Transcript Slide 1

1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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]
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
COORDINATE SYSTEM
sediment bed
z
y
x
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]
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CASE OF 1D, BEDLOAD ONLY, SEDIMENT APPROXIMATED AS UNIFORM
IN SIZE



s (1   p )x  1  gb x  gb
t
x  x
 1   q
s
b x
 qb
x  x
or thus
 1
water
qb
qb

(1   p )
t
x
qb

This corresponds to the original
form derived by Exner.
bed sediment + pores
1
x
4
x
x +x
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
2D GENERALIZATION, BEDLOAD ONLY
(Yes, this is still a course on 1D morphodynamics, but it is useful to know the
2D form.)

qb  qbx eˆ x  qby eˆ y
where
eˆ x , eˆ y
denote unit vectors in the
x and y directions.
 

(1   p )
 -  qb
t
sediment bed
z
y
x
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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].




s (1  p )x  1  s qb x  qb x  x  1  s Ds  Es x  1
t
or thus
water
Es
Ds
qb
qb
qb

(1   p )
 Ds  E s
t
x

bed sediment + pores
1
7
x
x
x +x
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
EVALUATION OF Ds AND Es
Let c( x, z, t ) 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,
Ds  v scb
where ccbb denotes the near-bed value of c .
c
z
cb
Ds  v s c b
Similarly, a dimensionless entrainment rate E can be defined such that
Es  v sE
Thus
qb

(1   p )
 v s c b  E 
t
x
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
2D GENERALIZATION, BEDLOAD + SUSPENDED LOAD

qb  qbx eˆ x  qby eˆ y
sediment bed
z
y
x
 

(1   p )
 -  qb  v s c b  E
t
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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 :



s (1   p )  base x  1  s qb x  qb
t
x  x
 1 D
The tectonic subsidence rate  (uplift rate ) is given as
    
base
t
Thus with the previously-presented evaluations for Es and Ds:
qb
 

(1  p )
    v s cb  E
x
 t

2D generalization:
 
 

(1  p )
    -  qb  v s (cb  E)

t


s
 Es x  1
qb
Es
Ds
qb
x

base
10
u
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
TRANSPORT RATE OF SUSPENDED SEDIMENT
Definitions:
z = upward normal coordinate from the bed [L]
u = local streamwise flow velocity averaged over turbulence [L/T]
c = 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]
H
qs   u c dz
0
uu
H
UH   u dz
0
H
UCH   u c dz  qs
0
yz
cc
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u
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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
sCUH
H
sEsx  D x
s s
x
1
sCUH
H

s x  c dz  sCUH x  sCUH x  x  s Es  Ds x
0
t
or reducing with the relation qs = UCH and previous evaluations for Es and Ds,
qs
 H

 v s E  c b 
 0 c dz  
 x
t 
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u
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REDUCTION: 1D EXNER FORMULATION IN TERMS OF TOTAL BED
MATERIAL LOAD
In most cases the condition c << 1 prevails, allowing the approximation
qs
 H

 v s E  c b 
 0 c dz  


t
x
The simplified form of the above equation can be combined with the Exner
equation of conservation of bed sediment,
qb
 

(1  p )
    v s cb  E
x
 t

to yield the following form for Exner:
q q
 

(1  p )     - b  s
x
x
 t

or defining total bed material load qt = qb + qs,
q
 

(1  p )
   - t
x
 t

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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
2D GENERALIZATION, TOTAL BED MATERIAL LOAD
Let v denote the local average velocity in the transverse (y) direction. Then

qs  qsx eˆ x  qsy eˆ y
H
H
0
0
qsx   u c dz , qsy   v c dz ,
Now

qb  qbx eˆ x  qby eˆ y



qt  qb  qs
Thus
sediment bed
z
y
x
 
 

(1  p )     -  qt
 t

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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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:
N
 f  1
i1
i
qbi(x, t) = volume bedload transport rate of sediment in the ith grain size range [L2/T]

 

s (1   p )fidzx  1  s qbi x  qbi

0
t
x  x
 1
qbi
qbi
x
Or thus:
1

qbi
 
(1   p )  fidz  
t 0
x
z'
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
ACTIVE LAYER CONCEPT
qbi
qbi
La

x
z'
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. 16
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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.
Fi ( x, t ) ,   La  z  
fi( x, z, t )  
 fi ( x, z) , z    La
Thus
 
 La
 


fidz  
fidz   fidz  fIi (  L a )  FiL a 

t 0
t 0
t La
t
t
where the interfacial exchange fractions fIi defined as
fIi  fi L
a
describe how sediment is exchanged between the active, or
surface layer and the substrate as the bed aggrades or
degrades.
17
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REDUCTION OF SEDIMENT CONSERVATION RELATION USING THE ACTIVE
LAYER CONCEPT contd.
Between
q
 
(1   p )  fidz   bi
t 0
x
and
 
 La
 








fidz  
fidz   fidz  fIi (  L a )  FiL a 

t 0
t 0
t La
t
t
it is found that
qbi

 

(1  p )fIi (  La )  FiLa   
t
x
 t

(Parker, 1991).
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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
N
qbT   qbi , pbi 
i1
qbi
qbT
The conservation relation can thus also be written as

q p
 

(1  p )fIi (  La )  FiLa    bT bi
t
x
 t

Summing over all grain sizes, the following equation describing the evolution
of bed elevation is obtained:
qbT

(1   p )

t
x
Between the above two relations, the following equation describing the
evolution of the grain size distribution of the active layer is obtained:
L 
q p
q
 F
(1  p )La i  Fi  fIi  a    bT bi  fIi bT
t 
x
x
 t
19
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
EXCHANGE FRACTIONS


fi zL ,
0

a
t
fIi  

Fi  (1  )pbi ,
0
t

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.
Stratigraphy (vertical variation of the grain size distribution of the substrate)
needs to be stored in memory as bed aggrades in order to compute
20
subsequent degradation.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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:
 

q p
 
(1  p )fIi  (  La )     FiLa    bT bi
x
 t
  t

To include suspended sediment, let vsi = fall velocity, Ei = dimensionless
entrainment rate, and cbi denote the near-bed volume concentration of
sediment, all for the ith grain size range, so that the relation generalizes to:
 

q p
 
(1  p )fIi  (  La )     FiLa    bT bi  v si cbi  Ei 
x
 t
  t

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,
 

q p
 
(1  p )fIi  (  La )     FiLa    tT ti
x
 t
  t

21
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
2D GENERALIZATIONS

qbi  qbxi eˆ x  qbyi eˆ y
 
 

 
(1  p )fIi  (  La )     FiLa     qbi  v si cbi  Ei 
 t
  t


qti  qbxi  qsxi eˆ x  qbyi  qsyi eˆ y
 
 

 
(1  p )fIi  (  La )     FiLa     qti
 t
  t

22
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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
median bed
material grain size
Long profiles showing
downstream fining and
gravel-sand transition in
the Kinu River, Japan
(Yatsu, 1955)
23
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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.
upstream
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.
downstream
24
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
FURTHER PROGRESS
q
 

(1  p )
    - b  v s cb  E
x
 t

Sediment approximated as uniform in size
 

q p
 
(1  p )fIi  (  La )     FiLa    bT bi  v si cbi  Ei  Sediment mixtures
x
 t
  t

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 c (ci ) of sediment in
suspension, so that cb (cbi ) can be computed.
•Specify the thickness of the active layer La.

The key flow parameter turns out to be boundary shear stress b .
25
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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.
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