The catchment: mechanistic model

Andrea CastellettiPolitecnico di Milano

NRMNRML09L09

2

Adriatic Sea

Fucino

VILLA VOMANO

PIAGANINI

PROVVIDENZA

CAMPOTOSTO

MONTORIO (M)

SAN GIACOMO (SG)

Irrigation district(CBN)

S. LUCIA (SL)

PROVVIDENZA (P)

3

Identifying the Model

Definining the components and the system scheme

Identifying the models of the components Aggregated model

4

Adriatic Sea

Fucino

VILLA VOMANO

PIAGANINI

PROVVIDENZA

CAMPOTOSTO

MONTORIO (M)

SAN GIACOMO (SG)

Irrigation district(CBN)

S. LUCIA (SL)

PROVVIDENZA (P)

5

The catchment

Reference section

6

Which output?

Reference section

Outflow from the catchment

7

... and the input?

Reference section

Outflow from the catchment

8

... and the input?

Precipitation Sunshine duration Temperature Air relative umidity Atmospheric pressure Wind velocity

Meteorological variables:

Describe and modulate energy and water exchanges between

atmosphere and the earth.

volume in the time interval [t, t+1)

average temperature in the interval [t, t+1)

How to proceed then?

9

The catchment: block diagram

When the model is particularly complex even the simple identification of the causal network might be too difficult.

The system is first decomposed into sub-components, then a causal network is constructed for each component.

BLOCK DIAGRAM

Like causal networks, block diagrams describe cause-effect relationships between relevant variables. However, at a higher conceptual level, at which some complex processes and variables are not yet considered.

10

Block diagram 1° step

Pt1

catchment

Tt1

dt1

air temperature

precipitation (solid and liquid)

outflow from the

catchment

11

Hydrograph

12

The water cycle

rainfallsnowfall

evaporation

total flow

infiltration

percolation

intercepted rainfall

evapotranspiration

capillary fluxhypodermic

flowdeep flow

surface flow

evaporation

snow

13

Block diagram2° step: functional components

snow pack

Tt1

qt1n

ground

drainage net

qt1s

dt1

inflow to the ground

outflow from the ground

Pt1

outfllow from the

catchment

14

Block diagram 3° step: orography

....band 1

Tt11

Pt11

qt11

band 2

Tt12

Pt12

qt12

band m

Tt1m

Pt1m

qt1m

+

Tt1 Pt1

snow pack

flow to the ground

qt1n

15

The lake Como catchment

16

Block diagram 4° step: sub-catchments

+

(c)

+

+

(c)

(a)

(b)(b)

(b) (b)

(a)

(a)(a)

The model of each sub-catchment is first identified, then combined with the other to form the aggregated model of the catchment.

The model of each sub-catchment is first identified, then combined with the other to form the aggregated model of the catchment.

17

Didactic scheme

COMPONENT Reservoir Catchment Other

components

TYPES of MODELS

BBNs Mechanistic

DETAILS

Mechanistic Campotosto

18

Didactic scheme

COMPONENT Reservoir Catchment Other

components

TYPES of MODELS

BBNs Mechanistic

DETAILS

Mechanistic Campotosto

Mechanistic intercept.1350

19

Mechanistic model

1. Model structure

Typical structure of rainfall/runoff models

air temperature

precipitation (solid and liquid)

Usually rainfall/runoff model have the following structure.

snow pack

Tt1

qt1n

ground

drainge net

qt1s

dt1

inflow to the ground

outflow from the ground

Pt1

outfllow from the

catchment

21

1. model structure

Mechanistic model

1. model structure

1a. snow pack

22

Snow pack: the state

• snow-pack depth

• density

• snow temperature

• water content of the snow

• color of the snow surface

• snow-pack depth

• density

• water content of the snow

Solid phase

(water equivalent)

Liquid phase

How is the state of the snow-pack made?

