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Water: WARSSS

Channel Processes: Bedload Transport

Bedload is that portion of the total sediment in transport that is carried by intermittent contact with the streambed by rolling, sliding, and bouncing. Bedload transport prediction has two broad approaches: those calculated by force, and those by power. The summary by Yang (1996) describes nine specific bedload formula approaches:

  1. Shear stress,
  2. Energy slope,
  3. Discharge,
  4. Velocity,
  5. Bed form,
  6. Probabilistic,
  7. Stochastic,
  8. Regression, and
  9. Equal mobility.

There appear to be as many approaches to bedload transport in the literature as there are varied stream conditions. An earlier summary of the status of sediment transport formulas by Vanoni (1975, pg 190) was:

"Unfortunately, available methods or relations for computing sediment discharge are far from satisfactory, with the result that plans for works involving sediment movement by water cannot be based strongly on such relations. At best these relations serve as guides to planning, and usually the engineer is forced to rely on experience and judgment in such work."

An excellent, detailed summary of sediment transport equations, including bedload relations, is presented in Reid and Dunne (1996). The application of any sediment transport relation relies on the understanding of its assumptions/limitations and the availability of measured data to properly calibrate the model. Nonetheless, some basic concepts are presented here as they may relate to the prediction methods.

When the flow conditions exceed the criteria for incipient motion, sediment particles on the streambed start to move. The transport of bed particles in a stream is a function of the fluid forces per unit area…the tractive force or shear stress (eqII11b), acting on the streambed. Under steady, uniform flow conditions, the shear stress is

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(eq.II-2)       Equation II-2 describs fluid forces exerting shear stress on the streambed.

where:
y is the specific weight of the fluid,
D is the mean depth, and
S is the water surface slope.

The gravitational force resisting particle entrainment, fg, is proportional to:

(eq.II-3)       equation II-3

where:
y(sub's') is the specific weight of sediment, and
d is the particle diameter.

The Shields relation (Shields 1936) with modifications by Graf (1971) developed relations associated with initiation of particle movement using the ratio of the fluid forces to the gravitational force, proportional to the dimensionless quantity called the critical dimensionless shear stress, t*(sub'c'), where:

(eq.II-4)       equation II-4

The dimensionless bed-material transport rate per unit width of streambed, Q*B is:

(eq.II-5)       II-6: equation for the dimensionless bed-material transport rate per unit width of streambed

where:
g is gravity, and
Qs is the volumetric transport rate per unit width of streambed determined from bedload samples.

The empirical function developed by Parker (1979) is

(eq.II-6)       Equation II-6. The empirical function developed by Parker (1979)

where:
t*(sub'c')is the threshold value of t* required to initiate particle motion.

Figure 7

The widely used application of critical shear stress by particle size class as initially developed by Shields (1936) is shown in Figure 7. (PDF, 160 kb, 1 p.)

Field measurements by the author have shown that for heterogeneous bed materials, larger particles are entrained at shear stress values much lower than indicated in this relation. A revised relation will be presented in the prediction section based on measured stream values.

As reported by Erman et al. (1988), Equation II-5 is identical to the Einstein bedload function (Einstein, 1950) but has a much simpler mathematical form. For the bed-material transport curve plotted in Figure 8, the threshold dimensionless shear stress was computed by a function derived by Andrews (1983) and was found to be in excellent agreement with observed threshold conditions in Sagehen Creek (Andrews and Erman, 1986).

Figure 8



Figure 8. Dimensionless transport rate of bed material in Sagehen Creek in relation to dimensionless shear stress. The 58 daily totals from 1982, 1983 and 1984 are in excellent agreement with the curve drawn from a function developed by Einstein (1950) and later modified by Parker (1979). (from Andrews and Erman, 1988) (left)



















Figure 9

Figure 9. Relation between the ratio of threshold particle diameter to the median particle diameter of subsurface bed material and the critical dimensionless shear stress (from Andrews and Erman, 1986)


The computed value for eqII03b for Sagehen Creek is 0.043. The limits of the curve shown in Figure 9 are within the range of Einstein's original data used by Parker (1979).

Andrews (1984) investigated critical dimensionless shear stress by developing a relationship using the ratio of a given size particle, di to subsurface eqII07a where eqII03b is equal to:

(eq.II-7)       Equation II-7

The values of eqII03b, derived from measured data where eqII07b values varied from 0.3 to 4.0, are shown in Figure 9. The threshold shear stress for median particles in the riverbed surface can be calculated using Equations II-4 and II-7. Surface and subsurface material sampling procedures are summarized by Bunte and Abt (2001).

Work by Andrews and Erman (1986) on Sagehen Creek, California, and additionally reported by Andrews and Nankervis (1995) produced the equation:

(eq. II-8)       Equation II-8

The importance of the entrainment calculation is not only significant for bedload transport but also for calculating the competence of the river to transport the largest sediment clasts made available by its catchment. The equations described by Bagnold (1980), Erman et al (1988), Einstein (1950), Parker (1979) and Andrews and Erman (1986) require some form of entrainment relation for initiation of particle motion. Various computational methods for eqII03b are summarized in Table 1 (Reid and Dunne 1996). Their summary included the statement:

"The uncertainties in estimating this critical stress…therefore seriously degrade efforts to predict transport rates and make worthwhile the expenditure of some effort to constrain these rates using local field evidence."

