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where z 0 is the elevation of the uncompactable layer of bed material.
By defining the bed thickness as H
=
z b
z o and the average dry density of the bed
ρ d = z b
as
z 0 ρ d dz
/
H , Eq. (11.36) can be written as
t (
H
ρ d ) =
0
(11.37)
The bed change due to consolidation is then determined by
H
H
ρ d ∂ ρ
d
t =−
(11.38)
t
11.3.2 Numerical solutions
Numerical discretization
Eqs. (11.31) and (11.33) can be discretized using the same numerical methods as
those for Eq. (7.43) and (6.53). Usually, the time-derivative terms are discretized
using the forward or backward difference scheme to establish an explicit or implicit
time-marching procedure. The convection terms should be discretized using an upwind
scheme, while the diffusion terms are discretized using the central difference scheme or
a similar scheme. The settling term in Eq. (11.31) and the exchange term in Eq. (11.33)
are often treated as source terms. The resulting algebraic equations can be solved using
the Gauss-Seidel, ADI, or SIP method introduced in Section 4.5.
In the 3-D model, the near-bed boundary condition (11.32) can be discretized using
a scheme similar to Eq. (7.54) or (7.55).
The bed change equation (11.34) is discretized in time as
t
z b =
p m (
D b
E b )
(11.39)
1
Eq. (11.38) can be discretized in time to calculate the bed change due to consoli-
dation. However, to consider the heterogeneous properties of bed materials deposited
in different times, the following multiple-layer model is often used. The bed soil from
the bed surface to the uncompactable layer is divided into a suitable number of layers
in the vertical direction, as shown in Fig. 11.12. Each layer is characterized by its dry
density and residence time. The top layer holds the newly deposited mud, and a bed
formation time (such as 2 hours) may apply to this layer. The evolution of dry densities
at other layers is determined by Eq. (11.23) or (11.24). The mass of sediment at each
layer is conserved during the consolidation process, i.e.,
t
ρ dj ) =
0
(11.40)
j
where
δ j and
ρ dj are the thickness and dry density of the j th layer of bed material.
 
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