Hydrology Basics
Module
Hydrology & Water Resources Management I
WS 2021/22
Tue: 11.30 – 13.00 hr, Room A 219
Institute for Hydrology and Water
Resources Management,
Leibniz University Hannover
Prof. Uwe Haberlandt
Golbarg Goshtasbpour, MSc.
Chapter 1: Introduction
Definition „Hydrology“:
“Hydrology is the science that treats the waters of the Earth, their
occurrence, circulation and distribution, their chemical and physical
properties, and their reaction with the environment, including their
relation to living things. The domain of hydrology embraces the full
live history of water on Earth” (Maidment, 1992).
2
Water Cycle:
Scheme of the global water cycle (www.dkrz.de, 2002)
3
Task 1: Understanding & Description
Heterogeneity in nature
Hydrological modelling
Abstraction for description
MIKE-SHI-Modell
(www.dhigroup.com, 2001)
(Source: Grayson & Blöschl, 2000)
4
Task 2: Assessment of water availability
Total actual renewable water resources per year and capita (WWAP, 2012)
< 500
strong water shortage
500 – 1000
water stress
1000 – 2000
water sensibility
5
Task 3: Design, Operation & Forecasts
Water level Maxau/Rhine [cm+PNP]
Maxau without retention measures
Flood May/June 1999 at gauge Maxau
Elbe Flood Dresden 2002
(Source: I. Lepenies)
Operational flood forecasting
Runoff
[m³/s]
Probability of none-exceedance
outside extrapol. range
HQ100
Planning & operation of dams
Estimation of design floods
6
Interval of recurrence
Task 4: Impact analysis (e.g. from climate and
land use change)
Flood risk Swiss 2005
Erosion & matter
emission
Water shortage, droughts
Diffuse pollution
7
Hydrology and its interrelation with other
disciplines
Economy
Planning Sociology
Hydromechanics
Physics
Biology
Chemistry
Mathematics
Meteorology
Ecology
Hydrology
Water supply
Geology
Waste water treatment
Soils Sciences
Hydraulic Engineering
Geography
….
Basic Natural
Sciences
Water Resources
Management
Applied
sciences
….
Social Sciences,
Engineering &
Specific applications
8
Methods: Observation
Climate station
Remote sensing
Discharge measurement
9
Methods: Experiment
Permeameter for measuring
hydraulic conductivity
Weighing Lysimeter for
measuring actual
evapotranspiration (Brandis)
Tilting channel
for hydraulic
experiments
10
Methods: theory, computer experiments
Event 1, basin outlet
Precipitation
Snowmelt
Sm  f  T  Tcrit 
Snow Storage

Inf  1 
Evapotranspiration


Runoff
S1
S1,max

  Inf max


Event 2, basin outlet
Etot  E  T  I  Min( D, S )
R1 
1
 S  RPerc  RInt
K1 1
Slow Storage
Groundwaterflow
3
16
Interflow
Percolation
6
Date
Fast Storage
Infiltration
9
0
02.03.1998 07.03.1998 12.03.1998 17.03.1998
R2 
1
S
K2 2
RPerc S1

RInt
S2
Discharge [m3s-1]
Sublimation
Canopy Storage
1  Pb
LAI max
Discharge [m3s-1]
I
Interception
12
12
8
4
0
21.01.1995
26.01.1995
31.01.1995
Date
05.02.1995
Scheme for the vertical structure of a simple hydrological water balance model
(Model SLURP)
11
Table of contents
1.
2.
3.
4.
5.
Introduction
Water- and energy balances
Precipitation
Evaporation
Streamflow
Description of
water balance
components
6.
Subsurface water
Soil and
groundwater
7.
8.
9.
10.
11.
Introduction to modelling
Models for runoff generation
Theory of hydrologic systems
Conceptual models
Models for flow routing
Approaches for
runoff calculation
from rainfall
12
Recommended Literature
Chow, V.T., Maidment, D.R. and Mays, L.W. (1988): Applied
Hydrology. McGraw-Hill*
Maidment, David R. [Editor] (1993): Handbook of hydrology.
New York: McGraw-Hill.*
Dyck, S. und Peschke, G. (1995): Grundlagen der Hydrologie.
Berlin: Verl. f. Bauwesen.
WMO (1992): International Glossary of Hydrology., WMO No.
385, available online in several languages at
http://webworld.unesco.org/water/ihp/db/glossary/glu/HINDEN
.HTM
* Available in students library at WATENV office
13
References



Maidment, D.R. (Editor), 1992. Handbook of Hydrology. McGraw-Hill Inc.
Grayson, R. and Blöschl, G., 2001. Spatial patterns in catchment hydrology:
observations and modelling. Cambridge Univ. Pr., 404 pp.
WWAP (World Water Assessment Program). 2012. The United Nations World
Water Development Report 4: Managing Water under Uncertainty and Risk.
Paris, UNESCO.
14
Chapter 2: Water- and Energy Balances
2.1 Total available water
2.2 Water cycle
2.3 Water balance
2.4 Catchment
2.5 Energy balance
Exercise 1, Part 1:
 Delineation of catchment boundaries/ areas
 Transformation of units of the water balance components
1
Cycles: 2.1 Available water
Total water = 1386 · 106 km3
Oceans
96.53%
Groundwater
1.69%
Freshwater part from
total water = 2,53 %
Groundwater
30.02%
Soil moisture
0.00%
Ice
1.76%
Soil moisture
0.18%
Lakes
0.01%
Atmosphere
0.04%
Swamps
0.00%
Biology
0.00%
Atmosphere
0.00%
Fig. 2.1a: Total water of the Earth
(Korzun 1978)
Rivers
0.00%
Lakes
0.26%
Biology
0.00%
Ice
69.47%
Swamps
0.03%
Rivers
0.01%
Fig. 2.1b: Fresh water parts of the
Earth (Korzun 1978)
2
Table 2.1: Water storages on the earth and residence times (Korzun, 1978 )
Hydrosphere
component
Volume
[1.000 km3]
Percentage of total
water [%]
Percent. of total
fresh water [%]
Residence
time
Oceans
1338000
96.54
0
2500 yr
Groundwater
23400
1.688
30.061)
1400 yr
Soil moisture
65
0.001
0.18
1 yr
Ice
24364
1.758
69.56
1600-10000 yr
Lakes
176,4
0.0127
0.261)
17 yr
Swamps
11,5
0.0008
0.03
5 yr
Rivers
2.1
0.0002
0.006
17 d
Biology
1.1
0.0001
0.003
Several h
Atmosphere
12.9
0.0009
0.037
8d
Total water
1385984.5
100
0
--
Fresh water
35029
2.53
100
--
1)
Only fresh water part of specific volume is considered
Residence
time:
storage capacity [km3 ]
TR [yr] 
flow [km3 /yr]
(2.1)
3
Cycles: 2.2 Water cycle
Introduction:
 Most fundamental principle of hydrology
 Endlessly proceeding cycle, characterised by changes in state
and location, combining atmosphere, land and oceans,
consisting of the main components evaporation, water vapour
transport, precipitation and runoff
 Water engine is fueled by solar energy (for evaporation) and
driven by gravity (for precipitation and runoff)
 Is a closed system; no water can be lost or emerge
 In hydrology usually only parts of the global system considered,
often catchments  these are open dynamic systems, for
which however also complete balances can be made
4
Water vapour transport
Transpiration
Precipitation
458 x 103 km3
(1269 mm)
Evaporation
505 x 103 km3
(1400 mm)
72 x 103 km3
(485 mm)
Evaporation
Precipitation
119 x 103 km3
(800mm)
River discharge
Land
Ocean
Total flow
47 x 103 km3
(315 mm)
Lake
Groundwater discharge
Percolation
Surface ~96%
Groundwater ~ 4%
Fig. 2.2: Scheme of the global water cycle (Time reference 1 year,
numbers from Korzun, 1978)
5
Table 2.2: Main components and fluxes in the water cycle (all in mm/∆t)
Term
Definition
Precipitation, P
Liquid or solid water precipitated from the atmosphere
Evaporation, E
Rate of liquid water transformation to vapour from open
water, bare soil, or vegetation surfaces
Transpiration, T
Rate of liquid water transformation to vapour from soil
through the plants (through stomata and cuticle)
Evapotranspiration, ET
Sum of evaporation and transpiration
Interception, EI
Storage of precipitation on plant surfaces (enters the
atmosphere as part of evaporation E)
Runoff R, Q
Water volume leaving a certain surface area, soil volume
or catchment per time interval
Infiltration, f
Water entering the soil zone through its surface
Percolation
Water passing through the soil into the deeper layers,
often entering the saturated zone
GW-Recharge
Water entering the groundwater zone
6
Cycles: 2.3 Water balance
Long term mean annual water balances:
Global:
P  ET
[mm yr -1]
(2.2)
577000 km3 yr -1  1130 mm yr -1, A  510  106 km2
Global
land:
Global
sea:
Germany
1961-90
(HAD, 2003)
P  ET  R [mm yr -1]
(2.3)
800  485  315 [mm yr -1], AL  149  106 km2
P  ET  R [mm yr -1]
(2.4)
1269  1400  131 [mm yr -1], AS  361 106 km2
P  ET  R
[mm yr -1]
779  481 298 [mm yr -1], A  0,357  106 km2
7
Water balance equation:
 with capability to consider any time period and spatial extent:
Basic:
Detailed:
P
ET
R
RO, RI, RG
Qin
Tr, Wi
∆S
P  ET  R  S [mm yr -1]
(2.5)
P  ET  RO  RI  RG  Qin  Tr  Wi  S [mm yr -1]
(2.6)
precipitation
evapotranspiration
runoff, general
surface runoff, interflow, groundwater runoff
(upstream) inflow to the system
transfer (in), withdrawal (out)
storage change
8
Climate zones:
- Comparing precipitation P and potential evapotranspiration
PET, climate zones of the earth can be defined
- Because of high temporal climate variability and many factors
affecting climate different classifications are possible
- The following is a simple classification:
Humid:
P > PET , 10 -12 months per year
Semi humid:
P > PET , 6 – 9 months per year
Arid:
P < PET , 10 -12 months per year
Semi arid:
P < PET , 6 – 9 months per year
9
Climate diagrams:  Characterising aridity depending on season
Roma (Rome)/Italy
41°54´N/12°29´E
46m
Month
Temp.
Pcp.
[°C]
[mm]
Jan
6.9
76
Feb
7.7
88
Mrz
10.8
77
Apr
13.9
72
May
18.1
63
Jun
22.1
48
Jul
24.7
14
Aug
24.5
22
Sep
21.1
70
Oct
16.4
128
Nov
11.7
116
Dec
8.5
106
Annual mean temperature
15.5 °C
Annual mean precipitation
Fig 2.3: Climate
diagram after Walter
& Lieth
880 mm
• Assumption: Evaporation depends only on temperature
• and: P(month) ≥ 2 * T(month)  humid, otherwise arid
10
Cycles: 2.4 Catchment
 A catchment (or watershed, or drainage basin) is the area of
land draining into a stream above a given cross section.
 Catchments are open dynamic systems for which the water
balance equation is valid (cp. Eq. 2.5 and 2.6)
 The associated catchment area AC in km2 is the measured area
over the basin in horizontal projection
 Catchments are separated by catchment boundaries or
watershed divides.
 Depending on hydrogeological conditions subsurface
catchments AC,u and surface catchments AC,o can be
distinguished
 In practice a catchment is often delineated from topographical
maps assuming no difference between subsurface and surface
catchments
11
AC,o
Catchment AC
AC,u
High permeability
Medium
permeability
Low
permeability
Rock
Rock
Aquifer
AAE2
C2
C1
A
AC1
E1
Contour line
C2
Control point
Q
Fig. 2.4: Catchment with surface and subsurface boundaries (upper fig);
delineation of catchment area from a topographical map (lower fig)
12
Cycles: 2.5 Energy balance
Climate, Weather
Evapotranspiration
Precipitation
P
ET
H
ET´
sensible heat flux
Energy balance
RN=ET´+ H + B
Water balance
P = ET + R  S
Net
radiation
RN
Vegetation
Runoff R
Soil
Storage
change S
Soil heat flux
B
Balance area
Location factors
Fig 2.5: Energy- and
water balances as
coupled system (Dyck &
Peschke, 1995)
Balance volume
13
Energy balance:
Rn  B  H  lET
Rn
B
H
lET
[W m-2 or kJ m-2 s-1]
(2.7)
net radiation (total incoming radiation)
soil heat flux
sensible heat flux (measured as temperature)
latent heat flux (measured as evaporation)
Relation between
water and energy
balance units:
flux=
flux density
density  heat of vaporization
(2.8)
lET [W m-2 ]
ET [mm d ] 
 l
-1
Latent heat of vaporization
of water (with water
temperature T in °C)
l  2501 2.37 T
[kJ kg-1]
(2.9)
14
Cycles: 2.6 References



