Geoscience Reference
In-Depth Information
with the boundary condition at the bed
ω
sf
c
s
∂
c
+
ε
z
b
=
D
b
−
E
b
(11.32)
∂
z
z
=
where
c
is the local sediment concentration;
D
b
is the deposition rate determined using
Eq. (11.14);
E
b
is the erosion rate determined using one of Eqs. (11.19)-(11.21); and
ω
sf
is determined using one of Eqs. (11.1), (11.2), (11.4), (11.6), (11.7), and (11.9).
In the depth-averaged 2-D model, the sediment transport equation is
E
s
,
x
h
∂
E
s
,
y
h
∂
∂(
hC
)
+
∂(
hU
x
C
)
+
∂(
hU
y
C
)
=
∂
∂
C
+
∂
∂
C
∂
t
∂
x
∂
y
x
∂
x
y
∂
y
+
E
b
−
D
b
(11.33)
where
C
is the depth-averaged sediment concentration.
One can see similarity between Eqs. (7.43) and (11.31) and between Eqs. (6.53) and
(11.33). The width-averaged 2-D and 1-D cohesive sediment transport equations are
similar to Eqs. (6.123) and (5.27), which are not repeated here. The difference between
cohesive and non-cohesive sediment transport models lies in the determination of
ω
sf
,
D
b
, and
E
b
, as well as bed consolidation.
The rate of change in bed elevation,
∂
z
b
/∂
t
, is determined by
p
m
)
∂
z
b
∂
(
1
−
=
D
b
−
E
b
(11.34)
t
Gibson
et al
. (1967) proposed a theory to describe the bed consolidation process.
The model based on this theory determines the evolution of the void ratio of a soil
layer using the following 1-D equation in the vertical direction at each horizontal
computational point:
1
d
de
1
ρ
f
g
k
1
∂
σ
∂
e
ρ
z
+
∂
e
k
d
de
∂
e
s
ρ
f
−
+
=
0
(11.35)
z
z
∂
t
+
e
∂
∂
1
+
e
∂
where
e
is the void ratio of bed soil,
z
is the reduced material coordinate deduced
z
(
σ
from the vertical coordinate
z
by
δ
z
=
δ
1
+
e
)
,
k
is the permeability, and
is the
effective stress.
Constitutive relationships for the permeability
k
and the effective stress
σ
as func-
tions of the void ratio are needed to close Eq. (11.35). These constitutive relationships
are rarely available; thus, it is difficult to apply this approach in the solution of real-
life problems. Therefore, a simpler approach is often used, in which the evolution of
bed density is determined by empirical functions, such as Eqs. (11.23) and (11.24).
The overall mass of sediment should be conserved in the consolidation process, i.e.,
z
b
z
0
ρ
∂
∂
d
dz
=
0
(11.36)
t
