Predicting fractional bed load transport rates: Application of the Wilcock-Crowe equations to a regulated gravel bed river



[1] Bed load samples from four locations in the Trinity River of northern California are analyzed to evaluate the performance of the Wilcock-Crowe bed load transport equations for predicting fractional bed load transport rates. Bed surface particles become smaller and the fraction of sand on the bed increases with distance downstream from Lewiston Dam. The dimensionless reference shear stress for the mean bed particle size (τ*rm) is largest near the dam, but varies relatively little between the more downstream locations. The relation between τ*rm and the reference shear stresses for other size fractions is constant across all locations. Total bed load transport rates predicted with the Wilcock-Crowe equations are within a factor of 2 of sampled transport rates for 68% of all samples. The Wilcock-Crowe equations nonetheless consistently under-predict the transport of particles larger than 128 mm, frequently by more than an order of magnitude. Accurate prediction of the transport rates of the largest particles is important for models in which the evolution of the surface grain size distribution determines subsequent bed load transport rates. Values of τ*rm estimated from bed load samples are up to 50% larger than those predicted with the Wilcock-Crowe equations, and sampled bed load transport approximates equal mobility across a wider range of grain sizes than is implied by the equations. Modifications to the Wilcock-Crowe equation for determining τ*rm and the hiding function used to scale τ*rm to other grain size fractions are proposed to achieve the best fit to observed bed load transport in the Trinity River.

1. Introduction

[2] River engineers and scientists have been studying the nature of bed load transport and working to develop reliable equations for calculating sediment transport rates for more than a century [e.g., du Boys, 1879]. Important applications of sediment transport predictions include estimating rates of reservoir sedimentation [Morris and Fan, 1998], stream restoration design [Skidmore et al., 2001], assessing aggradation and flood risks [Klaassen, 2006], and prescribing in-stream flow requirements [Andrews and Nankervis, 1995]. At their most fundamental levels, these applications require predictions of total sediment transport rates only. Efforts to achieve even this relatively modest goal, however, have met with only limited success. The poor performance of many bed load transport equations has been widely noted [Wilcock, 2001; Barry et al., 2004; Duan and Scott, 2008]. Under optimum conditions, the best performing bed load functions predict transport rates within a factor of 2 of the actual value about two thirds of the time [Ackers and White, 1973; Gomez and Church, 1989].

[3] A more ambitious application of predictive sediment transport equations is to support numerical models that aim to simulate geomorphic evolution. Models of this sort have been developed to investigate geomorphic dynamics in a variety of settings, including alluvial fans [Parker et al., 1998], deltas [Sun et al., 2002; Swenson et al., 2005], and river channels [Ferguson et al., 2001; Lisle et al., 2001; Cui, 2007]. In gravel bed rivers with mixed sediment particle sizes, differences in the transport rates of different particles size classes are likely to influence geomorphic evolution through processes such as longitudinal sediment sorting [Knighton, 1980; Fedele and Paola, 2007] and bed surface coarsening [Dietrich et al., 1989]. Simulation of these geomorphic processes requires bed material transport equations capable of predicting fractional transport rates, as well as total transport.

[4] At present, the Wilcock and Crowe [2003] equations are among the more widely used formulae for predicting fractional bed load transport rates in gravel bed streams. The Wilcock-Crowe equations (hereinafter referred to as WC) were developed using bed load transport information obtained in laboratory flume experiments with bed material sediments ranging in particle size from sand to coarse gravel [Wilcock and Crowe, 2003]. This paper presents a comparison between fractional transport rates predicted with the WC equations and fractional transport rates determined by intensive sampling of bed load transport during periods of sustained high flow in the Trinity River, a regulated gravel bed stream in northern California. Our objectives for this comparison are (1) to evaluate the accuracy of the WC equations for predicting total and fractional bed load transport rates in a relatively large gravel bed stream, (2) to calibrate the equations for optimal performance in the Trinity River, and (3) to consider the extent to which the calibration results may represent a general, rather than site-specific, improvement.

[5] The WC equations are shown to yield reasonable estimates of total bed load transport in the Trinity River. These equations, however, systematically under-predict the transport of the coarsest fractions in the bed load. Numerical experiments summarized later demonstrate that differences in the predicted mobility of the largest grains in the bed load can have a disproportionately large influence on the output of morphodynamic models. Thus it is critical that the transport equations used in these types of models predict the fractional transport characteristics of the larger size classes as accurately as possible.

2. Wilcock-Crowe Bed Load Equations

[6] The WC bed load equations are summarized briefly below (see Wilcock and Crowe [2003] for a complete description of the equations and their derivation). The WC transport function is

equation image
equation image

where Wi* is the dimensionless bed load transport parameter, the subscript indicates the ith particle size fraction of N size fractions (i = 1, 2, …, N), ϕ is defined as (τ/τri), τ is the shear stress acting on bed sediment particles, and τri is the reference shear stress for particle size fraction i. The reference shear stress is defined as the shear stress that produces a value of 0.002 for Wi*, which is related to the volumetric bed load transport rate for size fraction i (qbi) by

equation image

where s is the submerged specific weight of the sediment, g is gravitational acceleration, Fi is the fraction of the bed surface consisting of particle size fraction i, and u* is the shear velocity of the flow.

[7] Wilcock and Crowe [2003] proposed that the dimensionless reference shear stress for the geometric mean particle size on the bed surface (τ*rm) is a function of the fraction of sand on the bed surface (Fs), such that

equation image

This dimensionless reference shear stress is converted to dimensional form according to

equation image

where ρ is the density of water and Dsm is the geometric mean particle diameter on the bed surface. The ratio of τri to τrm is given as a function of the ratio of the particle diameter of the ith size class (Di) to Dsm:

equation image
equation image

3. Bed Load Sampling in the Trinity River

3.1. Study Area

[8] The Trinity River is the principal tributary to the Klamath River, which drains the Klamath Mountains of northern California and discharges into the Pacific Ocean (Figure 1). Streamflows in the Trinity River have been regulated downstream from Trinity and Lewiston dams since 1960. Current dam operations include annual spring flow releases with peak flows ranging between 42 and 311 m3/s, depending on the anticipated water yield upstream from the dams.