Dyck, S. & Peschke, G. (1995): Grundlagen der Hydrologie. Verlag für
Bauwesen, Berlin.
HAD,
2003.
Hydrologischer
Atlas
von
Deutschland,
Atlastafeln
Hydrometeorologie. Bundesministerium für Umwelt, Naturschutz und
Reaktorsicherheit. Offenbach und Berlin.
Korzun, V.I. (ed.) (1978): World water balance and water resources of the Earth.
Studies and Reports in Hydrology 75. Translation from Gidrometeoizdat,
Leningrad (in Russian), UNESCO, Paris.
15
Chapter 3: Precipitation
3.1 Formation of precipitation
3.2 Types of precipitation
Hannover (52°N, 9°O)
Nairobi/ Kenia (1°S, 36°O)
Niederschlag [mm/mon]
60
50
40
30
20
10
Jan
Feb
Mar
Apr
Mai
Jun
Jul
Aug Sep
Okt
Nov
180
160
140
120
100
80
60
40
20
0
Jan Feb Mar Apr
Dez
Mai
Jun
Monat
Jul Aug Sep Okt
Nov Dez
Monat
Mumbai/ Indien (18°N, 72°O)
Belo Horizonte/ Brasilien (20°S, 43°W)
800
350
Niederschlag [mm/mon]
3.4 Variables describing precipitation
70
0
Niederschlag [mm/mon]
3.3 Variability in space and time
Niederschlag [mm/mon]
80
700
600
500
400
300
200
100
0
300
250
200
150
100
50
0
Jan
Feb
Mar
Apr
Mai
Jun
Jul
Aug Sep
Okt
Nov
Dez
Jan
Feb
Mar
Apr
Mai
Jun
Monat
Jul
Aug
Sep
Okt
Nov
Dez
Monat
3.5 Precipitation measurement
3.6 Areal precipitation
3.7 Design rainfall
3.8 Snow
Exercise 1, Part 2 :
 Calculation of areal precipitation
1
Precipitation: 3.1 Formation of precipitation
Introduction:
 Precipitation is most important input for hydrological balances,
models and prognoses
 Precipitation is the liquid or solid water precipitated from the
atmosphere
 Generally it can be distinguished between falling precipitation
and deposited precipitation
 Falling precipitation: rain, snow, sleet (graupel) and hail
 Deposited precipitation: dew, fog, frost
2
Water in the atmosphere - terms and definitions:
 Absolute humidity a [kg/m3]: mass of water vapour per unit
volume of moist air
 Vapour pressure e [hPa]: partial pressure of water vapour; is a
measure for the content of water vapour in the air
 Saturation vapour pressure eS [hPa]: partial pressure of water
vapour for maximal moisture content of water in the air
 Saturation deficit d: difference between saturation vapour
pressure and actual vapour pressure [hPa] or difference
between maximal and actual moisture content of the air [kg/m3]
 Relative humidity U: ratio of vapour pressure e to saturation
vapour pressure eS
 Dew-point temperature τ [°C]: temperature, which need to be
reached in cooling the air in order to have saturated conditions
3
Saturated vapour pressure curve
es
[hPa]
d

40
es
30
20

d
e
Saturation deficit
dew-point
temperature
difference
dew-point
temperature

10
T [°C]
-10
0
10

20
30
T
Fig. 3.1: Saturated vapour pressure as function of the air temperature
4
Requirements for formation of precipitation:
1. Sufficient water vapour in the air,
2. Cooling of air mass below the dew-point temperature,
3. Availability of condensation nucleus (e.g. aerosols), on
which the water molecules can attach themselves
A) Coalescence T > 0°C:
Flow together of cloud particles mainly by collision 
forming of liquid water droplets
B) Bergeron-Findeisen-Process T < 0°C:
Sublimation of water vapour at ice particles; growing by
aggregation with other ice particles  forming of snow,
graupel, hail  can become rain during fall
5
Precipitation: 3.2 Types of precipitation
Convective precipitation:
 Develops from intensive rising of warmer air masses from
heated earth surfaces
 Expresses often high rainfall intensities with short duration and
limited spatial extend (thunderstorms)
 Typical is the high spatial rainfall variability
6
Stratiform/ frontal precipitation:
 Precipitation connected to advancing frontal systems:
 Warm front: warm air is rising above cold air
 Cold front: warm air will be overtaken and replaced by cold air
 Larger extent and duration of precipitation
 In mid latitudes largest contribution to total rainfall
7
Orographic precipitation:
 Develops by lifting of moist air mass at mountains and hills (or
along coast lines)
 Leads to precipitation with different duration and intensity
 Orographic mechanisms are the cause oft the higher precipitation
amounts in the mountains
8
Precipitation: 3.3 Variability in space and time
Global distribution in four zones:
1.
2.
3.
4.
Equatorial range (tropical range) with highest rainfall
amounts P > 1500 mm/yr (> 4 mm/d),
Following to the north and south the dry regions with little
rainfall up to 0 mm/yr for North Africa and Central Asia,
Following the humid regions in mid and higher latitudes with
rainfall between 300 and 2000 mm/yr,
Finally, following the regions at the poles with less rainfall
Precipitation regimes:
1.
2.
3.
Tropics type with 2 rainy seasons (Max. April & November)
Subtropics type with rain in winter
Temperate zone with rain all year
9
750
1500
3000
6000 mm/yr
Fig. 3.2: Global distribution of mean annual rainfall during 1991 - 95 calculated
from the CRU data set (New et al., 2000) by Gerten et al. (2005)
10
Fig. 3.3: Distribution of the
mean annual precipitation in
mm/yr for Germany during
1961-90;
Source: produced using raster
data of the DWD after MüllerWestermeier 2005) (URL:
http://imk-tornado.
physik.uni-karlsruhe.
de/~muehr/Karten/
regen6190jahr.png, access
date: 04.07.2015).
11
Fig. 3.4: Largest observed rainfall as function of duration in the world and in
Germany (DWD 2002; Dyck 1980; NOAA 2015; WMO 2009)
12
Precipitation in mm
PI [mm/h]
Event: 13/08/1990
PI [mm/h]
PI [mm/h]
Time [d]
Time [2 h]
Time [5 min]
Fig. 3.5: Spatial rainfall distribution of a
thunderstorm with about 1.75 hours
duration near Stuttgart/ Germany
Fig. 3.6: Temporal distribution of rainfall
for different time step aggregation
13
Precipitation: 3.4 Variables
Precipitation depth: P in mm for ∆t
Amount of water from precipitation at a certain location in water
depth over a horizontal area. The time period for rainfall
accumulation needs to be given as well!
Precipitation intensity/ rate: PI in mm/∆t
Precipitation per time at a certain location.
Duration of precipitation: D in ∆t
Time period in ∆t during which precipitation occurs.
Specific rainfall: rN in l·s-1ha-1, l·s-1km-2
Precipitation volume per time and area.
14
Precipitation: 3.5 Measurement
3.5.1 Point measurements :
Collector
surface
Non-recording rainfall gauge:
(Type Hellmann)
Vessel
• Measuring of the accumulated rainfall
volume over time
• Collector opening with 200 cm2 area,
located 1 m above ground, no heating
• Measurement period: often 1 day
(DWD: 7:30 previous day - 7:30 CET)
• Resolution: P=0.1 mm; D=1d
• Max. capacity 60 -70 mm
Funnel
Can
Fig. 3.7: Non-recording rainfall gauge
(Dyck & Peschke, 1995)
15
Precipitation: 3.5 Measurement
Recording rainfall gauges:
• Continuous recording of the rainfall sum
curve on a chart: a float principle
• Collector opening with 200 cm2 area,
located 1 m above ground, heating
possible
• Measurement period: 1 day or 1 month
• Resolution: P=0.1 mm; D=5 min
• Capacity: 400 mm
A – Collector, G –chamber with float
and siphon, S – receiver can for
storage, T-chart
Fig. 3.8: Recording rain gauge (Dyck
& Peschke, 1995)
16
Volume [mm]
Time [h]
Fig. 3.9: Rainfall depth mass curve on a daily chart from a recording rain gauge
•
•
•
•
The slope of the mass (sum) curve represents the rainfall rate
The vertical lines show the emptying of the chamber by the siphon
For storing the analogous data the sum curves need to be digitized
The for any time interval the rainfall rate can then be calculated
17
Precipitation: 3.5 Measurement
Digitally recording devices:
• Continuous registration
• Sensors: tipping-bucket,
weighting, drop counting
• Collector opening with 200 cm2
area, located 1 m above ground,
heating possible
• Measurement period: ≥ 1 month
possible
• Resolution: P < 0.1 mm; D = 1 min
• Remote transmission possible
• Operational use possible (e.g. for
flood forecasting)
Funnel
Modulator
Drop counter
Photo sensor
for drops
Drop
destroyer
Photo sensor for
tipping bucket
Tipping counter
Tipping bucket
Fig. 3.10: Rainfall measurement with
drop counter and tipping bucket
18
Measurement errors for point observations:
Always underestimation of rainfall:
1. Wind induced error
2. Losses from surface wetting
3. Evaporation from collectors
Magnitude of errors:
- 10 - 20 % for rainfall
- about 25 % for snow
Methods for correction of systematic error:
 In Germany method acc. to Richter (1995) for daily data is
often applied
Pcorr  P  b  P
b  f  location of station, type of precipitation
  f  type of precipitation
 Internationally different approaches are in use, see e.g.
Sevruk (2005)
19
Precipitation networks:
• Globally about 40,000 rainfall
gauges are registered (GPCC
archive: http://gpcc.dwd.de)
• This amounts to a density of 1
station / 3725 km2 for the land
• Most of these gauges are nonrecording gauges !
In Germany the DWD operates:
• 4000 non-recording gauges (1
station/ 90 km2),
• 200 recording gauges from the
“basic network” (obs > 10 years),
(1 station/ 1800 km2) and
• 1300 recording stations from the
new “Network 2000” which
includes the “basic network” and
many “new” stations with very
short records (1 station/ 275 km2)
Fig. 3.11: Recording stations from
“Basic network” with about 200
recording stations in Germany
20
3.5.2 Radar-rainfall observation
Fig. 3.12: Principle of radar rainfall observation (Source: Homepage DWD)
21
Radar-reflectivity Z is calculated from receiving power:
Z
Pr
r
C
k
Pr  r 2
C k2

