Geoscience Reference
In-Depth Information
The recurrence coefficients
F
L
−
1
,
G
L
−
1
, and
H
L
−
1
are obtained by eliminating
δ
Q
L
from Eqs. (5.60) and (5.68) in the reach between points
L
1 and
L
. Therefore, the
coefficients
F
,
G
, and
H
can be computed using Eqs. (5.91)-(5.93) in the first sweep
from node
B
to node
A
, consequently yielding
−
δ
Q
1,
AB
=
F
1,
AB
δ
h
1,
AB
+
G
1,
AB
+
H
1,
AB
δ
h
L
(5.94)
where
Q
1,
AB
denotes the discharge increment at point 1 of channel
AB
; and
F
1,
AB
,
G
1,
AB
, and
H
1,
AB
are recurrence coefficients known from Eqs. (5.91)-(5.93).
Similarly, a sweep from node
C
to node
A
along channel
AC
gives
δ
δ
Q
1,
AC
=
F
1,
AC
δ
h
1,
AC
+
G
1,
AC
+
H
1,
AC
δ
h
M
(5.95)
and a sweep from node
D
to node
A
along channel
AD
gives
δ
Q
1,
AD
=
F
1,
AD
δ
h
1,
AD
+
G
1,
AD
+
H
1,
AD
δ
h
N
(5.96)
The compatibility conditions of discharge continuity and equal water stages at node
A
are written as follows:
J
t
n
+
1
Q
n
+
1
1,
j
q
1
(
)
+
=
0
(5.97)
j
=
1
z
n
+
1
s
1,1
z
n
+
1
s
1,2
z
n
+
1
s
1,
j
z
n
+
1
s
1,
J
=
=
...
=
=
...
=
(5.98)
where
j
is the index of the channels that emanate from node
A
,
J
is the total number
of such channels and
J
t
n
+
1
=
3 for node
A
in Fig. 5.12, and
q
1
(
)
is the external inflow
(or outflow) to node
A
at time
t
n
+
1
.
Applying the Taylor series expansion to Eqs. (5.97) and (5.98) yields
J
J
t
n
+
1
Q
1,
j
+
q
1
(
)
+
1
δ
Q
1,
j
=
0
(5.99)
j
=
1
j
=
δ
h
1,1
=
δ
h
1,2
=
...
=
δ
h
1,
j
=
...
=
δ
h
1,
J
(5.100)
Substituting Eqs. (5.94)-(5.96) and (5.100) into Eq. (5.99) yields a linear algebraic
equation in terms of
J
+
1 unknowns:
f
(δ
h
A
,
δ
h
L
,
δ
h
M
,
δ
h
N
)
=
0
(5.101)
Eq. (5.101) is derived based on node
A
. Performing the above procedure for all
nodes in the network eventually leads to a system of linear equations for the stage