23

snow pack

Snowpack: variables

= solid phase of precipitation

= liquid phase of precipitation1

RtP

1S

tP1 tP

Solid phase of the snow-pack (water equivalent) ts Liquid phase of the snow-pack th

flow to the ground1 n

tq

average air temperature1 tT

State variables

Outputs:

Inputs:

1tP1tT

tt

t

sx

h

1ntq

24

Snowpacksolid phase dynamics

ts1 ts Pt1S -

melting

M

Tt+1

Tt+1min [ ]}max { 0,

meltingmelting

, ts

ts saturation to st

Net daily snow-meltNet daily snow-melt

M T

t1, h

t, s

t M T

t1, h

t, s

t

always non-negative

Assumption: snow-melt grows linearly with T.

mm of snow melt per °C and per day.

This approach is usually known as “degree-day”.

25

M

Tt+1

the frozen volume is always non-

negative

Snowpacksolid phase dynamics

melting - freezing

Tt+1-max [ 0, ]} ( )

- freezing- freezing

ts

min { ,th

- th

For the sake of simplicity let’s assume the same for melting and re-freezing.

For the sake of simplicity let’s assume the same for melting and re-freezing.

meltingmelting

saturation to

th

ts1 ts 1 StP -

Net daily snow meltNet daily snow melt

M T

t1, h

t, s

t M T

t1, h

t, s

t

26

M

Tt+1

Snowpacksolid phase dynamics

snow melt - freezing- freezing- freezing

ts

- th

snow meltsnow meltsnow melt - freezing

Net daily snow meltNet daily snow melt

1, , t t tM T h s 1, , t t tM T h sts1 ts 1 StP -

27

Snowpackliquid phase dynamics

45°

1 th th 1 RtP ts min{ , }

ts 1ntq

1 , , t t tM T h s

1th

th 1 RtP 1 , , t t tM T h s

flow to the ground

28

1ntq

th 1 RtP 1 , , t t tM T h s

Snowpackflow to the ground

45°

1 ntq - ts max{ 0 , }

- ts

th 1 RtP 1 , , t t tM T h s

29

min{ , }1 th th 1 RtP ts 1 , , t t tM T h s

Consistency check:it’s raining without snow-pack

System equations:

1 ntq - ts max{ 0 , } th 1 R

tP 1 , , t t tM T h s

ts1 ts 1 StP - 1, , t t tM T h s

It’s raining1 0S

tP

... without snow-pack 0ts

0th

30

min{ , }1 th th 1 RtP ts 1 , , t t tM T h s

Consistency check:it’s raining without snow-pack

System equations:

1 ntq - ts max{ 0 , } th 1 R

tP 1 , , t t tM T h s

ts1 ts 1 StP - 1, , t t tM T h s

It’s raining1 0S

tP

... without snow-pack 0ts

0th

1

1

1 1

0

0

t

t

n Rt t

s

h

q P

The flow to the ground is the very rainfall.

31

1. model structure

1a. snow pack

Mechanistic model

1. model structure

1b. ground

32

Ground

Evaporation

Surface flow

Flow to the ground

Hypodermic flow

Deep flow

Total runoffRoot zone

Water table

Soil

Percolation

Infiltration

1ntq

ground

1ctq

1tP

snow pack

1tT

33

, min MS

Ground the soil

Evaporation

Surface flow

Infiltration

MS

tS 1 ntq

1

M

tS

S

1 1max 0, ( ) t t te P T SK

1tS

1 1max 0, Mt tS S S

tS

1 tS

1ntq

Flow to the ground

Surface flow

1max 0, t MS S

t

M

SS

Degree of saturation

% of inflow retained by the ground

1 t

M

SS

34

Inflow retained by the ground

|1

100%-

γ = 1

γ >1

γ < 1

% of inflow retained by the soil

Degree of saturation t MS S

1 t

M

SS

35

Ground root zone

Infiltration

Percolation

trK rHypodermic

flow

1tr

St

SM

qt1n tr ( 1- K

r)

min K

prt, R

M

tr

min K

prt, R

M

Percolation

rt

RM

KP

36

Ground water table

Percolation

Deep flow

tfK f

1tf tf 1- fK min , p t MK r R

tf

37

Ground- drainage network

1 1max 0,

st t

r t f t

Mq S

K r K f

S

1 11 st t td dd qKd K

The total flow from the ground qst+1 is subject to a storing process in the drainage network.

r tK rHypodermic

flowtr

Deep flowf tK ftf

Total flow

Storing coeff.