1
Di6

V vol
[mm6 /m3 ]
(3.1)
receiving power [W]
distance [m]
radar device constant [W m5 mm-6]
parameter for target’s composition
V
D
radar volume [m3]
drop diameter [mm]
Z-R-relationship for calculating precipitation rate:
Z  a  Rb
R
Z
a, b
(3.2)
precipitation rate [mm/ hr]
radar reflectivity [mm6 m-3]
parameter
(DWD: a=256, b=1.42; Marshall-Palmer: a=200, b=1.60)
22
Fig. 3.13: Radar network for rainfall
estimation (Source: www.dwd.de)
Fig. 3.14: Radar precipitation for the Bode river
basin for 12 hours in 5-min time steps
23
Precipitation: 3.6 Areal precipitation
 „Areal precipitation is the precipitation depth averaged over a
specific region with a defined area„ (DIN 4049)
 Estimation can be made direct from point observations or in
two steps, first interpolating point rainfall on a raster and then
taking the average over all raster cells within the considered
region.
Methods:





Arithmetic mean,
Thiessen-Polygon method (nearest neighbour),
Inverse-Distance method,
Isohyetal method,
Geostatistical methods.
24
Arithmetic mean:
1 n
PA  Pi
n i 1
PA areal precipitation [mm/∆t]
Pi point precipitation at gauge i [mm/∆t]
n number of stations
(3.3)
Fig 3.15: Considered point
stations and catchment area for
estimation of areal precipitation
25
Thiessen-Polygon method (Nearest Neighbour):
PA 
with
n
w  P
i 1
i
wi 
(3.4)
i
Ai
n
A
i 1
i
(3.5)
PA
Pi
n
wi
Ai
areal precipitation [mm/∆t]
precipitation for station i [mm/∆t]
number of stations
weight for station i
area of influence for Station i [km2]
catchment
boundary
_._._._
Fig 3.16: Construction of the
Thiessen polygons
triangle net
_________
Thiessen
polygon
_ _ _ _
26
Inverse-Distance method:
Fig. 3.17: Setting up an orthogonales
raster for the catchment
Fig. 3.18: Calculating precipitation for
each raster using the four nearest
neighbours, one in each quadrant
27
Inverse-Distance method (cont.):
Pj 
4
w
i 1
i, j
 Pi
1 di2,j
wi , j  4
1

2
i 1 di , j
1 nR
PA 
Pj
nR j 1
(3.6)
(3.7)
(3.8)
Pj precipitation for raster point j [mm/∆t]
Pi precipitation at station i [mm/∆t]
wi,j weight for station i at the raster point j
wi,j weight for station i at the raster point j
di,j distance between station i and raster
point j [km]
PA areal precipitation [mm/∆t]
Pj precipitation at raster point j [mm/∆t]
nR number of raster points in the catchment
28
Isohyetal - method:
PA 
n
w  P
i 1
i
i
(3.9)
PA areal precipitation [mm/∆t]
Pi precipitation for the isohyetal
area i [mm/∆t]
wi weight for the isohyetal area i
n number of isohyetal areas
Fig. 3.19: Isohyetal map for a
catchment
29
Precipitation: 3.7 Design precipitation
 Defined precipitation load for the design of hydraulic structures
(e.g. reservoirs, storm sewers, levees, etc.)
 Design rainfall is defined by three variables at least: 1) rainfall
intensity/depth, 2) rainfall duration and 3) rainfall frequency
General procedure:
1.
Frequency analysis of historically observed precipitation 
fitting of probability distributions
2.
Estimation of extreme rainfall from the probability distributions
 design precipitation (storm)
3.
Calculation of design runoff from design precipitation using
hydrological models
4.
Calculation of design water levels using hydraulic models
30
Statistical estimation of extreme rainfall depth*1:
1.
Rainfall series will be characterised by depth P & -duration D
2.
Sampling of extreme precipitation depth  one value per year
for each duration class (annual series) or all values exceeding
a threshold (partial series)
3.
For each duration class frequency analysis is carried out 
probability distribution functions are fitted F(x) = Pne(X≤x)
4.
Estimation of design values from F(x) considering specific
return periods T = average time for return of the design or a
greater event (given usually in years)
T (x) 
1
1
1


Pe ( x )
1  Pne ( x ) 1  F ( x )
(3.11)
Pe exceedance probability
Pne non-exceedance probability
31
*1
See also course “Statistical methods in hydrology”
Return period T [yr]
Precipitation intensity [mm/d]
60
1.01
2
P60_GEV
5
...
10
25
50
100
P60_Obs
50
40
Design precipitation
P(50yr,60min) = 46 mm
30
20
Pne=0.98 
Pe=0.02
T=1/.02=50yr
10
0
0.01
0.5
0.8
0.9
0.96
0.98
0.99
Non-Exceedance probability Pne [-]
Fig. 3.20 Estimation of the “1 in 50 years” design precipitation with duration D =
60 min for station Dortmund-Nettebach (nobs = 69 years) using a GEV
(Generalised extreme value) distribution fitted to the annual series
32
Regionalisation of design precipitation:
 Design precipitation is valid only for the point from which the
historical observation is taken
 Transferring information from observed points to unobserved
locations is called regionalisation
1
0
?
2
3
 The simplest approach is to take the design values from
nearest observed neighbour for the target location
 However spatial variability of rainfall is usually higher than
network density, so some kind of interpolation is required
 For interpolation the methods introduced in chapter 3.6 can be
employed
33
Method KOSTRA applied in Germany (Bartels et al., 1997):
• Interpolation of design rainfall ! (without
snow) on a regular raster with cell areas
of 71.5 km2 from analysed point data
based on a geostatistical approach
using additional information from
topography etc.
• Return periods from T = 0.5 - 100 yr
• Based on 200 recording stations for D <
24 hr (period 1951 – 1980)
• Based on about 4000 non-recording
stations for D > 24 hr (1951-2000)
• For durations from D = 5 min - 72 hr
• Separate maps for summer, winter and
calendar year
Fig. 3.21: Design rainfall for T=100yr ,
D=60min, calendar year (DWD/ itwh, 34
2001)
Depth-duration-frequency (DDF) relationships:
50
T=100 yr
Rainfall depth
intensity
PI [mm]
[mm]
40
T=50 yr
T=20 yr
T=10 yr
30
T=5 yr
T=2 yr
20
T=1 yr
T=0,5 yr
10
0
0
30
60
90
120
150
Duration D [min]
Fig. 3.22: Depth-duration-frequency curves for Hannover derived from
KOSTRA (raster column:33 row: 37) for the calendar year
35
Areal design rainfall

Reduction factor [%]

