Geoscience Reference
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that the actual bed-load transport rate is equal to the transport capacity under the
equilibrium condition at every computational point (cross-section or vertical line), i.e.,
q
b
=
q
b
∗
(
U
,
h
,
τ
,
d
,
γ
s
,
...)
(2.145)
where
q
b
∗
is the equilibrium (capacity) bed-load transport rate, which can be
determined using an empirical formula introduced in Section 3.4.
Eq. (2.145) can be used to close the 1-D, 2-D, and 3-D sediment transport models.
For example, the depth-averaged 2-D model is closed using Eq. (2.145) for bed-load
transport rate, Eq. (2.86) for suspended-load concentration, and Eq. (2.90) or (2.91)
for bed change. This approach is often called the equilibrium (or saturated) sediment
transport model.
2.6.2 Formulation of non-equilibrium transport model
Because of variations in flow conditions and channel properties, the sediment trans-
port in natural rivers usually is not in states of equilibrium. Sediment cannot reach new
equilibrium states instantaneously, due to the temporal and spatial lags between flow
and sediment transport. Therefore, the assumption of local equilibrium transport is
usually unrealistic and may have significant errors in cases of strong erosion and depo-
sition. A more realistic and general approach is the non-equilibrium (or unsaturated)
sediment transport model, which is described below.
For only suspended-load transport, the bed change is attributed to the net sediment
flux at the lower boundary of the suspended-load zone and thus determined by
p
m
)
∂
z
b
∂
(
−
t
=
D
b
−
1
E
b
=
αω
s
(
C
−
C
∗
)
(2.146)
For only bed-load transport, Bell and Sutherland (1983) proposed a loading law
based on their analysis of laboratory tests:
∂
q
b
∂
q
b
q
b
∗
∂
q
b
∗
∂
=
K
l
(
q
b
∗
−
q
b
)
+
(2.147)
x
x
where
K
l
is the loading-law coefficient. However, because Eq. (2.147) is an observation
of steady bed-load transport, its application to unsteady total-load sediment transport
is not straightforward. In addition, the last term on the right-hand side of Eq. (2.147)
lacks a physical basis. Daubert and Lebreton (1967), Wellington (1978), Nakagawa
and Tsujimoto (1980), Phillips and Sutherland (1989), and Thuc (1991) used the
following more general bed-load exchange model near the bed:
p
m
)
∂
z
b
∂
1
L
b
(
(
1
−
=
q
b
−
q
b
∗
)
(2.148)
t
where
L
b
is the adaptation length of bed load. Eq. (2.148) is based on theoretical
reasoning similar to that of Einstein (1950) but for bed load at a non-equilibrium state.