Surface flow

1max 0, t MS S

38

1. model structure

1a. snow pack

1b. ground

Mechanistic model

2. Analysis of the model properties

39

Outflow from the catchment

Total flow

Water table

Roots

Soil

1 1(1 ) st d t d td K d K q

1 1 surf. flowst r t g t tq K r K f

1tr 1ntt

M

Sq

S

tr( 1- )rK min ,p t MK r R

tS1 n

tq

1 tS

MS

γ

tevap1 tS 1max 0, t MS S

1tS 1surf. flow t

Raining without snow pack

1tS

1surf. flow t

1stq

1td

1 (1 ) min ,t f t p t Mf K f K r R

1surf. flow t

1 1 n Rt tq P the inflow to the ground is the rainfall1 1 n Rt tq P the inflow to the ground is the rainfall

1R

tP affects dt+1

1ntq

1tS

1stq

The model is a improper one.

It can not be used for managing or forecasting.

It is uselles!

The model is a improper one.

It can not be used for managing or forecasting.

It is uselles!

40

The model is an improper one

Total flow+

Surface flow

EvaporationFlow to

the ground

Root zone

Water table

Soil

Percolation

Infiltration

41Outlfow from the

catchment

Total flow

Water table

Roots

Soil

Proper model

f

t1(1 K

f) f

t min K

prt, R

M

q

t1s K

rrt K

gf

t

1tr

St

SM

qt1n tr ( 1- ) Kr

min Kprt, R

M 1 surf. flow t

tS1 n

tq

1 tS

MS

γ

tevap1tS 1max 0, t MS S

1tS 1surf. flow t

qt1n P

t1R flow to the ground is rainfall qt1

n Pt1R flow to the ground is rainfall

qt1n

1tS

1tS

1tr 1surf. flow t

rt+1 does not

affect qst+1

1R

tP does not affect dt+1

1surf. flow t

dt1K

dd

t (1 K

d)q

t1s

42

1 1(1 ) st d t d td K d K q Outflow from the

catchment

Total flow

Water Table

Roots

Soil

tS1 n

tq

1 tS

MS

γ

tevap1 tS 1max 0, t MS S

1tS 1surf. flow t

1tr 1t

tM

Sy

S

tr( 1- )rK min ,p t MK r R1 surf. flow t

Rainining without snowpack (ground – proper model)

1 (1 ) min ,t f t p t Mf K f K r R

1 st r t g tq K r K f

1 1 n Rt tq P 1 1 n Rt tq P

1tS

1tr

2 1 1

2 2

1tr 2stq

2stq 2td

Rainfall is affecting only the outflow dt+2

1

The new model can be used in managemen and forecasting,

however ....

The new model can be used in managemen and forecasting,

however ....

43

Typical model performance

1 Aug 10 Aug 20 Aug 30 Aug 10 Sep 20 Sep 30 Sep

0

250

500

750

1000

1250

Infl

ow

(

m³/

s )

simulatedobserved

A one-day delay due to the model properties.

A one-day delay due to the model properties.

44

Solution to reduce the delay

There are two possible solutions:

1) Reducing the time step to a value smaller than the concentration time in the sub-catchment considered

(right solution)

2) Manipulating and transformin the model into a proper model. (wrong solution)

This latter solution is quite common in hydrology, but it precludes the use of the model in prediction.

45

Readings

IPWRM.Theory Ch. 5 + Ap. 5