Figure 1.

Location of the Trinity River in northern California.

[9] Lewiston and Trinity dams trap all but the finest sediment particles eroded from the upper part of the Trinity basin. Because occasional high streamflows capable of transporting some gravel and cobble bed materials downstream were released or spilled from Lewiston Dam in the decades following dam closure, the river bed became armored between Lewiston Dam and Rush Creek, the first major tributary downstream from the dam. Furthermore, the quantity of mobile gravel stored in the channel is believed to have decreased significantly over a distance of 25 km or more downstream since dam closure [U.S. Fish and Wildlife Service, 1999]. The Trinity River Restoration Program (TRRP) currently operates four sediment sampling stations on the mainstream Trinity River (Figure 2) that provide the sediment transport information necessary for restoration activities, including flow releases from Lewiston Dam, mechanical channel alterations, and gravel augmentation. In order of increasing distance from Lewiston Dam, the sampling locations are Trinity River at Lewiston (TRAL), Trinity River at Grass Valley Creek (TRGVC), Trinity River at Limekiln Gulch (TRLG), and Trinity River at Douglas City (TRDC). TRAL is about 3 river km downstream from Lewiston Dam and is downstream from one tributary that delivers significant quantities of sediment to the main stem. TRDC is more than 30 river km downstream from the dam and is downstream from at least six significant tributaries. Thus the four sampling locations are positioned to span conditions ranging from relatively low sediment supplies and low transport rates to considerably greater sediment supplies and transport rates.

Figure 2.

Map showing the locations of the four bed load sampling locations and significant tributaries to the Trinity River downstream from Lewiston Dam.

3.2. Bed Load Sampling

[10] Trinity River bed load samples are collected during annual spring high-flow releases from Lewiston Dam by Graham Matthews and Associates (GMA). Bed load sampling in large streams is inherently difficult and uncertain. The bed load samples presented herein were collected and analyzed according to practices recommended by the U.S. Geological Survey (USGS) [Edwards and Glysson, 1999], and an independent review and evaluation of the Trinity River sediment sampling program was conducted by the Sedimentation Laboratory, California Water Science Center, U.S. Geological Survey (L. A. Freeman, written communication, 2006).

[11] Bed load sampling was conducted with cable-suspended Toutle River 2 (TR-2) bed load samplers deployed from a cataraft attached via rollers to temporary cableways secured to trees on either side of the channel at each of the four sampling locations. The TR-2 sampler has an intake nozzle measuring 0.152 × 0.305 m and a 0.5-mm mesh bag. Sampling was conducted by lowering the sampler to the bed at 10–14 verticals, depending on sampling location and discharge, spaced 3.04 m (10 feet) apart across the width of the active channel following standard USGS equal-width increment procedures [Edwards and Glysson, 1999]. The sampler was placed on the river bed at each vertical for 30–60 s, depending on the transport rate. A video camera mounted on the sampler for a subset of samples verified expectations that the bed was in partial transport at the highest discharges, and that no bed forms were present.

[12] As reported in this paper, “bed load sample” refers to the composite of all material collected by lowering the sampler to the bed once at each vertical on the sampling cross section. Approximately 50 samples were collected during the peak and falling limb of the 2006 release, and approximately 20 were collected during the 2007 release at each of the four bed load sampling locations. The discharges at which samples were collected span a wide range of high flows, including the peaks of both releases (Figure 3). Transport rates were relatively high during much of these release periods, such that the total mass of these samples amounted to nearly 3.3 t. All this material was subsequently dried and sieved into 1/2-ϕ size classes. A description of methods and summaries of the sampled transport rates and sediment load computation are in the contractor's report to TRRP [GMA, 2007].

Figure 3.

Hydrographs of spring releases from Lewiston Dam in (left) 2006 and (right) 2007. Circles and squares indicate days when bed load samples were collected. Circles indicate samples used to construct sediment rating curves. Samples taken on the rising limb of the 2006 release (squares) were excluded when fitting rating curves.

[13] Although bed load sediment samples have been collected annually at three of the four current bed load sampling locations since 2002, this paper considers only those samples collected after the peak of the 2006 release and during the 2007 release (Figure 3). The analysis is limited to this time period in part because the channel changed substantially during 2006 and the ancillary data necessary for applying sediment transport equations to the earlier time periods are unavailable. Surveys of channel cross sections, water surface slopes, and pebble counts to characterize the bed surface texture at the sampling locations were obtained for the first time in 2006. In addition, water year 2006 was extremely wet, and relatively large discharges were released from Lewiston Dam earlier than expected to meet dam safety requirements. These early high flows prevented collection of survey and pebble count data prior to the spring release.

[14] Pebble counts obtained after the high flow release do not represent bed conditions during the rising limb of the release, especially with respect to the percent of sand and very fine gravel on the bed surface. Large quantities of these relatively fine particles were delivered to the Trinity River from tributary watersheds during the winter of 2006, including debris flows that reached the river from upland areas near Lewiston Dam that were burned by a wildfire in 1999 [Madej, 2007]. A reconnaissance of the streambed in the Lewiston area in March 2006 found that approximately 60% of the bed was covered with sand-sized sediment. Subsequent reconnaissance after the release showed that the majority of the sand had been flushed from the upper river, and postrelease pebble counts at the sampling locations contained relatively little sand. As indicated by equation (3), the fraction of sand on the bed is a critical parameter in the WC model. Comparisons between rating relations developed separately for the rising and falling limbs of the 2006 release confirm that the change in the surface sand fraction significantly altered bed load transport rates during the release (Figure 4). Substrate mapping conducted by the authors both prior to and after the 2007 release indicated that little new sand was delivered to the river during the preceding winter (water year 2007 was classified as a dry year for dam operation purposes) and that the 2007 release did not noticeably change bed conditions. Pebble count data obtained in the fall of 2006 was therefore considered suitable for modeling sediment transport during the 2007 spring release.

Figure 4.

Total transport rating curves fit to subsets of the 2006 and 2007 sample data from Trinity River at Lewiston (TRAL). The solid curve at upper left is fit to the 2006 rising limb data only, and the lower right solid curve is fit to the 2006 falling limb data only. The middle solid curve is fit to the 2007 samples only, and the dashed curve is fit to combined 2006 falling limb and 2007 data.