Areal rainfall intensities decrease with increasing areal extent; design
rainfall intensities are valid only for the point extent (gauge);
In engineering practice often depth-area curves are used for reduction
of point rainfall depending on the required regional design rainfall
Fig 3.23: Depth-area curves
for reduction of point rainfall
depending on duration
(Source: DVWK 1991)
Area AC [km²]
36
Precipitation: 3.8 Snow
Relevance for water resources management:
1. Stored water might generate extreme floods, especially when
melting and heavy rainfall occur at the same time 
2. Recharges groundwater and provides enough discharge for
water supply in summer 
Parameter:
 Snow depth d: level above earth surface (in cm),
 Snow density ρs, mass of snow per unit volume (kg/dm³),
 Water equivalent w  water volume stored in snow cover as ice
or water (mm)  most important parameter for planning
 Duration of snow cover: number of days with snow cover
 Extent of snow cover: area covered by snow
37
Measurement:
•
•
•
•
Heated precipitation gauges
Depth d  snow gauges
Water equivalent w  snow pillows; melting & weighting
Snow cover using remote sensing (satellites, aircrafts)
Calculation of snow melt:
• Detailed calculation based on energy balance
• Often simplified temperature index methods are applied (e.g.
degree-day-method) depending only on air temperature
• Differentiation between snow melt and meltwater leaching from
snow is required, since snow can store a significant amount of
liquid water, which not immediately infiltrates or becomes runoff
38
Precipitation: 3.9 References
 Bartels, H. et al., 1997. Starkniederschlagshöhen für Deutschland - KOSTRA.
Deutscher Wetterdienst, Offenbach am Main.
 DVWK, 1991. Starkniederschläge in der Bundesrepublik Deutschland. DVWK
Schriften, 97. Paul Parey, Hamburg und Berlin.
 Dyck, S. und Peschke, G. (1995): Grundlagen der Hydrologie. Berlin: Verl. f.
Bauwesen.
 Gerten, D., Haberlandt, U., Cramer, W. and Erhard, M., 2005. Terrestrial carbon
and water fluxes. In: M. Hantel (Editor), Observed Global Climate, LandoltBörnstein Handbook Series, Group V: Geophysics, Vol. 6 Springer, Berlin, pp.
12.1-12.15.
 HAD
(2003):
Hydrologischer
Atlas
von
Deutschland,
Atlastafeln
Hydrometeorologie.
Bundesministerium
für
Umwelt,
Naturschutz
und
Reaktorsicherheit., Offenbach und Berlin.
 New, M., Hulme, M. and Jones, P. (2000): Representing twentieth-century spacetime climate variability. Part II: Development of 1901-1996 monthly grids of
terrestrial surface climate. J. Climate, 13: 2217-2238.
 Richter, D. (1995): Ergebnisse methodischer Untersuchungen zur Korrektur des
systematischen Messfehlers des Hellmann-Niederschlagsmessers. 194, Berichte
des Deutschen Wetterdienstes, Offenbach am Main.
 Sevruk, B. (2005): Rainfall measurement: gauges. In: M.G. Anderson (Editor),
Encyclopedia of Hydrological Sciences. Wiley&Sons Ltd, Chichester, pp. 529-535.
39
Chapter 4: Evaporation
4.1 Introduction
4.2 Types of evaporation
4.3 Variability in space and time
4.4 Measurements of evaporation
4.5 Calculation of evaporation
4.5.1 Preface
4.5.2 Evaporation from water surfaces
4.5.3 Potential evaporation from land
4.5.4 Actual evaporation from land
4.5.5 Areal evaporation
4.6 References
1
Evaporation: 4.1 Introduction
 Evaporation is the direct transformation of liquid water to water
vapour below the boiling point of water
 Physical precondition is a vapour pressure gradient from the
water surface to the atmosphere (saturation deficit required)
 Evaporation is required to maintain the water cycle. Compared
to precipitation the evaporation process is continuous and
shows a strong diurnal cycle.
 Evaporation is considered a loss in the water budget  about
60 % of the precipitation input evaporates at the global scale
 Assessment of evaporation is especially important in arid areas
(P < PET) for quantifying water availability (e.g. irrigation,
reservoir operation, etc.)
2
Evaporation: 4.2 Types of evaporation
Evaporation E [mm/∆t]:
Evaporation according physical laws from open
water (EW), from bare soils (EB) and from plant
surfaces (Interception EI).
Factors affecting evaporation:
- Saturation deficit of the lower atmosphere,
- Air temperature,
- Solar radiation,
- Wind,
- Soil properties (moisture, type, temperature, etc.)
3
Interception EI [mm/∆t]:
Part of precipitation, which is intercepted on plant
surfaces (canopy) and evaporated without
reaching the ground
 Interception is part of evaporation.
Factors affecting interception:
- Meteorological factors (duration, depth, intensity and temporal
distribution of precipitation; wind, solar radiation; etc.
- Characteristics of the vegetation (plant species, age, season,
coverage, etc.)
Contribution to total evaporation (for mid-latitude climates):
- Agricultural crops: 6 - 12 % of annual precipitation Pyear
- Coniferous forest about 30 % of Pyear
4
Transpiration T [mm/∆t]:
 Evaporation of water from within plants through
stomata resulting from plant metabolism.
 Water is uptaken by roots, transported through the
plants and evaporated from the leafs
 Water transport is maintained by a tension gradient
 ψsoil 0.1 MPa < ψroot 0.5 MPa < ψleaf 2 MPa < ψair 30 MPa
Factors affecting transpiration:
- Meteorological conditions (solar radiation, temperature,
saturation deficit, wind, etc.)
- Vegetation properties (species, age, season, etc.)
- Soil properties (moisture, soil type, rooting depth, etc.)
- Chemism, (C02, light, O2, nutrients)
5
Evaporation: 4.2 Types of evaporation
Potential Evapo(transpi)ration (ETP) in mm/Δt:
Maximal possible evaporation for an idealized water
saturated surface under existing atmospheric
conditions allowing to use all available energy.
Actual Evapo(transpi)ration (ETR) in mm/Δt :
Real occurring evaporation from a surface for actual
water availability under existing atmospheric
conditions.
Usually is :
ETR  ETP
6
Evaporation: 4.3 Variability in space and time
750
1500
3000
6000 mm/yr
Fig. 4.1: Global distribution of mean annual actual evaporation ETR calculated
for 1991 to 1995 by Gerten et al. (2005) (cp. global P-distribution, Fig. 3.4)
7
Fig. 4.2: Distribution of the
mean annual potential
evaporation ETP in mm/yr
for Germany during 19611990 (Source: HAD, 2003)
8
Fig. 4.3: Distribution of
the mean annual actual
evaporation ETR in
mm/yr for Germany
during 1961-1990
(Source: HAD, 2003)
9
18
16
14
ETR [%]
12
10
8
6
4
2
0
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Month
Fig. 4.4: Seasonal distribution of actual evaporation ETR in % from mean annual
actual evaporation for Germany (DVWK, 1996)
10
Evaporation: 4.4 Measurement
Classification of methods:
1. Water balance methods: evaporation ET in mm/Δt is
determined as residuum from the water balance equation
requiring precipitation, runoff and change in storage as known.
2. Micrometeorological methods: latent heat lET in W/m2 is
obtained based on the energy balance equation; direct
methods measures lET directly (e.g. Eddy covariance method);
indirect methods measure other variables related to IET (e.g.
Flux-Profile or Bowen Ratio Energy Balance methods)
3. Remote sensing methods: also based on energy balance;
estimation of all terms leaving lET as unknown
4. Plant-physiological methods: determines transpiration T;
measuring sap flow or gas exchange directly at the plants
11
Evaporation pans
 Potential evaporation ETP can be measured using
evaporation pans based on the water balance
 Pans can be used as floating pans for measuring lake
evaporation (Fig. 4.5a) or as land pans for estimating potential
evaporation for land surfaces (Fig. 4.5b)
 The pan evaporation can be calculated from the difference
between precipitation and change in water volume in the pan:
ETPPan  P - SPan
(4.1a)
 The true potential evaporation from land surface is usually
lower then the observed pen evaporation because of oasis
effects; correction is possible using pan coefficients:
ETP  kPan  ETPPan
(4.1b)
12
Fig. 4.5b: Class A Pan for measuring ETP
from land (Source: Masoner et al., 2008)
Fig. 4.5a: Floating Pan for
measuring ETP from open water
(Source: Masoner et al., 2008)
Fig. 4.6: Differences
between evaporation from
floating pan and land pan
(Source: Masoner et al.,
2008)
13
Lysimeter
Measurement of ETR(Land) (or ETP(Land))
 Undisturbed natural soil column with growing vegetation;
surface area of A > 1 m2 and soil depth of 1 to 2 m
 They are isolated to prevent lateral water exchange. Drainage,
change in storage and precipitation needs to be measured.
 Using weighing lysimeters (Fig. 4.7) the change in water
storage can be measured by change in weight.
 Using non-weighting lysimeters the change in water storage
can be estimated by measurement of soil moisture.
 Actual evaporation is calculated as residuum from the water
balance equation:
ETR  P  RD  S
(4.2)
P  Precipitation, RD  Drainage, S  Change in storage
14
E
Raing
gauge
P
E
P
Root zone
Root zone
Non
root zone
Non
root zone
Cellar
Capillary fringe
Capillary fringe
Gravitational
water
Groundwater
Collector
Scale
Scale
Fig 4.7: Scheme of a weighting lysimeter
for cases with groundwater impact (left)
and without groundwater impact (right)
(from DVWK, 1996)
Fig. 4.8: Weighting lysimeter site for the
estimation of drainage, actual evaporation
and nitrate leaching (Brandis)
15
Evaporation: 4.5 Calculation
E (potential)
E (actual)
ET (actual)
4.5.1 Preface
1.
Temperature
2.
Solar radiation
3.
Wind
4.
Saturation deficit
6.
Soil properties
7.
Soil moisture at surface
8.
Vegetation (type, age, rooting depth, etc.)
9.
Soil moisture up to routing depth
10.
Nutrients, CO2
Energy
Aerodynamics
Fig. 4.8: Factors influencing evaporation
16
Overview of methods for calculating evaporation
Nr. Method
Surface
Type
Time
step
Semester
1
Dalton
water
ETP
day
Hydro-I
2
Haude
land
ETP
day ->
month
Hydro-I
3
Turc-Wendling
land
ETP
day
Hydro-I
4
Drying functions
land
ETR
day
Hydro-I
5
Penman
land
ETP
day
Hydro-II
6
PenmanMonteith
land
ETR
day
Hydro-II
7
FAO crop
reference
land
ETP
day
Hydro-II
For more methods see Shuttleworth (1993), Chow et al. (1988), DVWK (1996)
17
4.5.2 Evaporation from open water
Dalton-method  aerodynamic method:
EW  f (v )   eS (TW )  e 
EW
f(v)
es(TW)
e
- Evaporation from water [mm ꞏ d-1]
- Function of wind velocity [mm ꞏ d-1 ꞏ hPa-1]
- saturated vapour pressure for water temperature Tw [hPa]
- actual vapour pressure [hPa]
d  eS (T )  e
hPa
Magnus formula:
Wind function:
a,b,c
v
(4.3)
(4.4)
 Saturation deficit (cp. Fig. 3.1)
 17.62  T 

 243.12  T 
hPa 
eS (T )  6.11  exp 
(4.5)
Temperature T in °C
f (v )  a  b  v c
(4.6)
- empirical coefficients depending on climate and topography
- Wind speed at 2 m height [m ꞏ s-1]
18
Obtaining actual vapour pressure e:
Can be calculated from relative
humidity U in %, which can be
measured directly using hygrometers.
e  es 
U
100
Wikipedia, 2013
Can be calculated based on temperature
difference between wet and dry
thermometer using a psychrometer.
e  es    Td  Tw 
Td – dry bulb temperature, Tw – wet bulb temperature,
γ – psychrometer constant (γ=0.67 for elevation < 500
m.a.s.l., γ=p/1007 for elevations > 500 m.a.s.l. with air
pressure p in hPa; from Häckel, 2008)
19
Wikipedia, 2013
4.5.3 Potential evaporation from land
Haude-method  aerodynamic method:
ETPHaude  f   eS (TW )  e 14  f  d14
f
d14
[mm  d-1]
(4.7)
- empirical parameter [mm ꞏ d-1 ꞏ hPa-1]
- saturation deficit at 14.00 hr [hPa]
Month
f
October – March
April, May
June
July
August
September
0.22
0.29
0.28
0.26
0.25
0.23
Tab. 4.1: Parameter f of the
Haude-method for Germany
(DVWK, 1996)
20
Turc-Wendling methods  energy balance method
ETPTUWE 
RG
s
γ
l
ρ
fk
s
s 
 0.71 RG


 0.27  fk 
  l

(4.8)
- global radiation [W m-2 ]
- slope of the saturation pressure curve [hPa K-1]
- psychrometer constant = 0.655 [hPa K-1]
- latent heat of vaporization see Eq. (2.9) [KJ kg-1]
- density of water [kg m-3]
- coastal factor
(fk=0.6 for distance to coast < 50 km, otherwise fk=1.0)
with simplification
s
T  22
 2.3 
s 
T  123
with T in °C
21
Global radiation RG (Solar radiation)
 Incoming total short wave solar radiation at the Earth’s surface

S
RG  R0  aG  bG  
S0 

R0
aG
aG+bG
S
S0
[W  m-2 ]
(4.9)
- extraterrestrial radiation (at top of atmosphere)
- fraction of R0 on overcast days (S/S0 = 0)
- fraction of R0 on clear (for average climates aG=0,25 ; bG=0,50)
- bright sunshine hours [hr]
- total day length [hr]
For evaporation only absorbed shortwave radiation RS is effective:
RS  (1   )  RG
[W  m-2 ]
(4.10)
μ - Albedo (short wave solar reflection coefficient)
(e.g. short farm crops: 15 – 25%, new snow: 75-95%, old snow: 40-70%;
coniferous forest 5 – 12%, deciduous forest 15-20%, water 3 -20%)
22
Consideration of different vegetation for evaporation
 All methods so far have considered potential evaporation for
a reference surface of usually wet short grass  ETPGrass
grass reference evaporation in a wider sense
 Estimation of ETP for other surfaces requires corrections
because of different albedo, coverage, height, etc.
 Simplest correction method uses so called crop coefficients
ETPc  kc  ETPGrass
[mm  d-1]
(4.11)
ETPc
- crop specific potential evapotranspiration
ETPGrass - reference potential evapotranspiration
kc
- crop coefficient (0.2  kc  1.4) (see e.g DVWK, 1996)
Kc = 0.2
0.7
1.2
0.4
Fig. 4.8: Stages of crop grow: 1 initial stage, 2 – development, 3 –
mid season, 4, late season and
example for associated kc
(see Chow et al., 1988)
23
4.5.4 Actual evaporation from land
Actual evaporation can be calculated by reducing potential
evaporation depending on water availability i.e. soil moisture
content (SMC):
ETR  f ( )  ETPc
f ( )
ETPc
[mm  d-1]
(4.12)
- soil moisture dependent drying function
- potential crop specific evapotranspiration
f
θS
θFK
θ0
θWP
1
0
WP
O
FC
S
- SMC at saturation
- SMC at field capacity
- limiting SMC for ETP
- SMC at wilting point

24
4.5.5 Areal evapotranspiration ETRA



ETRA is the actual
evapotranspiration averaged for a
certain area or catchment
High heterogeneity of the land from
variability in land use, soils,
precipitation, which requires
structuring/ classification of areas
Structuring considers usually
hydrological similar subareas called
hydrotopes (or hydrological
response units, HRU) or raster cells
1
ETRA 
Atot
n
 A  ETR
i 1
i
i
[mm  t ]
-1
(4.13)
1
1
1
5
4
1
1
1
5
5
2
2
2
3
4
2
2
3
3
3
25
Evaporation: 4.6 References









