RELATIONS FOR THE CONSERVATION OF BED SEDIMENT

1
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.

• qb volume bedload transport rate per unit width
  L2T-1
• qs volume suspended load transport rate per
  unit width L2T-1
• qt qb qs volume bed material transport rate
  per unit width L2T-1
• gb ?sqb mass bedload transport rate per unit
  width ML-1T-1
• (corresponding definitions for gs, gt)
• bed elevation L
• ?p porosity of sediment in bed deposit 1
• (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

2
COORDINATE SYSTEM

• 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

3
ILLUSTRATION OF BEDLOAD TRANSPORT
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.
4
CASE OF 1D, BEDLOAD ONLY, SEDIMENT APPROXIMATED
AS UNIFORM IN SIZE
or thus
This corresponds to the original form derived by
Exner.
5
2D GENERALIZATION, BEDLOAD ONLY (Yes, this is
still a course on 1D morphodynamics, but it is
useful to know the 2D form.)
where
denote unit vectors in the x and y directions.
6
ILLUSTRATION OF MIXED TRANSPORT OF SUSPENDED LOAD
AND BEDLOAD
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.
7
CASE OF 1D BEDLOAD SUSPENDED LOAD
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.
or thus
8
EVALUATION OF Ds AND Es
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
Thus
9
2D GENERALIZATION, BEDLOAD SUSPENDED LOAD
10
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
2D generalization
11
TRANSPORT RATE OF SUSPENDED SEDIMENT
Definitions z upward normal coordinate from
the bed L local streamwise flow velocity
averaged over turbulence L/T local volume
sediment concentration averaged over turbulence
1 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 1
z
12
1D EQUATION OF CONSERVATION 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,
13
REDUCTION 1D EXNER FORMULATION IN TERMS OF TOTAL
BED MATERIAL LOAD
In most cases the condition ltlt 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,
14
2D GENERALIZATION, TOTAL BED MATERIAL LOAD
Let denote the local average velocity in the
transverse (y) direction. Then
Now
Thus
15
1D CONSERVATION OF BED SEDIMENT FOR SIZE
MIXTURES, BEDLOAD ONLY
fi'(z', x, t) fractions at elevation z' in ith
grain size range above datum in bed 1. 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
Or thus
16
ACTIVE LAYER CONCEPT
The active, exchange or surface layer
approximation (Hirano, 1972) Sediment grains in
active layer extending from ? - La lt z lt ? have
a constant, finite probability per unit time of
being entrained into bedload. Sediment grains
below the active layer have zero probability of
entrainment.
17
REDUCTION OF SEDIMENT CONSERVATION RELATION USING
THE ACTIVE LAYER CONCEPT
Fractions Fi in the active layer have no vertical
structure. Fractions fi in the substrate do not
vary in time.
Thus
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.
18
REDUCTION OF SEDIMENT CONSERVATION RELATION USING
THE ACTIVE LAYER CONCEPT contd.
Between
and
it is found that
(Parker, 1991).
19
REDUCTION contd.
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
20
EXCHANGE FRACTIONS
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. That is 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. Stratig
raphy (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.
21
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,
22
2D GENERALIZATIONS
23
WHY THE CONCERN WITH SEDIMENT MIXTURES?
Rivers often sort their sediment. An example is
downstream fining many rivers show a tendency
for sediment to become finer in the downstream
direction.
bed slope
elevation
Long profiles showing downstream fining and
gravel-sand transition in the Kinu River, Japan
(Yatsu, 1955)
median bed material grain size
24
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.
25
FURTHER PROGRESS
Sediment approximated as uniform in size
Sediment mixtures

• In order to make further progress, it is
  necessary to
• Develop a means for computing the bedload
  transport rate qb (qbi) as a function of the
  flow
• Develop a means for computing the dimensionless
  entrainment rate E (Ei) into suspension as a
  function of the flow
• Develop a model for tracking the concentration
  of sediment in suspension, so that
  can be computed.
• Specify the thickness of the active layer La.
• The key flow parameter turns out to be boundary
  shear stress .

26
REFERENCES FOR CHAPTER 4
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.