3.3. Surface Grain Sizes

[15] Pebble counts were conducted in the wetted channel on and immediately upstream from the four bed load sampling transects in December 2006 at a relatively small discharge of 8.5 m3/s. Approximately 100 pebbles were sampled and assigned to 1/2-ϕ classes, except for particles less than 4 mm in diameter, which were combined into a single class. The surface particle size distributions used in the fractional bed load transport calculations reported here are shown in Figure 5. As is expected downstream from a dam, Dm decreases with downstream distance, whereas Fs increases in the downstream direction (Table 1).

Figure 5.

Bed surface particle size distributions at the four bed load sampling locations.

Table 1. Water Surface Slope, Percent Variability in Slope Measurements (δS), Mean and 65th-Percentile Surface Particle Sizes, Fraction of Sand on the Bed Surface, and Reference Shear Stress at the Four Bed Load Sampling Locations
LocationaSδS (%)Dsm (m)D65 (m)Fsτrm (N/m2)
  • a

    TRAL, Trinity River at Lewiston; TRGVC, Trinity River at Grass Valley Creek; TRLG, Trinity River at Limekiln Gulch; TRDC, Trinity River at Douglas City.


4. Analysis Methods

4.1. Computed Bed Load Transport Rates

[16] Equations (1)(5) were coded in a spreadsheet to compute fractional bed load transport rates corresponding to all discharges for which a bed load sample exists at each sampling location. Transport rates were computed separately for 12 size fractions from 4 to 256 mm with 1/2-ϕ increments, and for a thirteenth size fraction defined as 0.5–4 mm. Transport rates for the twelve 1/2-ϕ size classes were merged into 1-ϕ size fractions for comparison with the bed load samples (although bed load samples were sieved into 1/2-ϕ size classes, these data were merged into 1-ϕ size fractions for analysis to minimize the number of samples with zero catches of the larger size fractions).

4.2. Estimating Boundary Shear Stress

[17] The WC equations require estimates of skin friction shear stress over the range of flows for which bed load transport rates are to be calculated. Only the portion of the total boundary shear stress acting on the bed particles, i.e., skin friction, contributes to sediment transport [Engelund and Hansen, 1967]. The skin friction shear stress, τ, is given by

equation image

where τT is the total boundary shear stress equal to ρgHS (H is the flow depth or hydraulic radius and S is the water surface slope) and τD is the form drag due to bed forms, channel curvature, bank irregularities, etc. We calculated the cross-sectional average skin friction shear stress using a relation suggested by Wilcock [2001] and following the general approach developed by Meyer-Peter and Müller [1948], Einstein [1950], and Engelund and Hansen [1967]:

equation image

Cross-sectionally averaged velocity, U, was calculated over a range of flows at each sample location using surveyed cross-section geometry and water surface elevations. Discharge across overbank areas is small compared with flow within the channel, so U was computed using only the flow area between the channel banks. At all four bed load sampling locations, at least four water surface levels spanning from base flow to nearly the maximum discharge sampled were available. The sets of discharge-velocity values were fit with empirical functions via least squares (r2 ∼ 0.99) to produce a velocity rating relation for each location. Water surface slopes through the sampling transects were estimated from water surface elevation profiles. The 65th-percentile surface particle diameters (D65) were estimated from the pebble count data.

[18] The magnitudes of τ computed by equation (7) are subject to significant uncertainty associated with potential errors in the input variables, as well as limitations on how well simple equations can relate flow roughness and velocity [Ferguson, 2007]. Of the input variables to equation (7), S is perhaps the most difficult to measure accurately in the field. Small surveying errors can produce significant errors in the measured values of S, and the actual values at a given channel location are likely to vary spatially and with discharge. Slope measurements were conducted at the bed load sampling locations via level surveys at multiple discharges during the release period over lengths of channel ranging from about 80 to 280 m (about two to six channel widths). Slope values used to compute τ, and the range of variability in the measurements at each sampling location, are given in Table 1. Where measurement variability is high, slopes measured at higher discharges and over shorter channel lengths were favored. For purposes of propagating errors in S to computed values of τ, we have conservatively assumed that S may be in error by a factor of 2. When raised to the 0.25 power, this translates to a potential error in τ of about 19%.

[19] Neglecting flow conveyance in overbank areas in computing U may result in overestimation of U and τ. The magnitude of this potential error source was evaluated by comparing U with alternative estimates of the flow velocity in the active channel determined from measurements obtained with an acoustic Doppler current profiler (ADCP) at the TRDC sampling location. ADCP velocities profiles were logged at each of the 11 sampling verticals over a range of discharges spanning 181–280 m3/s during the 2006 flow release as part of an investigation into the possibility of using the ADCP's bottom-tracking feature to quantify bed load transport rates [Gaeuman and Pittman, 2009]. Depth-averaged velocities at individual verticals were extracted by first fitting the profile data at individual verticals with the standard logarithmic velocity distribution,

equation image

where u(y) is the flow velocity measured at height y above the bed, y0 is the height above the bed where u(y) goes to zero, and κ = 0.4 is the von Karman constant. Fitted values of u* and y0 were then substituted into the vertically averaged form of the logarithmic velocity distribution to yield the depth-averaged velocity at vertical n:

equation image

The mean flow velocity in the active channel for each discharge as determined from ADCP measurements (Uadcp) is then obtained as the area-weighted average of equation imagen over all verticals.

[20] Comparison of Uadcp with U shows that the two estimates are in good agreement at lower discharges, but suggests that the mean flow velocity in the active channel may be overestimated by about 6% for the highest discharges (Figure 6). U is raised to the 1.5 power in equation (7), suggesting that τ may be overestimated by about 10% for discharges greater than about 250 m3/s.

Figure 6.

Mean flow velocity at Trinity River at Douglas City (TRDC) sampling transect determined from discharge and active channel area (U) versus mean flow velocity determined from acoustic Doppler current profiler (ADCP) data collected at individual verticals (Uadcp).