Chow, V.T., Maidment, D.R. and Mays, L.W. (1988): Applied Hydrology. McGrawHill.
Dyck, S. und Peschke, G. (1995): Grundlagen der Hydrologie. Berlin: Verl. f.
Bauwesen.
Dyck, S., (1980): Angewandte Hydrologie, Teil 2. VEB Verlag für Bauwesen,
Berlin, 544 pp.
DVWK (1996): Ermittlung der Verdunstung von Land- und Wasserflächen.
Merkblätter zur Wasserwirtschaft, 238/1996. DVWK, Bonn.
Gerten, D., Haberlandt, U., Cramer, W., Erhardt, M. (2005): Terrestrial Carbon and
Water Fluxes. In: Hantel (Ed.) Observed Global Climate. Landolt-Börnstein New
Series V/6, Springer, pp. 12.1 -12.15.
Häckl, H., 1991. Meteorologie. Eugen Ulmer KG, Stuttgart.
HAD
(2003):
Hydrologischer
Atlas
von
Deutschland,
Atlastafeln
Hydrometeorologie.
Bundesministerium
für
Umwelt,
Naturschutz
und
Reaktorsicherheit., Offenbach und Berlin.
Masoner, J.R., Stannard, D.I. and Christenson, S.C. (2008): Differences in
Evaporation Between a Floating Pan and Class A Pan on Land. JAWRA Journal of
the American Water Resources Association, 44(3): 552-561.
Shuttleworth, W.J. (1993): Evaporation. In: D.R. Maidment (Editor), Handbook of
hydrology. MacGRAW-HILL, New York, pp. 4.1-4.53.
26
Chapter 5: Runoff
5.1 Introduction
5.2 Variability in space and time
5.3 Stage-discharge-relationship
5.4 Measurement of discharge
5.5 Measurement of stage
5.6 Streamflow statistics and hydrographs
5.7 Design flow
5.8 References
1
Runoff: 5.1 Introduction
Runoff R:
Flow rate of water per time interval, draining a catchment, subcatchment or defined area on and below the surface (units: e.g.
mm/d). (Is applied here generally to describe the total flow from a
catchment, when origin or flow paths are less important.)
Discharge or streamflow Q:
Flow rate of water per time interval, passing a cross section of a
defined natural channel (units: e.g. m3/s)
Channel types:
1) Perennial: A channel which never dries.
2) Intermittent: A channel which dries at certain times in a year.
3) Ephemeral: A channel where water flows only after rainfall.
2
Runoff: 5.1 Introduction
Specific discharge q:
Streamflow QA at a certain point in a channel divided by the
drainage area AC of the catchment
q
QA
AC
[m3  s-1  km-2 ]
or
[l  s-1  ha-1 ]
(5.1)
 Allows comparing discharge from different catchments eliminating
the influence of the catchment area
 Specific discharge is related directly to precipitation and evaporation
and indirectly to climate, elevation, catchment size etc.
3
Runoff: 5.1 Introduction
Main flow components:
Surface runoff RO: Flow from the surface of poorly permeable
soils, temporarily saturated soils or permanently saturated soils.
Interflow RI: Rapid subsurface flow through pipes, macro-pores,
and the seepage zone in soils
Base flow RB: Return flow from groundwater. Base flow index (BFI):
Portion of base flow from total flow.
Quick flow/ direct flow RD: Sum of surface runoff and interflow.
(Is equal to the effective rainfall or to the flood flow volume, see
Chapter 10)
4
Runoff: 5.1 Introduction
EvapotransVerdunstung
piration,
ET ET
Precipitation
Niederschlag
PP
Surface
Oberflächensystem
system
OberflächenSurface
abfluss
runoff RROO
Soil
Bodensystem
system
ZwischenInterflow
abfluss
R RI
GrundwasserGroundwater
system
system
BasisBase
flow
abfluss
R RB
Verzögerung
Delay
(Flussbettspeicher)
(river bed storage)
HauptabflussMain flow
komponenten
components
Oberflächengewässer
River system
Main subsystems
Hauptteilsysteme
eines
Einzugsgebiets
of a catchment
Interzeption
Interception
EI
IE
I
Discharge
Durchfluss
QG
Q
B
Fig 5.1: Components and origin of flow in the catchment
5
Runoff: 5.2 Variability in space and time
750
1500
3000
6000 mm/yr
Fig. 5.2: Global distribution of calculated runoff (Gerten et al., 2005)
6
Runoff: 5.2 Variability in space and time
Runoff [%]
Runoff [%]
12
8
4
0
1
2
3
4
5
6
7
8
North America
16
14
12
10
8
6
4
2
0
Runoff [%]
Europe
16
1
9 10 11 12
2
3
4
5
1
2
3
4
5
6
7
Month
7
8
9 10 11 12
1
2
3
4
5
8
9 10 11 12
South America
14
12
10
8
6
4
2
0
1
2
3
4
5
6
7
6
7
8
9 10 11 12
8
9 10 11 12
Month
Runoff [%]
Africa
14
12
10
8
6
4
2
0
6
Month
Runoff [%]
Runoff [%]
Month
Asia
16
14
12
10
8
6
4
2
0
8
9 10 11 12
Month
Australia
14
12
10
8
6
4
2
0
1
2
3
4
5
6
7
Month
Fig. 5.3: Seasonal flow pattern (Shiklomanov, 1999)
7
Fig. 5.4: Distribution of
mean annual runoff R
in mm/yr for Germany ,
Period1961-90
(Source: HAD, 2003)
R  PCorr  ETR
Deficit areas:
R < 0, ETR > P
8
Runoff: 5.3 Stage-discharge-relationship
Objective
Continuous streamflow data for river sections
Problem:
Continuous measurement of discharge Q difficult
Solution:
1. Irregular measurement of Q and related water level, stage h
2. Derivation of stage-discharge-curve (rating curve) Q = f(h)
3. Continuous measurement of stage h is simple  transformation
into discharge Q = f(h)  continuous flow data Q
Discharge measurement often
indirectly based on velocity and
Continuity equation
Q v A
with
(5.2)
A  f (h)
9
Stage – discharge – curve (W-Q-relationship, rating curve)
h in m
Q  a  hb
observations
Discharge
Q
3
Q in m /s
(5.2)
discharge
h
stage
a, b parameter
log h
in m
Extrapolation ?
log Q  log a  b  log h
(5.3)
log Q
in m3/s
10
Runoff: 5.4 Measurement of discharge
5.4.1 Measurements using current meters:

Indirect method  calculation of Q from
velocity v
B h
Q
  v  x, y  dx dy
(5.4)
0 0

Current meter: propeller device with
rotating cups (Fig. 5.5)

Number of rotations per time n depends
on water velocity v and cup geometry
v  an  b

(5.5)
Calibration of Eq. (5.5) by towing the
meter through still water in a tank at a
series of known velocities.
Fig 5.5: Discharge measurement
11
using a current meter
5.4.2 Other measurements methods for Q:
 Volumetric measurement: only direct method to
measure discharge Q e.g. in l/s
 Tracer gauging: Indirect method, based on
dilution of a tracer along a measurement reach of
a river (good for rivers with high turbulence)
 Measurement weirs: Indirect method based on
water level measurements using predefined
rating curves according to weir characteristics
 Ultrasonic measurement: Flow velocity is
deduced from travel time differences of pulses
with and against flow direction
h
2
1
 Electromagnetic measurement: Water is
inducing a voltage according to its velocity when
flowing through a electromagnetic field (often
applied for pipes)
12
Runoff: 5.5 Measurement of stage
a datum for a certain point. The
datum of the gauge may be the
mean sea level or an arbitrary
datum chosen for convenience.
•
•
•
•
W
Stage
Stage is the water level above
PN
Datum
Stage is measured mainly for calculation of discharge using
rating curves.
Stage or water level can easier be continuously measured and
recorded as discharge.
The measurement stations which are used to calculate
streamflow are called streamflow gauges.
The measurements can be done manually (non-recording
gauges) or automatically (recording gauges).
13
Manual gauges, staff gauges




Staff gauges with 1 or 2 cm subdivision can be fixed vertical at river
bank or bridge or inclined often at staircases as step gauges
Readings are done usually once a day regularly e.g. at 7.00 hr
Manual gauges are used for comparison with recording gauges and for
reference during streamflow measurement
Water level is obtained from the staff gauges accurate to 1 centimetre
Water level = 235 cm
Fig. 5.6: Vertical staff gauge with example reading (left 1 cm, right 2 cm division) and step gauge (further right)
14
Recording gauges:
Analogue chart
recorder
Measuring hut
Recording instrument
Float cable
Faceplate with ventilation slots
HHW
Rock fill
Float
Slider for closing
the connecting pipe
outfall
structure
NNW
Connecting pipe
Stilling well
 100 cm
Concrete bearing
Digital shaft
encoder
Fig 5.7: Water level gauging site based on the float gauge principle (left);
analogue & digital recorders (photos to the right, OTT)
15
Digital bubble sensor
(OTT) including pump
mechanics, electronics
and data recorder
Fig 5.8: Water level gauging site based on the bubble gauge principle (left, Chow
et al. 1988); digital recorder (photo to the right, OTT)
16
Selection of a gauging site:



Site should be unaffected by variable backwater
conditions e.g. flood backwater, tidal effects, lakes,
tributaries, weeds
Downstream hydraulic control required: increase in
channel slope by weir, rock outcrop, etc.
Flow should be confined into a single channel; erosion &
sedimentation should be small; site accessible even during
high flows
Primary
control
Secondary
control
Downstream
control
Fig 5.9: Scheme of downstream controls for a gauging site (adapted from
Maidment, 1993)
17
Fig. 5.10: Streamflow gauge „Pionierbrücke“ at Sieber River
18
Fig. 5.11: Streamflow gauges Groß-Rühden (left), Derneburg (right) at Nette River
19
Runoff: 5.6 Streamflow statistics & hydrographs
5.6.1 Characteristic flow values




