FAST SOLUTION OF DIFFUSIVE SHALLOW WATER EQUATIONS …

Report
Corato G.2, Moramarco T.2, Tucciarelli T.1
1
2
Department of Hydraulic Engineering and Environmental Applications , University of
Palermo, Italy, Viale delle Scienze, 90128, Palermo, Italy
Research Institute for Geo-Hydrological Protection, CNR, Via Madonna Alta 126, 06128
Perugia, Italy
Four possible gauged station configurations
for discharge monitoring
I. Hydrometric river site with rating curve known
II. Hydrometric river site with unknown rating curve
400
2.2
350
2
1.8
1.6
1.4
1.2
Level observations
1
2/3/05
0.00
3/3/05
0.00
4/3/05
0.00
5/3/05
0.00
6/3/05
0.00
7/3/05
0.00
Jones Formula
(Henderson ,
1966), Fenton
(Fenton, 1999),
Marchi (1976)
300
250
200
150
100
VPMS model (old)
VPMS model (new)
Time (h)
0
10
Level observations
IV. Equipped river reach with
rating curve known at one of
ends (Rainfall-runoff modeling,
Rating Curve Model)
Negligible lateral flows
Level observations
Rating Curve
Significant
lateral flows
Level observations
Observed
50
0
III. Equipped River reach with level
observations only
(Dyrac (Dottori et al., 2009),
MAST (Aricò et al., 2009),
VPMS (Perumal et al., 2007; 2010)
Inflow
Feb. 1999
Discharge (m3/s)
Stage (m)
(m)
Stage
450
2.4
20
30
40
50
60
Level observations
III. Equipped River reach with level
observations only
Negligible lateral flows
Level observations
• In the context of the third configuration, the hydraulic DORA model
(Tucciarelli et al. 2000), based on the diffusive hypothesis, can be
applied.
• The model starting from observed stage hydrographs at channel ends,
allows there of estimating discharge hydrographs by using a calibration
procedure of Manning parameter based on the wave speed of the flood
computed through observed stages (Aricò et al. 2009)
• However the model application, besides the need of topographical data
of river sections, was found depending on the Manning’s roughness
calibration procedure that affected the model performances
Purposes

To address the minimum channel length, L, so
that the effects of the downstream boundary
condition on the computation of the upstream
discharge hydrograph is negligible

To propose a new procedure for Manning’s
calibration also for a real-time context by
exploiting instantaneous flow velocity
measurements carried out by radar sensors and
using the entropic velocity model
Using hydraulic modelling to optimize
Configuration III
Diffusive form of Saint Venant Equation:



H 

 H 1   R 2 / 3 A x 
=Q


t T x
H 
 n



x


Boundary conditions
Upstream
q(0,t) = qu(t)
Flow driven
h(0,t) = hu(t)
Water level driven
Downstream
h
=0
x x =L
or
 2h
2
x x =L
=0
Problem 1: Wath L?
In Configuration III each possible downstream
b.c. is an approximation of the physical one
We need a reach long enough to avoid a strong
estimation error of the discharge in the initial section
It’s possible to get a rough estimation of the
required length?
Syntetic Test
Hypothesis
1. Large rectangular channel, with constant bed slope
2. Linear variation of water depth in the upstream
section
Numerical model


H
5/3

H
 h

x
=Q


t x n
H



x


and

 dh

h(0, t ) = h0  
t

 dt x =0 
 2
 h
=0
 dt 2
x =L

Root hydraulic head gradient at upstream
section:
Qualitative behaviour
Dimensionless model
The previous problem can be solved numerically once
for ever using dimensionless variables, for the most
severe case of initially dry conditions
Dimensionless variables
=h
L
 = t
1/3
nL
 =x
L
Dimensionless equations



   5/3

   = 0

    



and

 d
(0,  ) = 0  
 d


 2
 h
=0
 d 2
 =1

with
0' =
d
d
=
 =0
dh
n
dt x =0 L2/3
and
 = -i  

 
 =0 
Error computation
The solution is function of L.
The reference solution is computed for L=
E = maxt
 dh