[21] According to Recking et al. [2008], the appropriate roughness height in a resistance model varies with the intensity of sediment transport. They suggest that the effective roughness height is 2.6 times larger when the bed sediments are fully entrained compared with conditions of no sediment transport. As the bed of the Trinity River was never fully mobile during the peak discharges sampled, we assume that the roughness height of 2D65 incorporated in equation (7) is potentially in error by a factor of 2. When raised to the 0.25 power, this implies another potential error in τ of about 19%.

[22] Inaccuracies in determining D65 from pebble counts also contribute to uncertainty in τ. According to Rice and Church [1996], pebble counts of 100 stones can be expected to determine the sizes of the 50th percentile and larger percentiles to within 0.3 ϕ with 95% confidence. An uncertainty of 0.3 ϕ is equivalent to 23% in arithmetic units. This is equivalent to an additional uncertainly in τ of about 5%.

[23] The four sources of error described above are independent of one another, and so may be combined as the square root of the sum of their squares [Gaeuman et al., 2003; Benjamin and Cornell, 1970]. The resulting net probable error in the values of τ input to the WC equations is estimated to be about ±29%. The implications of errors in τ of this magnitude are considered later.

4.3. Bed Load Rating Curves

[24] Fractional bed load transport rating curves were developed primarily to aid in the estimation of τri, as described in section 4.4. Sampled fractional transport rates were fit to a function of the form

equation image

where Qb is the bed load transport rate, Q is the water discharge, Qc is the water discharge when sediment transport begins, and a and b are fitting parameters.

[25] Fitting the measured transport rates for the >128 mm fractions requires special consideration. Transport of this fraction occurs at relatively high flows, and the majority of the bed load samples included in this analysis showed zero transport for this size class. In many instances, the zero-transport samples were collected when discharges were clearly too small to entrain large particles, so that their removal prior to curve fitting is appropriate. Discharge thresholds below which zero-transport samples were removed (QMIN) were defined according to the smallest discharges at which a particle larger than 128 mm has ever been captured at each sampling location (Table 2).

Table 2. Minimum Discharges at Which Bed Load Transport Rates >0 Have Been Detected for the >128 mm Size Class (Qb128) and Number of Samples Less Than the Minimum, Based on the 2006 and 2007 Samples Considered in This Paper and the Full Record of Samples at These Sampling Locationsa
Location2006 (Falling Limb) Through 2007All Years of Sampling Record
SamplesSamples Qb128 > 0Qmin (m3/s)Samples Q < QminYears SampledSamplesSamples Qb128 > 0QMIN (m3/s)Samples Q < QMIN
  • a

    A nonzero sample for the >128 mm class was obtained at a discharge of 100 m3/s at TRLG during the rising limb of the 2006 release. This was likely the result of bed disturbance during sampler emplacement or recovery.


[26] On the other hand, a large proportion of the bed load samples collected at higher discharges also showed zero transport in the >128 mm size class. This problem arises because conventional bed load sampling methods inherently violate the principals of geometric similarity. Geometric similarity for samples of different particle sizes implies that the same number of particles from each size class will be captured, and requires scaling the area of bed being sampled in proportion to the square of the grain size being considered [Wilcock, 1988]. Because the area of bed sampled with a conventional sampler is constant regardless of the sizes of the particles in transport, the sampled area of bed is vanishingly small compared with the area associated with geometrically similar samples of the largest particles. Consequently, the number of large particles available to be captured by the sampler is small, and the probability that none will be captured is high.

[27] Removing zero-transport samples collected at discharges greater than QMIN would result in rating relations that predict larger transport rates than are actually realized in the river (Figure 7). As the algorithm used to fit equation (10) involves a log transformation, which is undefined for zero values, we accommodated the zero-transport samples collected at discharges greater than QMIN by averaging transport rates within discharge increments and fitting the averaged data.

Figure 7.

Plot of fractional transport rates of the >128 mm fraction (open circles) versus discharge at TRAL. The lower curve incorporates zero-mass samples by averaging samples over discharge bins (shown by solid circles) and is the procedure used to derive rating curves for the >128 mm fractions presented in this paper. The upper curve represents the rating relation that would be obtained if zero-transport samples are excluded.

4.4. Estimating Reference Shear Stresses

[28] Following the method of Wilcock and Crowe [2003], we estimated the τri for each size fraction at each sampling location by plotting the sampled transport rates for size fraction i at each sampling location in dimensionless form (Wi*) versus shear stress. Equation (1) was plotted on the same graph for multiple values of τ spanning the range observed at the sampling locations, and τri was adjusted to achieve the best fit between equation (1) and the sample data. The value of τri for which equation (1) equals 0.002 was then read from the graph. Estimates of τ*rm were obtained for each bed load sampling location by interpolating τrm between the two nearest values of τri, and converting τrm to dimensionless form with equation (4).

[29] Fractional transport rating curves developed from the sample data and, in the case of >128 mm fractions, transport rates averaged over increments of discharge were plotted on the same graph to aid fitting. For reasons discussed above, fitting only the nonzero transport rates in this size fraction without consideration of the zero-transport samples would bias the values of the fitted transport function high and lead to underestimation of the reference shear stress for the fraction. In some cases, the appropriate fit is relatively obvious, whereas in other cases it is more subjective. We emphasized achieving a good fit in the vicinity of Wi* = 0.002 in cases where the cloud of sample point did not conform to the shape of equation (1). Examples of a relatively simple and a more subjective fit are given in Figure 8.

Figure 8.

Graphs illustrating method used to estimate τri. Open circles are dimensionless transport rates for (a) the 16–32 mm class and (b) the >128 mm class at TRAL plotted as a function of τ. Shaded triangles indicate transport rates computed with the WC equations, the solid curve is the fractional bed load transport rating curves fitted to the fractional sample data, and the cross indicates the fitted τ, τri coordinates for W* = 0.002. Solid circles in Figure 8b indicate dimensionless transport rates averaged across discharge bins that incorporate zero transport samples, and the vertical dash labeled QMIN indicates the shear stress corresponding to the minimum discharge for which a particle larger than 128 mm has ever been captured at TRAL.