Mean and extreme values which are
used as basic indicators to
characterise flow regime at a certain
streamflow gauge or for a catchment
Characteristic values for stage W,
discharge Q and specific discharge q
are published in the “Deutsches
Gewässerkundliches Jahrbuch” (DGJ)
Calculated data in Germany are based
on the calendar year and on the
hydrological year from 1st Nov to
31. Oct. (Winter: 11-4, Summer: 5-10)
Data for selected gauges are available
online at http://www.dgj.de/ (Germany)
or from Global Runoff Data Centre
http://grdc.bafg.de (Worldwide)
Fig 5.12: Main river basins in the20DGJ
Characteristic flow data in Germany based on daily time series:
HHQ
HQ
MHQ
MQ
MNQ
NQ
NNQ
highest ever observed value; date & time of occurrence need to
be given (e.g. period 1941-98, day 5.12.1990, time 11.14 hr)
highest observed value for a specific time segment (e.g. year or
winter season for the calendar year 1988)
arithmetic mean of the highest observed values of equal time
segments for a certain time period (e.g. mean value of annual
HQ for the period 1941 - 98)
arithmetic mean over all values of a certain time period (e.g.
mean over all daily flows for period 1941 - 98)
arithmetic mean of the lowest observed values of equal time
segments for a certain time period (e.g. mean value of annual
NQ for the period 1941 - 98)
lowest observed value for a specific time segment (e.g. year or
winter season for the calendar year 1988)
lowest ever observed value; date of occurrence need to be
given (e.g. period 1941-98, day 9.10.1947)
 Q for discharge,  W for water level, q  for specific discharge
21
6443 km2
River basin:
Weser
Gauge:
SCHWARMSTEDT
AE0:
DGJ-basin:
LEINE
River km:
6.2 km (above M.)
PNP: 21 m.o.s.l.
River:
LEINE
No:
48800301
Q in m3/s
Hydrological year 1998
Calendar year 1998
Year
Date
Winter
Summer
Year
Date
NQ
22.5
19.08.1998
23.0
22.5
22.5
19.08.1998
MQ
56.7
62.4
51.2
80.5
HQ
286
212
286
518
31.10.1998
bei W= 511 cm
bei W= 577 cm
1941 to 1998
1941 to 1998
10.2
8.50
8.50
20.6
27.0
22.3
21.7
MQ
61.5
81.6
41.7
61.7
MHQ
283
276
129
297
HQ
1200
1200
494
1200
NQ
8.50
MNQ
05.10.1947
11.02.1946
at W= 612 cm
04.11.1997
05.10.1947
11.02.1946
at W= 612 cm
Fig 5.13: Sheet from DGJ with characteristic flow values (http://www.dgj.de/)
22
5.6.2 Longitudinal river section
 The longitudinal section of a river relates characteristic flow
values of the river sections to their distance from the source
 Usually the characteristic mean flow values MHQ, MQ, NQ
increase with growing distance and growing catchment area
from the source and show positive jumps at tributaries
 The specific discharge Mq usually decreases in the direction
of the flow, corresponding to a reduction in runoff generation
with growing catchment area mainly depending on
meteorology and physical basin characteristics
 This typical behavior might be different in other regions or
countries, e.g. in karstic or arid environments
 The typical behaviour of hydrological longitudinal river
sections can be used to characterise a river from source to
mouth and to make plausibility checks of observed flow data
23
Fig 5.14: Longitudinal section of discharge along the Rhine downstream of Lake
Constance up to the Dutch border for the time period 1931 – 1990 (from HAD, 2003) 24
5.6.3 Hydrograph and flow mass curve:

The graphical representation of flow data in the sequence they
occur in time Q(t) at a certain point or gauge is called hydrograph

The mass curve is built by
consecutive summation of the
hydrograph over time :
SQ  t  
1000
6
2000
500
1500
400
300
1000
200
500
100
.0
.0
01
01
2.
19
19
Fig. 5.15: Hydrograph and
mass curve of daily discharge
at gauge Bad Düben (Mulde)
for calendar year 1981 with
MQ=96.8m3/s; HQ=859 m3/s
und NQ=18.6 m3/s
81
3.
19
81
01
.0
4.
19
01
81
.0
5.
19
81
01
.0
6.
19
01
81
.0
7.
19
81
01
.0
8.
19
81
01
.0
9.
19
01
81
.1
0.
19
81
01
.1
1.
19
01
81
.1
2.
19
81
0
81
0
1.
(5.16)
3
2500
600
.0
dt
3000
Sum Q in 10 m
Mass curve
700
3
0
3500
800
Q in m /s
 Q t 
Hydrograph
900
01
t
25
Time
5.6.4 Frequencies*1 and flow duration curve:
 Analysis of frequencies and empirical distribution of flows
 Applied usually on daily flow data
 Daily flows are not independent !
 Theoretical probability distributions may not be fitted to
consecutive daily flow data!
Binning of data:
 Frequency analysis requires countability
 For continuous variables binning (classification) is required
-
Class width/ -limits:
∆xk = xk - xk-1
-
Membership of xi to class k:
xk-1 < xi ≤ xk
-
Number of classes (rule of thumb):
M = 5 log10 N
26
*1 see also course “Statistical methods”
Frequencies
Absolute frequency:
M
nk Number of values per class with  nk  N
k 1
Relative frequency:
n
f( xk )  k
N
(5.18)
Absolute cumulative
frequencies for class k:
Nk   ni
Relative cumulative frequencies
for class k:
F ( xk )   f( xi )
k
(5.19)
i 1
k
(5.20)
i 1
27
Flow duration curve (FDC):

Representation of ordered Q - data from smallest to largest value
against the time in days it is exceeded (or not exceeded)  e.g.
easily applied for annual flow duration curves

Or representation of upper Q - class limits against relative
cumulative frequencies F (number of non-exceedance days) 
often applied to obtain average flow duration curves over
several years

Reference time period: often 1 year, 1 month; Δt often 1 day

Two types are possible: non-exceedance flow duration curve
with non-exceedance days Nne,k for x ≤ xk and exceedance flow
duration curve with exceedance days Ne,k for x > xk
28
Table 5.1: Frequencies of daily discharge at gauge Bad Düben (Mulde) for 1981
abs. cumulative freq.
Class limits
Q [m3/s]
class
lower
k
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
abs.
freq.
rel.
freq.
f (Q )
nk (Q )
18
30
40
50
60
70
80
100
120
140
160
200
250
300
400
600
800
859
0
19
48
52
37
37
15
37
22
14
26
30
16
4
4
2
1
1
0.0000
0.0521
0.1315
0.1425
0.1014
0.1014
0.0411
0.1014
0.0603
0.0384
0.0712
0.0822
0.0438
0.0110
0.0110
0.0055
0.0027
0.0027
Sum:
365
1
0
18
30
40
50
60
70
90
100
120
140
160
200
250
300
400
600
800
Exceedance [d]
Non-exceedance [d]
upper
NU , k (Q )
k
NÜ , k (Q )
0
19
67
119
156
193
208
245
267
281
307
337
353
357
361
363
364
365
Flow duration
curves Qk=f(Nk)
rel. Summenhäufigkeit
365
346
298
246
209
172
157
120
98
84
58
28
12
8
4
2
1
0
Non-exceedance [-]
FU (Q )
0.0000
0.0521
0.1836
0.3260
0.4274
0.5288
0.5699
0.6712
0.7315
0.7699
0.8411
0.9233
0.9671
0.9781
0.9890
0.9945
0.9973
1.0000
Exceedance [-]
FÜ (Q )
1.0000
0.9479
0.8164
0.6740
0.5726
0.4712
0.4301
0.3288
0.2685
0.2301
0.1589
0.0767
0.0329
0.0219
0.0110
0.0055
0.0027
0.0000
29
Absolute frequency [days]
60
50
40
30
Fig 5.16: Absolute frequencies nk
(Histogram) for daily discharge of the
gauge Bad Düben (Mulde) for 1981
(classification see Tab. 5.1)
20
10
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18
Class
1000
Q in m3/s
800
600
Exceedance curve
400
Non-Exccedance curve
200
0
0
50
100
150
200
Days
250
300
Fig. 5.17: Flow duration curves of
daily discharge of the gauge Bad
Düben (Mulde) for 1981
(see Tab. 5.1)
350
30
Runoff: 5.7 Design flow
Defined discharge for dimensioning of hydraulic structures (e.g.
reservoirs, dikes, sewage networks, bridges, etc.). Design flow is
defined using the return period T (see Chap. 3.7, design rainfall )
Variable
Application for design/ planning of
Peak discharge [m^3/s]
Weirs, spillway cross section, culverts, river geometry, etc.
Flood water level [m]
Dikes, dams, height of spillways, inundation areas,
damage functions
Flood volume [m3]
Flood storage capacity of dams and flood control
reservoirs, polders, inundation areas, etc.
Flood frequency
Hydro power, damage potential
Low flow characteristics
[m3/s, m3, d]
Water supply (Industry, agriculture, ecology, etc.), storage
capacity of dams
Low flow frequency
Hydro power, damage potential
31
Alternatives for estimation of design
flood characteristics
Statistical methods
(flood frequency)
From observed
flood time
series
Regionalisation of
observed floods
- Statistical Methods (1st Sem.)
- Hydrology II (2nd Sem.)
Deterministic methods
(R-R-Model + Statistic)
From design
storm (e.g. using
KOSTRA)
From synthetic
rainfall (e.g. from
rainfall model)
From continuous
observed rainfall
- Hydrology I (1st Sem.)
- Hydrology II (2nd Sem.)
32
Runoff: 5.8 References