Table 1. Equations for initiation of motion. eqII08a (dynes-cm-2) is calculated for particles of size di (cm) lying in a pavement or on a substrate with median grain size D50 (cm) or geometric mean diameter Dg (cm) (from Reid and Dunne 1996).

table01a table01b

A modified procedure that avoids sampling sub-pavement and uses field evidence to determine eqII03b requires a core sample at specific elevations and locations of depositional features such as point bars, as described by Rosgen (1996, pages 7-7 to 7-9, and 2001b). The objective of this core sample is to substitute for pavement/sub-pavement samples by obtaining Di (largest particle size of the bar sample) and, D50, (median size particle from the bar sample). The Di is substituted for the D50 of the bed of the riffle and the median diameter, D50 of the core sample of the bar is substituted for the sub-pavement eqII07a in order to calculate values of critical dimensionless shear stress, eqII03b (Equation II-7). If this resultant ratio of median diameters of bed versus bar sample is outside a range of 3-7, then the ratio of the Di from the bar sample is divided by the D50 of the bed, using the relations in Equation II-8, (Rosgen 2001b). This relationship agrees best with measured data if this ratio range is within 1.3-3.0. The bankfull shear stress value (eqII03b ) determined in this manner can be used for bedload transport relations and for calculation of competence. A rearrangement of Equation II-4 as shown in Equation II-9 provides an estimate of competence, or a calculation of the depth of flow and/or water surface slope required to move the largest particle of sediment made available for a given dimensionless shear stress. The largest particle required to be moved using this method is the largest clast measured on the surface of the lower 1/3 position of the point bar (Di).

(eq. II-9)       Equation II-9. A rearrangement of Equation II-4 as shown in here provides an estimate of competence or a calculation of the depth of flow and/or water surface slope required to move the largest particle of sediment made available for a given dimensionless shear stress.

Bagnold's approach (1980), involves the use of stream power, the mean rate of kinetic energy supply and dissipation along a stream channel.

Stream power (eqII10a), is defined as:

(eq. II-10)       Equation II-10. Definition of stream power.

Unit stream power or power per unit of streambed area () is defined as:

(eq.II-11)       eqII11

where:
eqII11b is defined in equation (2), and
u is mean velocity

Yang (1996) uses unit stream power per unit weight (eqII12a)

where:

(eq. II-12)       Equation II-12

The relation between unit stream power (equation II-11), and measured sediment transport rate by size classes is depicted in Figure 10 (Dunne and Leopold 1978) (PDF 160 kb, 1 p.).

The application of this relation is related to corresponding changes in sediment transport by particle size as related to changes in stream channel depth, velocity and/or slope…the variables influencing unit stream power. Thus, increases in width/depth ratio due to streambank instability or direct disturbance would show potential sediment deposition, filling of pools and other related channel consequences.

Figure 10

Bagnold's (1980), transport relation (ib) using stream power is shown as:

(eq. II-13)     Equation II-13.

where:
condition for equation II-13

Where ib is transport rate of bedload by immersed weight per unit width, i(b)* is a reference value for ib, eqII10a is stream power per unit bed area (equation II-11), eqII13b is threshold unit stream power, Y is depth of flow, Y* is a reference value of Y (0.1m), D is mode size of bed material, D* is reference value of D (0.0011m), eqII13c is excess density of solids, eqII13d is Shields threshold criterion, g is gravitational acceleration, and eqII13e is density of liquid. Dimensionless shear stress value eqII03b as shown in Equation II-8 is essentially the same as eqII13d in Equation II-13.

After testing 7 bedload equations on 22 streams with measured hydraulic and sediment data, Lopes et al. (2001a) determined the best equations overall were those of Schoklitsch (1962) and Bagnold (1980). A comparison of various bedload models for the Chippewa River in Wisconsin against measured bedload is shown in Figure 11 (Lopes et al. 2001a) (PDF 119 kb, 1 p.). Gomez and Church (1989) concluded that prediction of bedload under limited hydraulic information is best accomplished by using equations based on the stream power concept. However, after testing 12 various bedload equations in gravel-bed rivers for 410 bedload events, they concluded that none of the equations performed consistently due to the limitations of the data and the complexity of the sediment transport phenomena.

Figure 11

A reference dimensionless bedload sediment rating curve developed for a specific stream type/stability (Figure 12) (Troendle et al. 2001) can be used to develop a localized bedload sediment rating curve. To convert from dimensionless values, measured bedload and concurrent streamflow discharge must be obtained at the bankfull stage for a stream reach that is of the same type/stability relation. The reference curve should represent a range of stable stream types (Rosgen, 1994) and stability ratings (Pfankuch 1975, Rosgen 2001b). Multiplication of the values of both bankfull stage bedload and stream discharge times the appropriate bedload sediment ratios (SB/SBbkf) and corresponding discharge ratios (Q/Qbkf) for the appropriate reference curve generates a dimensioned bedload rating curve. The bankfull discharge values in the equation were converted from an instantaneous discharge to mean daily discharge (Figure 12). Reference dimensionless bedload rating curves are constructed from actual bedload and streamflow measurements over a wide range of flows for specific stream types and stability ratings in the hydro-physiographic province being assessed.

Figure 12

Figure 12. Dimensionless bedload transport for all historical B3 streams plotted over the pooled model for reference streams (from Troendle et al, 2001).

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