Reservoir modelling Non-linear reservoirs

The linear reservoir
Qin (n)
Reservoir modelling
Non-linear reservoirs
General equations:
Paul Torfs
dS
= Qin (t) − Qout
dt
Qout = F(S)
S(n)
linear reservoir:
F(S) = k S
Qout(n)
Sumova 2008
The non-linear reservoir
Sumova 2008
The power reservoir
4
I
Qin (n)
but many other formula’s are
“thinkable”
3
Q = a Sb
F(S) = k1 S + k2 S 2
Q2
(a = 1)
F(S) = k S b
S(n)
F(S) = k
..
.
Qout(n)
dS
= Qin (t) − Qout
dt
Qout = F(S)
(eb S − 1)
..
.
1
0
0
1
2
3
S
F(0) = 0?, F increasing?
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b = 1.2
b = 1.0
b = 0.8
many other formula’s are used
in practice
Sumova 2008
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b < 1: more than linear for small S, less then linear for big S
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b > 1: less than linear for small S, more then linear for big S
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The broken linear reservoir
The broken linear reservoir
Q
Q
Sc
Sc
Sc
if S ≤ Sc
S
Sc
if S ≥ Sc
Q = k1 S
Sumova 2008
S
Q = k1 Sc + k2 (S − Sc )
Sumova 2008
The finite reservoir
The finite reservoir
Qin
Smax
Qin
Smax
Qout
Qov
but should defined with the help of an
overflow flux:
(
0
if S < Smax
Qov =
max(Qin − Qout , 0) if S = Smax
Qout
reservoir with maximal content is not a special form
Sumova 2008
Sumova 2008
2.0
The shape of a reservoir
I
typical “outflow” is given by
function S ↔ Q
1.5
The shape of a reservoir
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linear reservoir: straight cylinder
R(h)
h
h = water level in reservoir
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let the reservoir by cylindrical
R(h) = radius for level h
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“change” formula:
0.5
S(n)
Qout(n)
0.0
I
0.0
0.2
0.4
0.6
0.8
1.0
1.2
outlet with fixed diameter 1
Q=h
1.0
Q
Qin (n)
I
what is the “form”
corresponding to an arbitrary
function as the one on the left?
π R2 (h) =
dS
dS
1
=
=
dQ
dh
dQ
dS
S
Sumova 2008
Numerics: explicit scheme
dS
= Qin (t) − F(S(t))
dt
S[n + 1] − S[n]
= Qin [n] − F(S[n])
∆t
1.0
S[n + 1] = S[n] + ∆t Qin [n] − F(S[n])
0.0
0.0
0.5
0.5
h
1.0
1.5
1.5
2.0
2.0
The shape of a reservoir
Q
Sumova 2008
0.0
0.2
0.4
0.6
S
0.8
1.0
1.2
−0.4
−0.2
0.0
0.2
0.4
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non-linearity of F causes no problem whatsoever
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works as well and as easily for system of many reservoirs
R
Sumova 2008
Sumova 2008
Numerics: implicit scheme
Numerics: implicit scheme
S[n + 1] − S[n]
= Qin [n + 1] − F(S[n + 1])
∆t
dS
= Qin (t) − F(S(t))
dt
S − S[n]
∆t
Qin − F (S)
Q
S[n + 1] − S[n]
= Qin [n + 1] − F(S[n + 1])
∆t
F(S)
S[n + 1] =?
S
a “general” non-linear
relation
non-linear problem to solve!
S[n + 1]
Sumova 2008
Numerics: implicit scheme
S
Sumova 2008
Numerics: implicit scheme
S[n + 1] − S[n]
= Qin [n + 1] − F(S[n + 1])
∆t
dS
= Qin (t) − F(S(t))
dt
S[n + 1] − S[n]
= Qin [n + 1] − F(S[n + 1])
∆t
S[n + 1] =?
S − S[n]
∆t
Q
Qin − F (S)
F(S)
S
a non-linear (and
non-continuous) relation
with an “if” statement
S
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non-linear problem to solve!
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when with “if”-statements even more difficult/impossible
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for system of many reservoirs: very difficult/impossible in
general
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Recession analysis
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Emptying a broken linear reservoir
∂S
(t) = −k S(t)b
∂t
in general:
1.0
0.9
0.8
0.7
0.6
S 0.5
0.4
0.3
0.2
0.1
0.0
∂S
(t) = −F(S)
∂t
is difficult to solve
I
some examples can be done:
∂S
(t) = −k S(t)b
∂t
S(0) = S0
i1/(1−b)
h
S(t) = S0 1 − k (1 − b) S0b−1 t
b = 1.4
b = 1.0
b = 0.6
0
1
2
3
4
5
t
Sumova 2008
Non-linear recession analysis
Sumova 2008
Non-linear recession analysis
Selecting the recession limbs
2.5
The data: only Q
Qout
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Sumova 2008
Non-linear recession analysis
Non-linear recession analysis
1.0
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we want S-Q scatter plots to fit
non-linear relations
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but only Q given
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for each recession limb i we do have:
dS
= 0 − Q(t)
dt
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2
0
1
−log(Q)+log(Q(t))
5
6
“Classical” linear recession: no succes
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0.5
1.0
or discrete:
Si [n] = Si [n − 1] − Q[n]
1.5
S
5
10
15
I
t
so we do known most of S but only do
not know Si [0] for each recession limb
Sumova 2008
Sumova 2008
Non-linear recession analysis
Non-linear recession analysis
what we know with Si [0] = 0 for each limb
(so each curve should be shifted to the right)
1.0
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(so each curve should be shifted to the right)
so assume we want to fit:
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Q(t)
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Si(t)−Si(0)
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Qi [n] ∼ F(Si [n] + Si [0], p1 , p2 , . . .)
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Si(t)−Si(0)
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Q(t)
make shift part of the fitting
what we know
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i = number of recession limb
n = pos in limb
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Sumova 2008
Non-linear recession analysis
Non-linear recession analysis
after the fitted shifting
fitted Q-S relation
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Sumova 2008
Non-linear recession of Isel river
6
Non-linear recession of Isel river
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●
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1500
0
day
Sumova 2008
500
1000
1500
Sumova 2008
Non-linear recession of Isel river
3.5
Non-linear recession of Isel river
2.0
●
3.0
●
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t
●
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●
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●
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2.0
●
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1.5
●
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1.0
●
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Q(t)
1.5
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0.5
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●
●
−20
●
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●
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●
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−5
0
Si(t)−Si(0)
Sumova 2008
Non-linear recession of Isel river
3.5
3.5
Non-linear recession of Isel river
Sumova 2008
●
●
●
●
●
3.0
3.0
●
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●
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10
15
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S
Sumova 2008
Sumova 2008