Dyck, S. und Peschke, G. (1995): Grundlagen der Hydrologie. Berlin: Verl. f.
Bauwesen.
Gerten, D., Haberlandt, U., Cramer, W. and Erhard, M. (2005): Terrestrial carbon
and water fluxes. In: M. Hantel (Editor), Observed Global Climate, LandoltBörnstein Handbook Series, Group V: Geophysics, Vol. 6 Springer, Berlin, pp.
12.1-12.15.
HAD
(2003):
Hydrologischer
Atlas
von
Deutschland,
Atlastafeln
Hydrometeorologie. Bundesministerium für Umwelt, Naturschutz und
Reaktorsicherheit., Offenbach und Berlin.
Maidment, David R. [Editor] (1993): Handbook of hydrology. New York: McGrawHill.
Shiklomanov, I.A. (1999): World water resources and their use. SHI/UNESCO,
pp. http://webworld.unesco.org/water/ihp/db/shiklomanov/.
33
Chapter 6: Subsurface water
6.1 Introduction
6.2 Soil water
6.2.1 Classification of soils
6.2.2 Soil water properties
6.2.3 Soil water potentials
6.2.4 Water retention curve
6.2.5 Water supply for plants
6.3 Groundwater
6.3.1 Groundwater occurrence
6.3.2 Groundwater recharge
6.3.3 Momentum equation
6.3.4 Principle of continuity
6.4 References
1
SubWat: 6.1 Introduction
Subsurface water: Water in the void space of the lithosphere;
flows beneath the land surface
Soil water: Subsurface water located in the pores of the lithosphere
with soil air, herbal and animal life (unsaturated zone, vadose zone)
Groundwater: Subsurface water,
which entirely fills the pores of the
lithosphere and only flows by
gravity force and frictional force
caused by flow
(saturated zone)
Fig. 6.1 Distinction of soil water
and groundwater
2
Importance of soil water:
 Unsaturated zone is responsible for flow generation processes
like groundwater recharge or direct flood runoff
 Important for transport and reaction of chemicals (water quality)
 Water content in soil determines the actual evapotranspiration
and limits the plant available water
Importance of groundwater:
 30% of the freshwater resources are groundwater (see Fig. 2.1)
 High water quality of groundwater
 Provides about 75% of water supply in Germany
3
Table 6.1 Water extraction of the public water supply companies in Germany classified by
the type of water (Federal Statistical Office of Germany)
Water extraction for public water supply
Water supply
companies
Year
Total
Groundwater
and spring
water
Water from
bank filtration
and enriched
groundwater
Water from
rivers, lakes
and reservoirs
Million m3ꞏa-1 (% of the whole water extraction)
Number
Germany
2001
5260
5409
4011 (74.1)
708 (13.1)
690 (12.8)
2004
5043
5372
3953 (73.6)
714 (13.3)
705 (13.1)
2007
4833
5128
3581 (69.8)
874 (17.1)
673 (13.1)
State Lower Saxony
2001
313
539.1
476.6 (88.4)
2.4 (0.4)
60.1 (11.2)
2004
293
536.4
473.7 (88.3)
2.7 (0.5)
60.0 (11.2)
2007
232
533.2
470.7 (88.3)
together 62.5 (11.7)
4
SubWat: 6.2 Soil water
6.2.1 Classification of soils
Soil texture: Soil fractions according to particle size of grains
and portions in mass-% of mineral components
Rough distinction by grain size in:
– Fine soil: Clay, Silt and Sand
diameter < 2 mm
– Coarse soils: gravel and stone
diameter ≥ 2 mm
Loam (L): mixture of the 3 fractions of the fine soil
5
Tab. 6.2 Fractions of fine soil
(DIN 4220, 2008, German standard)
Symbol
Particle size
(μm)
Clay
T
< 2.0
Fine clay
fT
< 0.2
Medium clay
mT
0.2 ≤ d < 0.6
Coarse clay
gT
0.6 ≤ d < 2.0
Silt
U
2.0 ≤ d < 63
Fine silt
fU
2.0 ≤ d < 6.3
Medium silt
mU
6.3 ≤ d < 20
Coarse silt
gU
20 ≤ d < 63
Sand
S
63 ≤ d < 2000
Very fine sand
ffS
63 ≤ d < 125
Fine sand
fS
63 ≤ d < 200
Medium sand
mS
200 ≤ d < 630
Coarse sand
gS
630 ≤ d < 2000
Denomination
Tab. 6.3 Coarse soil (DIN 4220, 2008,
German standard)
Name
Symbol
Particle size
(mm)
Gravel
G
2.0 ≤ d < 63
Fine gravel
fG
2.0 ≤ d < 6.3
Medium
gravel
mG
6.3 ≤ d < 20
Coarse
gravel
gG
20 ≤ d < 63
Stone
(cobble)
O
≥ 63
Stone
fO
63 ≤ d < 200
Block
mO
200 ≤ d < 630
Big block
gO
≥ 630
6
Soil textural triangles
Numerals:
2 low
3 medium
4 high
Fig. 6.2a Soil textural triangle for fine
soils from USDA (United States
Department of Agriculture) (from
Rawls et al., 1993)
Fig. 6.2b Soil textural triangle for fine
soils in Germany (DIN 4220, 2008,
German standard)
7
Particle size distribution
 Soil consists of a mixture of particles with different sizes
 The particle size distribution is used to characterise soils and can
be obtained by sieve analysis or sedimentation analysis
d10
d60
Fig. 6.3 Particle size distribution with characteristic grain
diameters d10 and d60
8
Porosity and density
VP
Fig. 6.4 Volume and mass relations in soils (Hillel, 2004)
9
Total porosity n (volume):
VP
Vt
n 
VP
 100
Vt
[%]
(6.1)
Volume of void space: water and air (L3)
Total volume: void space and solid matter (L3)
Dry bulk density ρb:
b 
Ms
Vt
[g/cm3 ]
(6.2)
Ratio of mass of solids Ms (soil dried at 105 °C) to total soil volume;
mineral soil: 1.25 gꞏcm-3 < ρb < 1.85 gꞏcm-3
Density of solids ρs:
s 
Ms
Vs
[g/cm3 ]
(6.3)
Ratio of mass of solids Ms and volume of solids Vs (without voids);
2.6 < ρs < 2.75 gꞏcm-3, mean value about 2.65 gꞏcm-3
10
Tab. 6.4 Typical values for porosity and density for
different soil textures (from Dyck & Peschke, 1995)
Soil
n
[%]
ρb
[g/cm3]
Sand
Loam
Silt
Clay
Organic
30…45
28…55
40…55
50…65
60…95
1.2…1.6
1.2…1.9
1.2…1.5
0.9…1.3
0.15…0.5
Soil horizons: layers with specific physical characteristics, which differ
from the layers above and beneath, located nearly parallel to the soil
surface
Soil types: describes whole soil profile; soils evolved at different places
by identical or similar evolution; have similar pedogenic properties and a
typical horizontal succession and characteristic
11
Fig. 6.5 Schematic representation
of a hypothetical soil profile (Hillel, 2004)
Fig. 6.6 Descriptive terminology for soil
profile horizons (Hillel, 2004)
12
6.2.2 Soil water properties
Water content ΘM
(by mass percent):
M 
Mt  Ms
M
 100  w  100
Ms
Ms
Mw
mass of water [kg]
Water content ΘV
(by volume percent):
V 
Vw
 100
Vt
Vw
Soil moisture Θ:
   V  h  0.01
s 
(6.4)
(6.5)
volume of water [cm3]
h
Degree of saturation s:
[%]
[%]
[mm]
(6.6)
soil depth [mm]
V
n
(6.7)
13
Types of soil water:
Seapage (gravitational) water:
Water which moves downward to the groundwater zone by gravity
force; recharges groundwater
Adsorption water:
Water, which is firmly bound at the surface of particles (by van-derWaals-forces, electrostatic forces)
Capillary water:
Water, which is lifted or kept by capillary forces (adhesion and
cohesion forces) in small pores
Rough estimation of
Capillary height hc:
hc 
0.3
d
[cm] (6.8)
d – diameter of “model” pore [cm]
14
6.2.3 Soil water potentials
Pressure below
groundwater table (GWT)
pL - atmospheric pressure
g - gravitational acceleration
Pressure at GWT :
Pressure in unsaturated
soil zone :
pW  pL   gh
(6.9)
ρ – density of water
h – depth below GWT
pW  pL
with h  0
pW  pL   gh  pL  
(6.10)
(6.11)
Water in unsaturated zone has negative pressure relative to
atmospheric pressure  suction Ψ
15
Potential Φ:
 General characterisation of energy of a water particle
 Work required, to move water from a reference point to a target
point in the soil (energy of length units)
Gravitational potential: Potential of a water particle at height z
above a reference level often GWT (positive sign)
Matrix potential: Potential of the attachment of water to the solid
matter (soil matrix) (negative sign); absolute value is equal to the soil
water suction ψ
Total potential in soil:
  z
p

z
g
g
[m]
(6.12)
Water movement always from location of higher potential to
location of lower potential!
16
6.2.4 Water retention curve
 Is very important for the
characterisation of
hydrological
properties of a soil!
 Also called pf-curves or
suction-saturationcurves
 pF-value: logarithm of
suction head measured
in cm or hPa
Fig. 6.7 Water retention curves
for sand, silt and clay ψ = f(wV)
FC
PWP
n
AWC
17
Characteristic values of water retention curves:
Field capacity FC [Vol.-% or mmꞏdm-1]:
Amount of soil moisture which is held against gravity force
 pF-values from 1.8 to 2.5 (≈ 60 to 320 hPa)
Permanent wilting point PWP [Vol.-% or mmꞏdm-1]:
Minimum soil moisture contents required, that plants don‘t welk;
 pF-value 4,2 (≈ 1,6ꞏ104 hPa).
Available water capacity AWC [Vol.-% or mmꞏdm-1]:
Amount of soil moisture that is available for plants
AWC  FC  PWP
[Vol-%]
(6.13)
18
Tab. 6.5 FC, PWP and AWC for different soil textures
Field capacity FC
(mmꞏdm-1)
Permanent
wilting point PWP
(mmꞏdm-1)
Available
water capacity
AWC (mmꞏdm-1)
Sand
13.5
3.5
10.0
Loamy sand
21.0
3.0
18.0
Sandy loam
25.5
4.5
21.0
Loam
36.0
10.5
25.5
Clay
40.0
18.0
22.0
Moor
74.0
40.0
34.0
Soil texture
19
6.2.5 Water supply for plants
Effective rooting depth:
 Plants take up water from soil through their roots
 Rooting depth depends on soil type, plants, growing
state and available water
 Effective rooting depth (Re) can be used to
calculate the plant available water or the water
available for transpiration in mm
 Different methods can be used to estimate Re
 As a first approximation, the soil depth in which 90
percent (by weight) of the roots lie, can be taken as
the effective root zone for irrigation purposes (FAO)
 Re (grassland) < Re (arable land) < Re (forest)
20
Re
Equal areas
Fig. 6.8 Determination of the
effective root zone Re
according to AD-HOC-AG
Boden (2005)
21
Tab. 6.6 Effective root zones for field cultures (AD-HOC-AG Boden, 2005, extract)
Effective root zone Re (dm)
Soil texture
(symbols)
gS, gSms, gSfs
Ss, mS, fS, mSgs, mSfs
Sl2, Su2, Su3, Su4
Sl3, St2
Sl4, St3,Slu
Ls2, Ls3, Ls4, Lt2, Lt3, Lts, Uu,
Us, Tu2, Tl, Tt
Uls, Ut2, Ut3, Ut4, Lu, Tu3, Tu4
Range of dry bulk density ρb (gꞏcm-3)
ρb1 – ρb2
ρb3
ρb4 – ρb5
7
8
9
10
13
5
6
7
8
9
5
6
6
7
8
13
10
8
14
11
9
In case of grassland 2 dm are to be withdrawn from the table value; for deciduous forest the
table values have to be multiplied by 1.5.
ρb dry bulk density (gꞏcm-3): ρb1: very low < 1.25, ρb2: low 1.25 – 1.45,
ρb3: medium 1.45 – 1.65, ρb4: high 1.65 – 1.85, ρb5: very high > 1.85
22
22
Available water in the root zone:
Available water capacity of the effective root zone AWCRe:
AWCRE  AWC  Re
[mm]
(6.14)
Plant available water AWP :
AWP  AWCRE  CR  t
[mm]
(6.15)
With CR  rate of capillary rise in mmꞏd-1 and ∆t  time interval
of capillary rise (d):
Can be taken from tables like Tab 6.7 depending on texture class
and distance to groundwater table
23
Tab. 6.7 Rate of capillary rise (AD-HOC-AG Boden, 2005, extract)
Soil
texture
(Symbols)
Rate of capillary rise CR (mmꞏd-1)
Distance between groundwater table and the bottom
of the effective root zone (dm)
2
3
10
11
12
13
14
15
17
20 25
Ss
>5 >5
5
3.5 2.2 1.1 0.6 0.3 0.2 0.1
–
–
–
–
–
–
–
–
Sl2
>5 >5 >5 3.1 1.7 1.0 0.6 0.4 0.3 0.2 0.1
–
–
–
–
–
–
–
Ls2
>5 >5 >5 2.6 1.6 1.3 1.0 0.7 0.4 0.2 0.1
–
–
–
–
–
–
–
Uu
>5 >5 >5 >5 >5 >5
Uls
>5 >5 >5 >5 4.6 3.2 2.3 1.7 1.3 1.0 0.8 0.6 0.5 0.4 0.3 0.2 0.1
–
4.9 2.0 1.2 0.8 0.5 0.4 0.3 0.2 0.2 0.1 0.1
–
Tt
1
4
5
6
7
>5
8
>5
9
>5 5.0 4.1 3.4 2.8 2.4 2.0 1.5 0.9 0.4
–
–
–
–
–
–
24
SubWat: 6.3 Groundwater
6.3.1 Groundwater occurrence
Different layers in underground water:
Aquifers: Saturated layers which are able to transmit and store
considerable quantities of gw, solids of the layer are sand, gravel,
weathered limestone or fractured sandstone
Aquitards: In comparison with aquifers they are of much lower
permeability and therefore transmit smaller quantities of gw
Aquifuges: Impermeable layers, unable to transmit or store gw
Aquicludes: Layers on the upper or lower boundary of a
groundwater system with very low permeability, unable to transmit
significant quantities of gw, barriers for gw flow
25
Unconfined groundwater:
Upper boundary of the aquifer is dependent on flow; water level in the gwobservation well is the very same as the water table in the aquifer
b
K
depth of the layer filled with gw (L)
hydraulic conductivity (L∙T−1)
Fig. 6.9 Unconfined groundwater
26
Confined groundwater
Upper boundary is independent on flow;
water level in the observation well is above
the aquifer boundary and below the ground
surface
Fig. 6.10 Confined groundwater
Artesian groundwater:
Upper boundary is independent on flow;
water level in the observation well is above
the ground surface (spring)
Fig. 6.11 Artesian groundwater
27
6.3.2 Groundwater recharge (GWR)
Groundwater recharge is seapage water volume which reaches
the groundwater table [mm/ ∆t, l s-1 km-2]
Fig. 6.12 Factors influencing groundwater recharge (from HAD, 2003)
GWR  P  ETR  RO  RI
[mm/Δt]
(6.16)
The main period for groundwater recharge in Central Europe is
the winter season, since in summer ETR > P
28
Fig. 6.13 Mean annual
groundwater recharge,
Period1961-90 Source:
Hydrological Map of
Germany, HAD, 2003
29
6.3.3 Momentum equation
Darcy’s law (1856): Specific discharge (Darcy’s velocity) q is
proportional to the hydraulic gradient
1D: q  -kf  Ihy
Q
h
 -k f 

