1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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.nasa.gov/mrsid/mrsid.pl
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
BANKFULL SHIELDS NUMBER OF RIVERS
The estimate bf50* of bankfull Shields number and dimensionless bankfull
discharge Q̂ were introduced and defined in Chapter 3; where D50 refers to a bed
surface median size,
bf 50 
Hbf S
RD50
, Q̂ 
Qbf
2
gD50 D50
The diagram below, also introduced in Chapter 3, suggests that rivers evolve
toward some relatively narrow range of bankfull Shields number.
1.E+01
1.E+00
bf 50
Grav Brit
Grav Alta
Sand Mult
Sand Sing
Grav Ida
1.E-01
1.E-02
1.E-03
1.E+02
1.E+04
1.E+06
1.E+08
1.E+10
1.E+12
1.E+14
3
Q̂
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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).
sand  bed : bf 50  1.86
gravel  bed :
1.E+01
bf 50  0.0487
1.E+00
bf 50
1.E-01
Gravel
Gravel Average
Sand
Sand Average
1.E-02
1.E-03
1.E+02
1.E+04
1.E+06
1.E+08
Q̂
1.E+10
1.E+12
1.E+14
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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.
sand  bed : bf 50  1.86
gravel  bed :
bf 50  0.0487
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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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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
Qbf2
C f 2 2  gHbf S
Bbf Hbf
momentum balance


s bf 50
Q tbf  Bbf qtbf  Bbf RgD D  t  
bf 50 
Hbf S
 const .  f orm
RD 50


 nt
c
bed material transport
form* = channel-formative
Shields number
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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.
Qbf2
Cf 2 2  gHbf S
Bbf Hbf


s bf 50
Q tbf  Bbf qtbf  Bbf RgD 50 D50  t  
bf 50 


 nt
c
Hbf S
 const .  f orm
RD 50
The nondimensionalizations of Chapter 3 are now re-introduced along with a new
one for Qtbf;
Qbf
Hbf
Bbf
Qtbf
Q̂ 
, Ĥ 
, B̂ 
, Q̂t 
2
2
D50
D50
gD50 D50
RgD 50 D50
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
NON-DIMENSIONALIZATION OF GOVERNING EQUATIONS
The three governing equations are made dimensionless with the hatted variables;
2
 Q̂ 
  ĤS
Cf 

 B̂Ĥ 


s f orm
, Q̂t  B̂  t  


 nt
c
,
ĤS
 f orm
R
Solving for B̂ , S and Ĥ as functions of Q̂ and Q̂t yields the following results;
B̂ 
S

1

s f orm
t  


 nt
c
(Rf orm )3 / 2

  
Ĥ 

s f orm
t  
t

s f orm
Q̂ t
 nt
c
C1f/ 2 Q̂
 nt
c
C1f/ 2 Q̂




Q̂ t
Rf orm
Q̂ t
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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).
B̂ 
S

1

s f orm
t  


 nt
c
(Rf orm )3 / 2

  
Ĥ 
 t s f orm  c
t

s f orm
 c
Rf orm


Q̂ t
Q̂ t
nt
C1f/ 2 Q̂
nt
C1f/ 2 Q̂
Q̂ t
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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
EH
t 
Cf
,
EH  0.05
n t  2 .5
,
,
s  1 ,
c  0
in which case the relations take the form
B̂ 

Cf
 EH 

2 .5

f orm
Q̂ t
R 3 / 2C1f/ 2 Q̂ t
S
 EH f orm Q̂
 EH ( f orm )2 Q̂
Ĥ 
1/ 2
RC f  Q̂ t
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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
 
qt  qb  P RgD 50 D50 
 1.5
where
P  11.2
,
nP  4.5
,
 
1 
 


c





nP
c  0.03
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CASE OF GRAVEL-BED RIVERS contd.
Adapting the previous formulation with the transport relation of Parker (1979) leads to
the following results;
B̂ 
1
Q̂ t

3/2
P (  f orm ) rf orm
Q̂ t
R3 / 2
S
Prf ormC1f/ 2 Q̂
P f ormrf ormC1f/ 2 Q̂
Ĥ 
R
Q̂ t
where for form* = 0.0487
rf orm

 
 1   
 f orm 


c
4 .5
 0.0135
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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,
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J.F.K. Competition Volume: 73-80.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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 .
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