5. Results

5.1. Comparisons of Transport Rates

[30] The bed load transport rates computed with the WC equations are in good agreement with the transport rates measured in the Trinity River, compared with the uncertainty often associated with bed load transport predictions [Ackers and White, 1973; Gomez and Church, 1989]. Calculated and measured total bed load transport rates at all four locations are compared in Figure 9. Over all locations, 68% of the predicted values are within a factor of 2 of the measured rates. Large differences between the predicted and measured values occur primarily at the smallest transport rates, <0.5 kg/s. Differences between the predicted and measured total bed load transport rates at the jth sampling location are expressed here in terms of transport rate error ratios (ɛj), defined as

equation image

where TWj is the sum of total transport rates at location j computed with the WC equations and TSj is the sum of total transport rates determined by bed load sampling for all transport pairs. This metric is comparable to the error ratios reported by Gomez and Church [1989]. Values of ɛj larger than 1 indicate that transport rates predicted by the WC equations are larger than the sampled rates, whereas values less than 1 indicate that the predicted rates are smaller than the sampled rates. Values of ɛj at TRAL, TRGVC, TRLG, and TRDC were found to be 1.04, 1.06, 1.27, and 0.74, respectively.

Figure 9.

Total transport rates predicted with the WC equations versus total transport rates determined from bed load sampling.

[31] Calculated and observed fractional bed load transport rates for each location are compared in Figure 10. Predicted bed load transport rates for individual size classes are within a factor of 2 of the observed values for 31% of the samples at TRAL, 43% at TRGVC, 53% at TRLG, and 51% at TRDC. The pattern of deviations from perfect correspondence shows little in the way of consistent bias across all four sampling locations with one notable exception: Transport of the >128 mm size class invariably falls below the 1:1 line. That is, transport rates for the >128 mm class computed with the WC equations are consistently smaller than the sampled transport rates for all samples for which the >128 mm transport rate is greater than zero.

Figure 10.

Fractional transport rates predicted with the WC equations versus fractional transport rates determined from bed load sampling. Dashed ellipses indicate the plotting positions occupied by the >128 mm class.

[32] Under-prediction of transport in the >128 mm class is also evident in fractional transport rate error ratios (ɛij), which are computed in the same manner as ɛj except that ɛij values are computed for each size class i using the proportions of TWCj and TSj in size class i to form the ratio

equation image

where FWij is the sum of transport rates for fraction i at location j as computed with the WC equations and FSij is the sum of transport rates for the same fraction as determined by bed load sampling. Three of the four locations plus the combined transport data from all four locations show ɛij values for the >128 mm class that are considerably less than 1; that is, the proportion of the predicted transport in the >128 mm class is about 1/3 to 1/16 of the proportion found in the bed load samples (Table 3). Other trends suggested in Table 3 include apparent tendencies for the WC equations to under-predict the transport of particles smaller than 16 mm and to slightly over-predict the transport of particles between 32 and 64 mm.

Table 3. Fractional Bed Load Transport Rates as a Percent of the Total Sampled (TSj) or Computed (TWj) Transport Rates and Fractional Transport Rate Error Ratios (ɛij)a
LocationFraction (mm)
  • a

    Where the subscript j is omitted, variables refer to fractional transport sums and error ratios compiled across all four sample locations.

TRAL% of TSj4.42.44.610.222.942.213.3
% of TWj3.806.521.547.519.90.8
TRGVC% of TSj30.
% of TWj26.701.611.737.420.81.8
TRLG% of TSj50.
% of TWj24.
TRDC% of TSj33.47.811.315.418.011.03.2
% of TWj41.22.01.714.426.913.50.3
All locations% of TS28.95.57.811.817.522.95.6
% of TW25.81.43.314.436.317.71.2

[33] Interpretation of the results reported in Table 3 is complicated by the fact that the majority of the bed load samples contain no particles in the >128 mm class. For example, ɛij for the >128 mm class at TRLG is greater than 1 even though the sampled transport rates plotted in Figure 10 are larger than the corresponding predicted rates. This apparent contradiction arises because the sum of the predicted rates for the class incorporates a large number of small but nonzero transport rates, whereas just two nonzero bed load samples were collected for that size class at that location. Zero-transport samples are, on the other hand, legitimate sample results that must be included in the analysis, at least for samples taken at discharges large enough to cause significant entrainment of the size class. The smallest discharge at which bed load samples contained any particles larger than 128 mm at each of the four sampling locations during the 2006 to 2007 study period (Qmin) are given in Table 2. Using these discharges as a rough filter to define the transport threshold for the >128 mm class, particles larger than 128 mm were captured in 11 of 30 samples taken when the discharge was sufficient to transport that size class at TRAL, seven of 32 samples at TRGVC, two of 27 samples at TRLG, and six of 30 samples at TRDC. Combining fractional transport rates across all locations provide a larger statistical sample size. All 26 samples containing particles in the >128 mm class indicate transport rates larger than the rates predicted from the WC equations. Fractional transport error ratios (ɛi) based on the combined fractional transport data are presented in Table 3.

5.2. Fractional Reference Shear Stresses

[34] Figure 11 shows the values of τ*rm determined for each of the four bed load sampling locations plotted as a function of Fs. Values of τ*rm determined from the Trinity data are 15–57% larger than those predicted by equation (3). These differences are approximately equal to or larger than the probable error bounds on the estimates for three of the four sampling locations. Uncertainty margins notwithstanding, the Trinity τ*rm values do not conform well to the shape of equation (3). No decrease in τ*rm with increasing Fs is apparent once values of Fs exceed 0.04 (Figure 11).

Figure 11.

Values of τ*rm determined from bed load samples versus bed surface sand fraction. Error bars indicate ±29%. The relation for τ*rm presented by Wilcock and Crowe [2003] is shown for comparison.

[35] Figure 12 shows values of τri/τrm determined from the bed load samples plotted as a function of Di/Dsm, along with equation (5). The trend of the Trinity River data is consistent with the trend of equation (5) for Di/Dsm ≤ 1 but deviates from the equation for Di/Dsm > 1. Specifically, the inflection to a larger slope defined by equation (5b) appears to be shifted right to a larger value of Di/Dsm in the Trinity data. This result is insensitive to uncertainty in the estimated values of τ used to compute Wi* because the ratio form cancels any errors in the absolute shear stress magnitudes.