- x Hx =0  n,
, i, L  - limL 
 dt

0.006
0.005
 dh

- x Hx =0  n,
, i, L 
 dt

i=10-2
i=10-3
i=10-4
E
0.004
0.003
0.002
0.001
0.000
0.E+00
2.E-07
4.E-07
6.E-07
d/d
8.E-07
1.E-06
1.E-06
A priori estimation of L
Relative Error
Ed =

max
qmax - q

qmax
=
Hmax -
x =0

Hmax


Hmax
E
i
x =0
Procedure
1. given Ed, compute E from the above equation
2. from the previous graph, the corresponding value:
0' =
dh
n
dt x =0 L2/3
L
Problem 2: Wath n?
In Configuration III the calibration of n is
carried out using the stage hydrograph of the
downstrem section
We need a long enough reach to estimate
the wave celerity
In present method n can be estimated using
a single mean velocity measurement
Calibration
Manning coefficent was determined minimizing
the follow objective function:
Err (n) =
qcomp (tcal , n) - qobs (tcal )
qobs (tcal )
where qcomp(tcal,n) is the computed discharge at
the instant tcal in which measurement is carried
out, while qobs is the observed discharge.
Problem 2: Estimatimation of calibration discharge
Entropic Method
To develop a practical and simple method for estimating discharge during high floods,
Moramarco et al. (JHE, 2004) derived from the entropy formulation proposed by Chiu
an equation applicable to each sampled vertical:
umax v 
y
y 

M
u(y) =
ln 1  e - 1
exp 1 
M
D
h
D
h





Gauged site: M estimated through the recorded pairs of (um, umax)
u m = (M)u max
um
eM
1
(M) =
= M
u max e - 1 M
If the measurement is carried out in the upper part of the flow area, umaxv is
sampled for each vertical. Anyway, to drastically reduce the sampling period it is
possible to consider only the upper portion where umax typically occurs and
assuming that the behaviour of the maximum velocity in the cross-sectional flow
area can be represented through a parabolic or elliptical curve.
Umax i (ms-1)
Problem 2: Estimatimation of calibration discharge
Entropic Method
a)
b)



2
Umax i (ms-1)
umax v ( x) = umax
 x
1 - 
 xS
c)
measured
d)
elliptical
parabolic
Gauged Section: M.te Molino (Tiber River) – 28/11/05 ore 11:30
h=8.2 m
riva DX
riva SX
a)
c)
b)
d)
Study Area: Upper Tiber Basin
Pierantonio (1805 km2)
Event
December 1996
April 1997
November 1997
February 1999
December 2000
November 2005
qpM [m3/s] tph [h] hpM [m] Duration [h]
380.53
429.44
308.17
427.93
565.89
779.03
22.5
32.5
18.5
21.5
74
30.5
4.74
5.07
4.22
5.06
5.92
7.1
49.5
74.5
45
59.5
100
64
Ponte Nuovo (4135 km2)
Event
qpM [m3/s] tpq [h] hpM [m] Duration [h]
November 2005
1073.2
32.75
8.52
70
December 2005
804.23
82.16
7.33
115
Test case 1: Pierantonio
L estimation
Mean bed slope i = 1.6x10-3
Typical Manning = 0.046 sm-1/3
dh
= 2.67 x10-4
m/s (observed during Nov. 05)
dt x =0
Ed = 0.05
 '0

8 x10-7
E = Ed i = 0.0015
(from diagram)
3/2
 dh

dt x =0 
L=
 '

0




200 m
Test case 1: Pierantonio
L=
200 m
L = 20000 m
Event
Qmax Error [%]
December 1996
April 1997
November 1997
February 1999
December 2000
November 2005
2.96
-3.59
-3.51
-2.26
-4.69
-4.66
Test case 1: Pierantonio
Discharge Estimation Results
Cal. Time [h] 12
15
Man [sm-1/3] 0.051 0.050
Qmax err [%] 2.39 3.70
18
0.051
0.90
Cal. Time [h] 10.5 20.5
Man [sm-1/3] 0.051 0.058
Qmax err [%] 20.60 6.92
25.5
0.061
1.66
Test case 2: Ponte Nuovo
L estimation
Mean bed slope i = 0.85x10-3
Typical Tiber Manning = 0.042 sm-1/3
dh
= 4.89 x10-4
m/s (observed during Nov. 05)
dt x =0
Ed = 0.05
 '0

3.6 x10-7
E = Ed i = 0.0015
(from diagram)
3/2
 dh

dt x =0 
L=
 '

0




400 m
Test case 2: Ponte Nuovo
Discharge Estimation Results
Cal. Time [h]
Man [sm-1/3]
Qmax err [%]
15
0.04
11.7
22
0.044
1.85
24
0.045
-1.36
Cal. Time [h] 10.5 130.5
Man [sm-1/3] 0.045 0.043
Qmax err [%] -2.8
2.8
Conclusions

The effect of downstream boundary condition over the
upstream stage hydrograph computation has shown
that short channel lengths are enough to achieve good
performance of the diffusive hydraulic model

The coupling of the hydraulic model with the entropic
velocity model turned out of great support for an
accurate calibration of Manning’s coefficient

The developed algorithm can be conveniently adopted
for the rating curve assessment at ungauged sites
where the standard techniques for velocity
measurements fail, in particular during high floods

Based on the proposed procedure, discharge
hydrographs can be assessed in real-time for whatever
flood condition.

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