SECTION VI:
FLOOD ROUTING
Consider
the watershed with 6 sub-basins
Q1 = QA + QB (Runoff from A & B)
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Q2 =
(QA + QB)2
(Routed runoff from Q1)
+
QC+ QD
+ (Direct runoff from C & D)
What causes attenuation?
1)
Storage
What causes flow lag?
1)
Flood wave travel time (celerity)
- f(length, depth, friction, slope)
Consider a short reach or reservoir (pond)
Continuity Eq.
Qin - Qout = S/ t
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and 2) Friction
See Appendix Figure 4.2 Bedient
Simple Hydrologic Flood Routing
Simple methods based on lumped continuity equation
f(x))
Qi - Qo = S/ t
Problem: Qi is known, need to solve for Qo
What about S?
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("lumped" not
To solve the flood routing problem a relationship between S and Q is
needed. Either:
(See handout, USGS rating curve)
2)
Q = f(S) or S = f(Q) directly - Less common than (1)
3)
Solve for Q(y) from momentum equation
(e.g., Q = kym, kinematic ), S(A), A = cross-sectional area
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RESERVOIR ROUTING - simplest hydrologic method - sometimes
called "Pond Routing"
"Linear" Reservoir
Qo = kS
k = [1/time] = routing coefficient
=f (channel geometry)
Qo = outflow linearly related to storage
then
Qin - Qout = S/ t
(Homework #’s 4.1, 3, 4, 5, 7)
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Qin - Qout = d(k-1Qout)/dt
k
dQout/dt + k Qout = k Qin(t)
Can solve analytically
(exponential solution)
f(time)
Solution:
Constant = Qout at t=0
Need functional form for Qi (i.e., Qi(t)) so that it can be integrated over
time. For example, could assume Qi = f(sine function)
or
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could use numerical (discrete step) methods
Finite Difference Method of Continuity Equation
Qin + Qo = ds/dt
Subscript 1 implies beginning of a time step
Known:
Qi1, Qi2, Qo1, S1
Subscript 2 implies end of a time step
Unknown: Qo2, S2
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Modified Puls Method (Finite Difference Continuity Eq.)
- also called Storage Indication Method (book)
- collect knowns and unknowns on opposite sides of the equation
To solve: use a relationship between S and Q from rating curve (stagedischarge relation), weir equation, uniform flow assumption, or other
information.
Can construct a table or graph of Qo = f(S + ( t/2) Qo)
Book has good example (p. 256 - 260)
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Consider a river reach shown
Consider uniform flow
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Manning's Equation for uniform flow Seq1 = Sbed
A = cross-sectional area
L = length of channel reach
and S = L A = reach storage
Then:
R = Hydraulic radius = Area/ Wetted perimeter
For rectangular channel (y = depth, b = bottom width),
Area = y b
WP = 2y + b
Rh = yb/(2y + b)
y/2 for b = 2y
Qo = Constant y2/3 S
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Muskingum Method For Flood Routing
S = prism storage + wedge storage = KQo + Kx (Qi - Qo)
Two Parameters K, x
x is not distance
Can substitute into continuity
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Can solve analytically as before
if x = 0, linear reservoir
In general, 0
x
0.5
note x is higher for more regular
("improved") channels
x is lower for more natural channel
e.g., natural irregular channels x ~ 0.15
concrete lined, trapezoidal channel 0.3
if x = 0.5, pure translation
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x
0.4
Finite differences applied to continuity Equation
Muskingum Eq.
For a linear reservoir x = 0
Must have storage and flow data to evaluate parameters
Plot graph shown for trial values of x, keep trying with different x, until
loop narrows (approximates a line)
Slope = K ,
S = K [ x Qi + (1 - x ) Qo]
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When storage is a maximum dS/dt = 0 = Qi - Qo,
substitute S = K [ Qo + Kx(Qi - Qo)]
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If we had these at the same time, we could solve
Use two slope values and solve for x
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In the real river reach the parameters are a function of the flow, Q.
For this case need parameter estimation for several flow rates (i.e.,
variable parameter, K(flow), x(flow), Muskingum Method).
Probably better to just go to full St. Venant (dynamic) equations.
Another method is
Muskingum-Cunge Method
- A better way for parameter estimation (relate them to physical
parameters of the channel) ref Cunge 1969 (Ref 66)
let Muskingum parameter, x, now be
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(x in the following eqs reps. distance)
Taylor Series
Remember:
and
Substitute for Q(x + x, t) and dQ(x + x, t)/dt in continuity equation
(neglect d3Q/dt dx2 term):
Use continuity equation of the form:
Define:
(Muskingum-Cunge)
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Solve for:
LHS
Combine d2Q/dx2 terms:
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Muskingum - Cunge
This is a form of the advection - diffusion equation (the bracketed term
represents a sort of flow wave diffusivity, D).
[RHS]
Likely to be curved:
Numerical Diffusivity
e.g., c = f(flow)
Also, approximate c by flood profiles:
Hydraulic Flood Routing
Large "Dynamic" Rivers
"Dynamic" - subjected to rapid fluctuations in flow requiring inclusion
of acceleration terms in equations of flow
* Must use St. Venant Eqns to adequately describe flow (p. 237-8)
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