Figure 12.

Reference shear stress ratios as a function of relative grain size. The dashed and solid curves depict equation (5) and equation (13), respectively. Equation (13) is presented in section 6.2.

6. Discussion

6.1. Sensitivity of Transport Computations to Input Errors

[36] Two types of input errors have the potential to alter the computational results obtained from the WC equations: errors in the magnitudes of τ at the sampling locations and errors in specification of the particle size distribution of the bed surface.

[37] Uncertainties in τ were examined by computing total transport rates and >128 mm transport rates using τ values corresponding to the upper and lower range of uncertainty in the original estimates. These values are defined by τu = 1.29τ and τl = τ/1.29. Total and fractional transport error ratios show that neither increasing nor decreases in the input shear stresses significantly reduces the tendency to under-predict >128 mm transport rates (Figure 13). Use of τl, however, causes the WC equations to under-predict total transport across all sampling locations by a factor of 3.3 (ɛj = 0.3), whereas use of τu results in over-prediction of total transport by a factor of 2.6.

Figure 13.

Graph showing the effect of increasing or decreasing values of τ input to the WC equations on transport rate error ratios for total transport (ɛj) and fractional transport of the >128 mm class (ɛij) across all sampling locations.

[38] The sensitivity of the computed fractional transport rates to uncertainty in the observed bed surface particle size distribution was found to have some potential to account for differences in the measured and computed transport rates for coarse fractions. This possibility was investigated by computing total and >128 mm transport rates using two alternative bed surface size distributions. One distribution simulated an error in specifying the surface fraction of particles larger than 128 mm at TRAL by increasing Fi for the 128–181 mm and 181–256 mm classes by 50%, from a total percentage larger than 128 mm of 12% as determined by pebble count to a total of 18% (fractions near the middle of the distribution were reduced slightly to maintain a cumulative total of 100%). The other distribution simulated an error in specifying the percent of the surface covered by the sand fraction at TRAL by increasing Fs from the measured value of 1% to 5%.

[39] The 50% increase in Fi for the coarsest particle fractions at TRAL was found to decrease total transport by about 10% and increase transport rates of >128 mm particles by a factor of about 1.4 across the full range of discharges. As expressed by transport error ratios, the effect of these differences would be to increase ɛij for the >128 mm class at TRAL from 0.06 to about 0.09. The simulated fivefold increase in Fs at TRAL caused total transport rates to increase by a factor of about 3 at moderate to high discharges. Increasing Fs also caused transport rates for the >128 mm fraction to increase by a factor of up to 5 at the highest discharges when these sizes are in transport. Together, these differences increased ɛij for the >128 mm class to about 0.1. Thus increasing Fs did not materially increase the proportion of the total transported sediment that is larger than 128 mm.

[40] Overall, it appears that errors in the input parameters are unlikely to be responsible for the differences in the relative magnitudes of fractional transport rates reported herein. It is plausible that the WC equations could be forced to approximate the total transport rates measured in the Trinity River while also reproducing the observed fractional transport rates of the coarsest fraction, but only through careful tuning of the input variables.

6.2. Deviation From the WC Hiding Function

[41] Wilcock and Crowe [2003] quantified differences in the relative transport rates of different size fractions with equation (5), the “hiding function” that determines the degree to which equal mobility among the different size fractions is realized [Parker and Klingeman, 1982]. The value of the exponent b expresses the degree to which equal mobility is achieved by the particle entrainment and transport processes. An exponent of zero in a relation of the form of equation (5a) produces a flat line for which the value of τri is independent of particles size; this is the definition of equal mobility. Conversely, an exponent of 1 corresponds to an upward sloping line for which the value of τri is exactly proportional to particles size or, in other words, perfect size selectivity. The results presented in Figure 12 indicate that bed load transport in the Trinity River more closely approximates equal mobility across the full range of grain sizes than is implied by equation (5).

[42] The WC hiding function is somewhat unusual in that it incorporates a substantial increase in the value of b for particles larger than Dsm, and values of b exceed the range reported elsewhere when Di/Dsm exceeds about 1.5. Parker and Klingeman [1982] reported a comparatively subtle upward curvature in log space based on field data (b = 0.071 for Di/Dsm ≤ 1 and b = 0.104 for Di/Dsm > 1), but b is more commonly reported as a single value over the full range of grain sizes. Estimates of b reported in previous studies include 0.24 [Ashworth and Ferguson, 1989], 0.12, [Ferguson et al., 1989], 0.0953 [Parker, 1990], 0.113 [Andrews, 1994], and 0.006–0.058 [Andrews and Nankervis, 1995]. In general, b is typically found to be in the range 0–0.3 [Wilcock et al., 2001].

[43] It should be noted that some of the studies cited here formulated the relation in terms of (τ*ri/τ*rm), in which case the equivalent exponent is equal to (b − 1). The values reported herein are presented as b, irrespective of the form used by the original authors. In addition, relations like equation (5a) can be developed with respect to either the surface particle size distribution or the subsurface size distribution. The values of b associated with surface mobility differ substantially from those associated with subsurface mobility because the mobility of the surface layer is regulated by grain-scale hiding effects, whereas the mobility of the subsurface is regulated by the overlying surface layer [Parker and Klingeman, 1982; Wilcock and Crowe, 2003]. All the estimates for b cited above are referenced to the surface layer.

[44] Equation (5b) can be modified to better fit the Trinity information by inserting a scaling parameter in the term containing Di/Dsm and adjusting the remaining parameters to

equation image

This fit yields values of the b ranging between 0.09 for small Di/Dsm and 0.3 for the largest particle size class. Alternatively, the Trinity data can be fit reasonably well (r2 = 0.76) with a power function equivalent to equation (5a):

equation image

The coefficient in equation (14) should be equal to 1 so that the function passes through the 1, 1 plotting position. The fitted value of 1.06 is comfortably close to that expectation, and replacing it with 1 has a negligible effect on the statistical fit. With either fitting approach, values for b derived from the Trinity data lie well within the range of values found elsewhere.

