APPROXIMATE FORMULATION FOR SLOPE AND BANKFULL GEOMETRY OF RIVERS

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CHAPTER 24 APPROXIMATE FORMULATION FOR SLOPE AND
BANKFULL GEOMETRY OF RIVERS
In all previous chapters in the e-book with the
exception of Chapter 15, rivers have been treated
as if they have specified channel widths, and
also as if they are independent of their
floodplains. Further progress on large-scale
morphodynamics calls for the relaxation of these
constraints. In this chapter the principles in
Chapter 3 and the relations of sediment transport
are used to develop an approximate formulation
for the prediction of bankfull geometry of
streams. No code is presented in this chapter.
The formulation is, however, used extensively in
subsequent chapters.
Small meandering stream and floodplain flowing
into the Sangamon Rivers, Illinois
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BANKFULL GEOMETRY AND FLOODPLAINS
Alluvial rivers create their own bankfull
cross-sections and floodplains through the
interaction of flowing water and sediment erosion
and deposition. Rivers establish their bankfull
width and depth through the co-evolution of the
river channel and the floodplain. The river bed
and lower banks are constructed from bed material
load. The middle and upper banks are usually
constructed predominantly out of wash load,
although they usually contain some bed material
load as well. As the river avulses and shifts,
this material is spread out across the floodplain.
The establishment of bankfull depth is equivalent
to the construction of a floodplain of similar
depth.
Minnesota River and floodplain south of the Twin
Cities, Minnesota. Flow is from bottom to
top. Image from NASA website https//zulu.ssc.nas
a.gov/mrsid/mrsid.pl
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BANKFULL SHIELDS NUMBER OF RIVERS
The estimate ?bf50 of bankfull Shields number
and dimensionless bankfull discharge were
introduced and defined in Chapter 3 where D50
refers to a bed surface median size, The
diagram below, also introduced in Chapter 3,
suggests that rivers evolve toward some
relatively narrow range of bankfull Shields
number.
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BANKFULL SHIELDS NUMBER OF RIVERS contd.
Within a considerable amount of scatter, the
diagram allows the following approximate
estimates of bankfull Shields number for
gravel-bed and sand-bed streams based on averages
(e.g. Parker et al., 1998 Dade and Friend, 1998).
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BANKFULL SHIELDS NUMBER OF RIVERS contd.
The theory behind the evolution of river
cross-section toward a quasi-equilibrium bankfull
Shields number is outlined in Parker (1978a,b).
The theory is, however, incomplete. At this
point the numbers below serve as useful empirical
results. The considerable scatter in the data is
probably due to a) differing fractions of wash
load versus bed material load in the various
rivers, b) differing amounts and types of
floodplain vegetation, which encourages
floodplain deposition and c) different hydrologic
regimes. Paola et al. (1992) were the first to
propose the assumption of constant bankfull
Shields number in modeling the morphodynamics of
streams. The general form of their analysis is
used in the succeeding material. Their results
are also applied to basin deposition in a
succeeding chapter.
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SIMPLE THEORY FOR BANKFULL CHARACTERISTICS OF
RIVERS

• The formulation given here is based on three
  relations
• a resistance relation describing quasi-normal
  bankfull flow
• an example sediment transport relation
  describing transport of bed material load at
  quasi-normal bankfull flow
• a specified bankfull Shields criterion.
• While varying degrees of complexity are possible
  in the analysis, here the problem is simplified
  by assuming a constant friction coefficient Cf
  and a sediment transport relation of generic form
  (with assumed constant ?s, ?t and nt). Where the
  subscript bf denotes bankfull flow, the
  governing equations are

