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CHAPTER 4: RELATIONS FOR THE CONSERVATION OF BED SEDIMENT. This chapter is devoted to the derivation of equations describing the conservation of bed sediment. Definitions of some relevant parameters are given below. . q b = volume bedload transport rate per unit width [L 2 T -1 ]
RELATIONS FOR THE CONSERVATION OF BED SEDIMENT
This chapter is devoted to the derivation of equations describing the conservation of bed sediment. Definitions of some relevant parameters are given below.
p = porosity of sediment in bed deposit 
(volume fraction of bed sample that is holes rather than sediment: 0.25 ~ 0.55 for noncohesive material)
g = acceleration of gravity [L/T2]
x = boundary-attached streamwise coordinate [L]
y = boundary-attached transverse coordinate [L]
z = boundary-attached upward normal (quasi-vertical) coordinate [L]
t = time [T]
x = nearly horizontal boundary-attached “streamwise” coordinate [L]
y = nearly horizontal boundary-attached “transverse” coordinate [L]
z = nearly vertical coordinate upward normal from boundary [L]
Double-click on the image to see a video clip of bedload transport of 7 mm gravel in a flume (model river) at St. Anthony Falls Laboratory, University of Minnesota. (Wait a bit for the channel to fill with water.) Video clip from the experiments of Miguel Wong.
rte-bookbedload.mpg: to run without relinking, download to same folder as PowerPoint presentations.
This corresponds to the original form derived by Exner.
(Yes, this is still a course on 1D morphodynamics, but it is useful to know the 2D form.)
denote unit vectors in the x and y directions.
Double-click on the image to see the transport of sand and pea gravel by a turbidity current (sediment underflow driven by suspended sediment) in a tank at St. Anthony Falls Laboratory. Suspended load is dominant, but bedload transport can also be seen. Video clip from experiments of Alessandro Cantelli and Bin Yu.
rte-bookturbcurr.mpg: to run without relinking, download to same folder as PowerPoint presentations.
Es = volume rate per unit time per unit bed area that sediment is entrained from the bed into suspension [LT-1].
Ds = volume rate per unit time per unit bed area that sediment is deposited from the water column onto the bed [LT-1].
Let denote the volume concentration of sediment c in suspension at (x, z, t), averaged over turbulence. Here c = (sediment volume)/(water volume + sediment volume).
In the case of a dilute suspension of non-cohesive material,
where cb denotes the near-bed value of c .
Similarly, a dimensionless entrainment rate E can be defined such that
CASE OF 1D BEDLOAD + SUSPENDED LOAD, ADDITION OF TECTONICS (SUBSIDENCE OR UPLIFT)
The analysis below is based on Paola et al. (1992). Conserve bed sediment between some base level z = base(x, t) and the bed surface :
The tectonic subsidence rate (uplift rate ) is given as
Thus with the previously-presented evaluations for Es and Ds:
z = upward normal coordinate from the bed [L]
= local streamwise flow velocity averaged over turbulence [L/T]
= local volume sediment concentration averaged over turbulence 
H = flow depth [L]
qs = volume transport rate of suspended sediment per unit width [L2/T]
U = vertically averaged streamwise flow velocity [L/T]
C = vertically flux-averaged volume concentration of sediment in suspension 
(mass of sediment in control volume)/t =
net mass inflow rate of suspended sediment
+ mass rate of entrainment of sediment into suspension
– mass rate of deposition onto the bed
or reducing with the relation qs = UCH and previous evaluations for Es and Ds,
REDUCTION: 1D EXNER FORMULATION IN TERMS OF TOTAL BED MATERIAL LOAD
In most cases the condition << 1 prevails, allowing the approximation
The simplified form of the above equation can be combined with the Exner equation of conservation of bed sediment,
to yield the following form for Exner:
or defining total bed material load qt = qb + qs,
Let denote the local average velocity in the transverse (y) direction. Then
fi'(z', x, t) = fractions at elevation z' in ith grain size range above datum in bed . Note that over all N grain size ranges:
qbi(x, t) = volume bedload transport rate of sediment in the ith grain size range [L2/T]
The active, exchange or surface layer approximation (Hirano, 1972):
Sediment grains in active layer extending from - La < z’ < have a constant, finite probability per unit time of being entrained into bedload.