6.3. Bed Surface Condition and τ*rm

[45] It is possible that deviations of the empirical values of τ*rm shown in Figure 11 from the WC relation reflect errors in the magnitudes of the shear stresses estimated for the Trinity River. If so, the apparent success of the WC equations in correctly predicting total transport rates is an artifact of those errors. Reducing τ*rm to values commensurate with equation (3) would require reducing estimated shear stresses by an amount similar to the maximum probable error. As pointed out earlier, such a reduction causes the WC equations to under-predict the total transport rate across all sampling locations by a factor of more than 3 and intensifies the tendency for the equations to under-predict transport rates for the >128 mm class.

[46] Alternatively, it seems reasonable to accept that the values of τ*rm are indeed larger in the Trinity River than is anticipated by equation (3). There is substantial evidence that the value of τ*rm for a sediment mixture with a given mean particle size is influenced by a number of additional factors, including differences in the particle size distribution of the mixture, differences in the arrangement of particles on the streambed, and hydraulic factors. For example, Duan and Scott [2008] derived an entrainment function in which τ*ri for all size fractions is a function of flow depth and the fraction of particles in each of the size classes, as well as Di. Kirchner et al. [1990] showed that water-worked beds require significantly larger shear stresses to initiate particle entrainment than unworked beds due to changes in the distribution of friction angles and the degree of particle protrusion.

[47] Furthermore, Kirchner et al. [1990] note that flume experiments characteristically show significant sediment transport at relatively small shear stresses near the lower bounds of the expected range of critical shear stresses. The observations of Wilcock et al. [2001] represent conditions in a sediment recirculating flume, where the bed load input rate is equal to the bed load transport rate and the system is, by definition, in equilibrium. Under these conditions, easily entrained grains with low friction angles are continuously replenished throughout the flume run. The situation in the Trinity River is quite different, in that sediment supplies from upstream are limited by the dams and the bed is exposed to shear stresses less than or near the threshold needed for significant sediment transport much of the time. Exposure to relatively low shear stresses tends to condition the bed by moving easily entrained particles to more stable positions [Paphitis and Collins, 2005].

[48] None of these additional factors is captured in equation (3), which casts τ*rm as a function of the fraction of sand (Fs) on the bed surface only. That this approach worked well with the laboratory data used to develop the WC equations is not surprising, in that Wilcock and Crowe [2003] analyzed the transport rates of five sediment mixtures produced by adding varying amounts of sand to a single gradation of gravel. Thus the differences in the particle size distributions of the five sediment mixtures were entirely determined by the fraction of sand added to the gravel.

[49] It is suggested here that Fs alone provides an incomplete description of the bed surface condition and that other choices of parameters for scaling τ*rm are worth considering. A relatively simple alternative is the geometric standard deviation of the sediment particle size distribution as presented by Parker [1990]:

equation image

Differences in the fraction sand on the bed are clearly contained in σsg through differences in Fi when i indexes the sand fraction, but σsg also incorporates information regarding the remainder of the surface grain size distribution.

[50] We initially evaluated the potential for using σsg for this purpose by reanalyzing the laboratory data used to derive the WC equations, which was first reported by Wilcock et al. [2001] and can be obtained from an archival Web site (see Wilcock and Crowe [2003] for details). Values of τ*rm determined by Wilcock and Crowe [2003] were plotted as a function of σsg computed from the laboratory surface grain size data, and a good fit could be obtained (Figure 14) with

equation image

A logistic function was chosen to fit these data because is it consistent with expectation expressed by Wilcock and Crowe [2003] that τ*rm asymptotically approaches a minimum value in the neighborhood of 0.021 for large σsg and must take a value appropriate for a uniform sediment (i.e., 0.036) as σsg approaches zero. The same functional form can be fit to the values of τ*rm and σsg computed from the Trinity River data (Figure 14) with

equation image

Equation (16a) retains the assumption embedded in equation (3) that τ*rm approaches a minimum value of 0.021 when σsg is large and a maximum value of 0.0365 when σsg is small. No data, however, are available to assess whether the maximum value of 0.0365 is valid. Likewise, minimum and maximum values of τ*rm cannot be specified for equation (16b) outside the range of the Trinity data, so the flattening of the curve indicated in equation (16b) and Figure 14 is hypothetical.

Figure 14.

Plots of τ*rm presented by Wilcock and Crowe [2003] versus σsg computed from the surface grain size data reported by Wilcock et al. [2001], and values of τ*rm and σsg determined from bed load samples and surface pebble counts in the Trinity River. The plus sign in Figure 14b indicates that data from two locations (TRGVC and TRLG) plot on top of one another.

[51] Figure 14 implies that variability in τ*rm depends on multiple variables, such that no function of a single parameter is likely to be generally applicable to all streams. Equation (16a) approximates the entrainment threshold for the mean particles size in a recirculating flume, whereas equation (16b) represents current conditions in the Trinity River, i.e., sediment-limited conditions downstream from a dam. Other relations would hold for other conditions. We hypothesize that quantification of these conceptual relationship requires a means for quantifying bed surface packing, as well as a parameter quantifying the grain size distribution of the bed surface.

6.4. Significance of Coarse-Fraction Transport for Modeling

[52] The coarsest size fractions in the bed load are a small proportion of the total bed load flux. Thus errors in predicting coarse-fraction transport rates have little significance for applications requiring estimates of total bed load transport rates. These errors may become important, however, when fractional transport rates are incorporated into morphodynamic models. The potential for relatively small differences in the transport rates assigned to the coarsest size fractions in the sediment supply to produce large differences in the output of morphodynamic models is demonstrated here using an early version of the “spawning gravel refresher,” a simple one-dimensional numerical model of stream channel evolution. This model is being developed as part of the River Restoration Toolbox available through the National Center for Earth Surface Dynamics [Wilcock and Parker, 2006].