momentum balance
bed material transport
?form channel-formative Shields number
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SIMPLE THEORY FOR BANKFULL CHARACTERISTICS OF
RIVERS contd.
Let surface median size D50, submerged specific
gravity R and friction coefficient Cf be
specified. The equations below provide three
constraints for five parameters bankfull
discharge Qbf, bankfull volume bed material load
Qtbf, bankfull width Bbf, bankfull depth Hbf and
bed slope S. Thus if any two of the five (Qbf,
Qtbf, Bbf, Hbf and S are specified the other
three can be computed. The
nondimensionalizations of Chapter 3 are now
re-introduced along with a new one for Qtbf
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NON-DIMENSIONALIZATION OF GOVERNING EQUATIONS
The three governing equations are made
dimensionless with the hatted variables Solvin
g for , S and as functions of and
yields the following results
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WHAT THE RELATIONS SAY
Slope doubling the water discharge halves the
slope doubling the bed material load doubles the
slope. Width doubling the bed material load
doubles the width doubling the water discharge
without changing the bed material load does not
change width (but slope drops and depth increases
instead). Depth doubling the water discharge
doubles the depth doubling the bed material load
halves the depth (but channel gets wider and
steeper).
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CASE OF SAND-BED RIVERS
The Engelund-Hansen (1967) relation can be used
for the case of bed-material load of sand-bed
rivers. It is obtained by means of the
evaluations in which case the relations take
the form
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CASE OF GRAVEL-BED RIVERS
In a gravel-bed river the sand load can be
treated as wash load, and the volume bed material
load Qt can be equated to the volume bed load Qb.
The effect of form drag due to bedforms can be
ignored as a first approximation. The
Meyer-Peter and Müller (1948) relation and its
corrected version due to Wong (2003) and Wong and
Parker (submitted) both have critical Shields
numbers ?c that are so high (0.047 and 0.0495,
respectively) that the median surface size D50 of
a gravel-bed river can barely be expected to move
at ?form 0.0487. An alternative form is the
Parker (1979) approximation to the Einstein
(1950) relation, which takes the form where
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CASE OF GRAVEL-BED RIVERS contd.
Adapting the previous formulation with the
transport relation of Parker (1979) leads to the
following results where for ?form
0.0487
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REFERENCES FOR CHAPTER 24
Dade, W. B. and Friend, P. F., 1998, Grain-size,
sediment transport regime, and channel slope in
alluvial rivers, Journal of Geology, 106,
661-675. Einstein, H. A., 1950, The Bed-load
Function for Sediment Transportation in Open
Channel Flows, Technical Bulletin 1026, U.S.
Dept. of the Army, Soil Conservation
Service. Engelund, F. and E. Hansen, 1967, A
Monograph on Sediment Transport in Alluvial
Streams, Technisk Vorlag, Copenhagen,
Denmark. Meyer-Peter, E. and Müller, R., 1948,
Formulas for Bed-Load Transport, Proceedings, 2nd
Congress, International Association of Hydraulic
Research, Stockholm 39-64. Paola, C., Heller, P.
L. and Angevine, C. L., 1992, The large-scale
dynamics of grain-size variation in alluvial
basins. I Theory, Basin Research, 4,
73-90. Parker, G., 1978a, Self-formed rivers with
stable banks and mobile bed Part I, the
sand-silt river, Journal of Fluid Mechanics,
89(1),109-126. Parker, G., 1978b, Self-formed
rivers with stable banks and mobile bed Part
II, the gravel river, Journal of Fluid Mechanics,
89(1), pp. 127-148. Parker, G., 1979, Hydraulic
geometry of active gravel rivers, Journal of
Hydraulic Engineering, 105(9), 1185-1201.
Parker, G., Paola, C., Whipple, K. X. and
Mohrig, D., 1998, Alluvial fans formed by
channelized fluvial and sheet flow. I Theory,
Journal of Hydraulic Engineering, 124(10),
985-995. Wong, M., 2003, Does the bedload
equation of Meyer-Peter and Müller fit its own
data?, Proceedings, 30th Congress, International
Association of Hydraulic Research, Thessaloniki,
J.F.K. Competition Volume 73-80.
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REFERENCES FOR CHAPTER 24 contd
Wong, M. and Parker, G., submitted, The bedload
transport relation of Meyer-Peter and Müller
overpredicts by a factor of two, Journal of
Hydraulic Engineering, downloadable at
http//cee.uiuc.edu/people/parkerg/preprints.htm .