approach. European Commission, DG XII, Environment and Climate Program, Despite a general trend of decreasing bed load transport Climatology and Natural Hazards Unit (contract ENV4-CT ), efficiencies with increasing relative roughness the data show a and by the Swiss Federal Office for Education and Science (BBW ). James Bathurst and Gordon Grant provided constructive great variability of the coefficient A of several orders of mag- reviews which substantially helped to improve the manuscript. I also nitude. Some of this variability for a given stream is probably acknowledge Brian McArdell's support with editing. related to relatively low discharges within the range of Q 1Qc to Q --- 5Qc where a larger variability of bed load trans- References port intensities can be expected than for higher flows. Another important factor is certainly also the changing sediment supply Ackroyd, P., and R. J. Blakely, En masse debris transport in a mountain stream, Earth Surf. Processes Landforms, 9, , conditions which can effect the variability in transport effi- Amt ffir Wasserwirtschaft, Untersuchungen in der Natur fiber die ciency both between different streams and among different Bettbildung, Geschiebe und Schwebstoffffihrung, Mitteilung 33, 113 events in the same stream. pp., Bern, Switzerland, The analysis presented here used a value of/3 = 1.5 in the Andrews, E. D., Marginal bed load transport in a gravel bed stream, general bed load transport equation (10). A similar analysis Sagehen Creek, California, Water Resour. Res., 30(7), , using a value of/3 = 2 in (10) yields similar results to those Bagnold, R. A., The flow of cohensionless grains in fluids, Philos. presented in Figures 4-8, and the conclusions are qualitatively Trans. R. Soc. London, Ser. A, 249, , similar. Bagnold, R. A., An empirical correlation of bedload transport rates in flumes and natural rivers, Proc. R. Soc. London, Ser. A, 372, , Summary and Conclusions If no substantial form roughness is present, an effective shear stress-based bed load transport equation of the form of (2) is equivalento an effective discharge-based approach such as (3) and (7). Such an empirical formula, derived from flume laboratop), experiments, was compared with bed load transport data from 19 mountain streams, including gravel bed rivers, boulder bed streams, and steep torrents. For the majority of the streams with channel slopes steeper than , no direct flow depth information is available, necessitating the use of a discharge-based bed load transport equation. Using double mass plots, summed bed load volumes were plotted against summed effective runoff volumes. The results show that (1) bed load volumes are roughly linearly proportional to effective runoff volumes and (2) by introducing the bed slope factor, the data plot closer together. These findings are in agreement with predictive bed load transport formulas considered in this study. Despite this general agreement, there is a large variation of the coefficient in these semiempirical formulas when comparing sediment transport data from different streams, especially from smaller and steeper streams, both with each other and with laboratory data. Using empirical equations of mean flow velocity, an estimate could be made of representative flow depths for the field data, although in most cases no direct measurements were available. When plotting the coefficient of the bed load transport formulas against a relative flow depth, it appears that a part of the decrease in bed load transport efficiency for the smaller and steeper streams is associated with decreasing values of relative flow depth. All the lower efficiency streams are grouped within the range of relative flow depths smaller than This pattern roughly corresponds to an increase in flow resistance and a distortion of the logarithmic velocity profile within a similar range of decreasing relative flow depths. Bagnold, R. A., Transport of solids by natural water flow: Evidence for a worldwide correlation, Proc. R. Soc. London, Ser. A, 405, , Bathurst, J. 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Rickenmann, Department of Water, Soil and Rock Movements, Swiss Federal Research Institute WSL, Zfircherstrasse 111, CH-8903 Birmensdorf, Switzerland. (rickenmann@wsl.ch) (Received August 9, 2000; revised May 31, 2001; accepted August 10, 2001.)