[53] The model implements a uniform flow approximation, the Wilcock-Crowe transport equations, and sediment continuity to evolve a channel with an equilibrium bed slope and surface grain size distribution under a given flow and sediment feed regime. The channel is assumed to be rectangular and of constant width. Inputs include a sediment feed rate and particle size distribution, a subsurface sediment size distribution, an initial surface layer size distribution, and an annual hydrograph. The version of the code used for this analysis was provided by Gary Parker in 2006. Members of Parker's group at the University of Illinois continue to improve the model and add functionality, such as the capacity to store the stratigraphic information associated with substrate deposited during a simulation (E. Viparelli et al., manuscript in preparation, 2009).

[54] The particle size distribution of the sediment feed, substrate, and initial surface were assigned the same value for all runs (Table 4). The initial bed slope was set to 0.0037 and channel width was set to 40 m. An input hydrograph consisting of a constant water discharge of 200 m3/s for 60 d/a was cycled repeatedly until an equilibrium condition was reached. For the initial model run (run 1), the sediment feed rate was set to 5000 t/a. These input parameters were chosen to loosely conform to Trinity River values to ensure that their relative scaling is realistic. The run came to equilibrium with a bed slope of 0.00367 and a surface substrate characterized by a geometric mean size of 114 mm, a median particle size of about 181 mm, and the grain size distribution given in Table 4.

Table 4. Input and Output Particle Size Distributions for Morphodynamic Model Runs
Percent Finer Than (mm)Feed and SubstrateInitial Bed SurfaceFinal Bed Surface, Run 1Final Bed Surface, Run 2Final Bed Surface, Run 3

[55] The model was altered slightly for subsequent runs by adding three lines of code that decreases τ*ri for the largest size class (181–256 mm) by 10%. The new code consists of an if-then statement that multiplies the value of ϕ for the 181–256 mm size class by 1.11 prior to computation of W* via equation (1). This has the effect of decreasing τ*ri for 181–256 mm particles by 10%, but changes nothing else in the model.

[56] A second model run (run 2) began with the same input parameters as run 1 but was conducted with the altered code. The model eventually came to equilibrium with a bed slope of 0.0032 and a bed surface with a geometric mean particle size of 103 mm, a median particle size of about 145 mm, and the grain size distribution given in Table 4.

[57] A third set of runs (run 3) used the altered code and the same initial input parameters as the runs 1 and 2, but the sediment feed rate was increased until the model converged with an equilibrium channel slope of 0.00367, as in run 1. The required sediment feed rate was found to be 37,000 t/a.

[58] Additional model runs confirmed that changes in the mobility characteristics of finer size fractions have relatively little effect on the morphodynamic response of the model. For example, decreasing τ*ri for the 45–64 mm size fraction by 10% made almost no difference in the equilibrium slope or bed surface grain size distribution.

[59] This sensitivity analysis demonstrates that the output from morphodynamic models can be highly sensitive to relatively small differences in the mobility of the largest particles in the bed load, even when those large particles represent a small proportion of the total bed load. In this case, increasing the mobility of the largest 4% of the bed load distribution by 10% produced a 13% decrease in equilibrium channel slope and a 10% decrease in the mean particle size on the equilibrium bed surface. Moreover, restoring the bed slope to the value attained with the original particle mobility parameters required a 7.4-fold increase in the sediment supply rate.

7. Conclusion

[60] A comparison between fractional transport rates and loads predicted with the WC equations and fractional transport rates and loads determined from an intensive bed load sampling effort indicates that the WC formulation is a relatively good predictor of total bed load transport in the Trinity River. Total bed load transport rates computed with the WC equations are within a factor of about 1.3 of the total bed load transport rates derived from bed load sampling, despite the inherent uncertainty associated with estimating boundary shear stresses in the field.

[61] The WC equations consistently under-predict the transport of particles larger than 128 mm. At two of the four sampling locations, predicted transport rates for that size class are underestimated by an order of magnitude or more. These discrepancies cannot be attributed solely to errors in estimating boundary shear stresses. The coarsest particles represent a small portion of the sediment transported, so their underestimation is of little significance in the context of total loads. The mobility of the largest clasts in the sediment supplied to the channel nonetheless has a substantial effect on the equilibrium bed surface texture that develops under a given flow and sediment regime. This represents a feedback that has the potential to materially alter predictions of subsequent transport rates and morphodynamic evolution.

[62] The downstream variations in bed surface particle size, sand fraction, and τrm in the Trinity River are consistent with the differences expected for locations downstream from a dam. Dm becomes smaller, Fs increases, and τrm decreases with downstream distance. The τ*rm and σsg show less longitudinal variability, except for at TRAL, where τ*rm is especially large and σsg is approximately half of the values determined for the other three sampling locations. TRAL is nearest to Lewiston Dam and is upstream from any significant natural bed material supply. No downstream trends are apparent, however, in the relationships between τrm and τri.

[63] Bed load transport in the Trinity River more closely approximates equal mobility across a wide range of particle sizes than is implied by the WC equations. A better fit to the Trinity River bed load sample information was obtained through a modification of the WC hiding function that relates τrm and τri. The modified hiding function predicts values of τri/τrm similar to those predicted by the WC hiding function for Di/Dsm ≤ 1, but a steep increase in τri/τrm with grain size for Di/Dsm > 1 predicted by the WC relationship is delayed to values of Di/Dsm near 3 in the modified equation.

[64] Values of τ*rm as represented in the WC equations are solely a function of the fraction of sand on the streambed. This formulation appears to be inappropriate for the Trinity River for two reasons. First, the WC equations were derived using sediment mixtures that differed only in the amount of sand added to the gravel, whereas the attributes of natural fluvial sediments can vary across the entire particle size distribution. We suggest casting τ*rm as a function of the geometric standard deviation of the bed surface particle size distribution, and show that this parameter accounts for the variability in τ*rm in the laboratory data used to derive the WC equations at least as well as the fraction of sand. Second, values of τ*rm determined from the Trinity bed load samples are significantly larger than those computed with the WC equations, and their range (0.031–0.052) exceeds the maximum that can be attained using the WC formulation (0.036). The larger dimensionless reference shear stresses found in the river are likely related to differences in bed surface organization caused by sediment supply limitations and the preferential entrainment of unstable particles.


[65] The authors thank Peter Wilcock for clarifying details regarding the equations used in this study and Blair Greimann for instructive discussions regarding alternative approaches to scaling particle entrainment by size fraction.