Sediment grains below the active layer have zero probability of entrainment.
Fractions Fi in the active layer have no vertical structure.
Fractions fi in the substrate do not vary in time.
where the interfacial exchange fractions fIi defined as
describe how sediment is exchanged between the active, or surface layer and the substrate as the bed aggrades or degrades.
REDUCTION OF SEDIMENT CONSERVATION RELATION USING THE ACTIVE LAYER CONCEPT contd.
it is found that
The total bedload transport rate summed over all grain sizes qbT and the fraction pbi of bedload in the ith grain size range can be defined as
The conservation relation can thus also be written as
Summing over all grain sizes, the following equation describing the evolution of bed elevation is obtained:
Between the above two relations, the following equation describing the evolution of the grain size distribution of the active layer is obtained:
where 0 1 (Hoey and Ferguson, 1994; Toro-Escobar et al., 1996). In the above relations Fi, pbi and fi denote fractions in the surface layer, bedload and substrate, respectively.
The substrate is mined as the bed degrades.
A mixture of surface and bedload material is transferred to the substrate as the bed aggrades, making stratigraphy.
Stratigraphy (vertical variation of the grain size distribution of the substrate) needs to be stored in memory as bed aggrades in order to compute subsequent degradation.
1D GENERALIZATIONS: TECTONICS, SUSPENSION, TOTAL BED MATERIAL LOAD
To include tectonics, make the transformation - base in the above derivation (or integrate from z’’ = 0 to z’’ = - base, where z’’ = z’ - base) to obtain:
To include suspended sediment, let vsi = fall velocity, Ei = dimensionless entrainment rate, and denote the near-bed volume concentration of sediment, all for the ith grain size range, so that the relation generalizes to:
Repeating steps outlined previously for uniform sediment, if qtT denotes the total bed material load summed over all sizes and pti denotes the fraction of the bed material load in the ith grain size range,
Rivers often sort their sediment. An example is downstream fining: many rivers show a tendency for sediment to become finer in the downstream direction.
Long profiles showing downstream fining and gravel-sand transition in the Kinu River, Japan (Yatsu, 1955)
median bed material grain size
WHY THE CONCERN WITH SEDIMENT MIXTURES ? contd.
Downstream fining can also be studied in the laboratory by forcing aggradation of heterogeneous sediment in a flume.
Downstream fining of a gravel-sand mixture at St. Anthony Falls Laboratory, University of Minnesota (Toro-Escobar et al., 2000)
Many other examples of sediment sorting also motivate the study of the transport, erosion and deposition of sediment mixtures.
Sediment approximated as uniform in size
Hirano, M., 1971, On riverbed variation with armoring, Proceedings, Japan Society of Civil Engineering, 195: 55-65 (in Japanese).
Hoey, T. B., and R. I. Ferguson, 1994, Numerical simulation of downstream fining by selective transport in gravel bed rivers: Model development and illustration, Water Resources Research, 30, 2251-2260.
Paola, C., P. L. Heller and C. L. Angevine, 1992, The large-scale dynamics of grain-size variation in alluvial basins. I: Theory, Basin Research, 4, 73-90.
Parker, G., 1991, Selective sorting and abrasion of river gravel. I: Theory, Journal of Hydraulic Engineering, 117(2): 131-149.
Toro-Escobar, C. M., G. Parker and C. Paola, 1996, Transfer function for the deposition of poorly sorted gravel in response to streambed aggradation, Journal of Hydraulic Research, 34(1): 35-53.
Toro-Escobar, C. M., C. Paola, G. Parker, P. R. Wilcock, and J. B. Southard, 2000, Experiments on downstream fining of gravel. II: Wide and sandy runs, Journal of Hydraulic Engineering, 126(3): 198-208.
Yatsu, E., 1955, On the longitudinal profile of the graded river, Transactions, American Geophysical Union, 36: 655-663.