Ages
l
q
(6.17)
Q
Ages
kf
Ihy
h
Δl
specific discharge, Darcy’s
velocity (L∙T−1)
discharge (L3∙T−1)
cross sectional area (L2)
hydraulic conductivity (L∙T−1)
hydraulic gradient (–)
hydraulic head (L)
length of flow (L)

3D: q  -kf  grad h
with
 h  h  h  
j
k  (6.18)
grad h   i 
y
z 
 x
Reference level
Fig. 6.14 Darcy’s experiment
30
Hydraulic conductivity
kf, (L1∙T−1):
 Saturated hydraulic
conductivity  GW-zone 
f(rock type)
 Unsaturated hydraulic
conductivity  unsaturated
zone  f(soil type, soil
moisture)  k(Θ)
Tab. 6.8 Saturated hydraulic conductivity
for different rock types (Busch et al., 1993)
Unconfined
rocks
Values of kf (m∙s−1)
Sandy gravel
3 ∙ 10−3 - 5 ∙ 10−4
Pebbly sand
1 ∙ 10−3 - 2 ∙ 10−4
Middle sand
4 ∙ 10−4 - 1 ∙ 10−4
Silty sand
2 ∙ 10−4 - 1 ∙ 10−5
Sandy silt
5 ∙ 10−5 - 1 ∙ 10−6
Clayey silt
5 ∙ 10−6 - 1 ∙ 10−8
Silty clay
≈ 10−8
Transmissivity T (L2∙T−1):
Ability of aquifer to transport water horizontally over depth m
T 
 kf dm
(6.18)
m
homogenious : T  k f  m
n
nonhomogenious : T   kf ,i  mi
(6.19)
i 1
31
Hydraulic head h (L):
Energy per unit weight of water in length units; equals its potential
h  z  hp  z 
h
z
hp
p
ρw
g
p
w  g
(6.19)
Hydraulic head (L)
Elevation head (L)
Pressure head (L)
Water pressure (M∙L∙T−2∙L−2)
Density of water (M∙L−3)
Acceleration due to gravity (L∙T−2)
Fig. 6.15
Definition of hydraulic head
32
32
Fig. 6.16 Hydraulic head in unconfined groundwater flowing towards a pumping well
33
Hydraulic gradient Ihy:
Total head loss over flow length for flow between two points
P1 and P2 (see Fig. 6.14)
Ihy 
h h1  h2

 sin  ! (  ) (6.20)
l
l
For small angels α: sin α = tan α
→ projection of l can be used
Ihy  tan  
h1  h2
x1  x2
(  ) (6.21)
Vertical flow without
impounding:
α = 90°  sinα = 1
 slope Ihy = 1
 q=-kf
Fig. 6.17 Sketch for hydraulic gradient
34
Pore water velocity:
 For the calculation of travel times e.g. considering chemicals
Darcy’s velocity q cannot be used, since it is related to the total
cross section area (rocks + voids); it is a fictive velocity
 For that we use the pore water velocity va
 It can be estimated using the porosity:
s q
va  
t n
(6.22)
s
t
q
n
flow distance (L)
travel time along s (T)
Darcy’s velocity (LꞏT-1)
porosity (-)
The pore water velocity can be estimated more
accurate using tracer experiments.
35
6.3.4 Principle of continuity
The change of flow through a representative elementary volume
(REV) is equal to the change in mass within the REV per time (mass
conservation)
Q
i
i

1 M

w t
(6.23)
or
M VW

 w
t
t
Qi
Fig. 6.18 Representative
elementary volume (REV)
Inflow or outflow of water in
coordinate direction i (L3ꞏT−1)
ρw Density of water (MꞏL−3)
ΔM Change of mass in the element (M)
Δt Time interval (T)
36
Storage in the aquifer:
Specific storage coefficient S0 [L−1]:
Volume of water ΔVw released from or taken into storage per unit
volume V of a porous medium per unit change in hydraulic head Δh
S0 
VW
V  h
(6.24)
Storage coefficient S [-]: Integral of S0 over depth m; volume of
water released from or taken into storage per unit surface area of
an aquifer per unit change in hydraulic head
S   S0
(6.25)
m
37
Storage coefficient for confined aquifers:
The storage coefficient S depends only on compressibility of
mineral grains and water in the aquifer
S  SE  W  g  m    n   
(6.26)
SE Elastic storage capacity (-)
α Compressibility of mineral grains (M−1ꞏLꞏT2)
β Compressibility of water (M−1ꞏLꞏT2)
m depth of aquifer (L)
n porosity (-)
Storage coefficient for unconfined aquifers:
It is the volume of water a aquifer will store or drain per unit area
and unit rise or decline of water table. Depends mainly on porosity.
S  n  SE
with SE  n 
Sn
(6.27)
38
Same drawdown by
depth s in both
aquifers
Fig. 6.19
Storage in unconfined and
confined groundwater
(after Langguth, Voigt, 2004,
changed)
Unconfined aquifer: Large quantity
from draining pores
Confined aquifer: Only small quantity
from elastic storage
39
SubWat: 6.4 References







AD-HOC-AG
Boden,
(2005):
Bodenkundliche
Kartieranleitung.
E.
Schweizerbart´sche Verlagsbuchhandlung, Stuttgart.
DIN 4220 (2008): Bodenkundliche Standortbeurteilung – Kennzeichnung und
Ableitung von Bodenkennwerten. Beuth Verlag, Berlin.
Dyck, S. und Peschke, G. (1995): Grundlagen der Hydrologie. Verl. f. Bauwesen,
Berlin.
HAD (2003): Hydrologischer Atlas von Deutschland. Bundesministerium für
Umwelt, Naturschutz und Reaktorsicherheit., Offenbach und Berlin.
Hillel, D. (2004): Introduction to environmental soil physics. Elsevier Academic
Press, San Diego, USA.
Langguth, H.-R., Voigt, R. (2004): Hydrogeologische Methoden. 2. Auflage,
Springer-Verlag, Berlin, Heidelberg.
Rawls, W.J., Ahuja, L.R., Brakensiek, D.L., Shirmohammadi, A., 1993. Infiltration
and soil water movement. In: D.R. Maidment (Editor), Handbook of hydrology.
MacGRAW-HILL, New York, pp. 5.1 - 5.51.
40
Chapter 7: Basics of modelling
7.1 Introduction
7.2 Phases of rainfall-runoff transformation
7.3 References
1
Modelling: 7.1 Introduction
What is a model?
Areal precipitation
Catchment
Runoff Q [m³/s]
Precipitation [mm/h]
 A model is an idealised description of reality or in fact of our
perception of reality.
 A model is valid only within certain bound of application i.e.
certain processes (e.g. floods), certain scales (point,
catchment), certain locations (e.g. climates), certain time, etc.
Time t [h]
Time t [h]
Fig 7.1: Classical objective of
hydrological modelling: estimation of
discharge hydrographs from
observed rainfall for catchments
(Rainfall-Runoff-Modelling)
2
Why model?
 Limitation of hydrological measurement techniques
 Extrapolation in space and time by modelling required
 Extrapolations for ungauged basins, where measurements
are not available
 Extrapolations into the future, where observations are not
possible
 Generally models are required for operational and strategic
decision making as well as research purposes.
3
Classification according to target variables:





Rainfall-runoff models  floods  chap. 8-11 !
Water balance models  water balance components
Ecohydrological models  nutrients, vegetation, etc.
Ground water models …
See before
Classification according to purpose:
 Forecast  real time forecast, operational model (e.g. floods)
 Prognosis (Simulation)  planning, design, impacts
 Research  formalisation and test of research hypotheses
4
Modelling: 7.2 Phases of rainfall runoff transform.
 Usually rainfall-runoff-modelling is divided into 3 phases :
I. Runoff generation (Chap. 8):
 Which part of precipitation becomes flood runoff ?
 Separation of precipitation in effective rainfall and losses.
II. Runoff concentration (Chap. 9,10):
 In which temporal distribution appears the effective rainfall at the
outlet of the catchment?
 Includes basically all vertical and lateral processes involving
transformation of surface runoff, interflow and base flow.
III. Flood routing (Chap. 11):
 How travels and changes a flood wave along a river course?
5
Runoff generation
Prediction
Runoff concentration
Flood routing
Prediction
Direct
runoff
Input
Input
Output
system
identification
Land phase
(vertical)
Catchment area
Output
system
identification
Land phase
(horizontal)
River bed phase
Stream
River basin
Fig. 7.7: Main phases of rainfall-runoff transformations considered in modelling (after
Dyck & Peschke, 1995)
6
Modelling: 7.3 References


Dyck, S. und Peschke, G. (1995): Grundlagen der Hydrologie. Berlin: Verl. f.
Bauwesen
Grayson, R. and Blöschl, G. (2001): Spatial patterns in catchment hydrology :
observations and modelling. Cambridge Univ. Pr., 404 pp.
7
Chapter 8: Models for runoff generation
8.1 Introduction
8.2 Runoff coefficient
8.3 SCS method
8.4 Φ-index method
8.5 Horton’s method
8.6 Comparison of methods
8.7 References
1
R generation: 8.1 Introduction
Objective:
 In a wider sense: Separation of precipitation into several flow
components RO, RI und RB (see Fig. 5.1)
 In a narrower sense: Which part of total rainfall P becomes direct
runoff (RD=RO+RI)? This part is called effective rainfall Peff (or
excess rainfall). The rest of P is called losses or abstractions.
Q
Total rainfall Ptot
Effective rainfall Peff,tot (blue)
Direct runoff RD
Separation of base flow RB
Fig. 8.1: Effective rainfall and
direct runoff for a rainfall runoff
event
Time t
2
Factors influencing runoff generation:
Climatic factors:
 Precipitation: form (rain, snow), intensity, duration, temporal and
spatial distribution  generation of RO, RI, RB
 Evapotranspiration: temperature, radiation, wind, air pressure,
humidity  generation of RB
Physiographic factors:
 Vegetation: type, age, density, etc.
 Topographical characteristics: elevation, slope, etc.
 Soil properties: soil moisture content, hydraulic conductivity,
porosity, groundwater table, etc.
3
Runoff generation process:
Generation of direct runoff depends on the infiltration dynamics,
which describes water entering the surface and soil water
dynamics involving retention and transport of water within the soils.
Methods:
1. Simple abstraction methods (e.g. runoff coefficient, Φ-index)
2. Empirical time dependent functions (e.g. Horton)
3. Physically based infiltration and soil water balance models
(Richard’s equation, Green-Ampt, etc.)
Governing equations:
Continuity (1-D):
Darcy (1-D):
v z  k   

z
(8.1a)
v z


 e  z, , t 
z
t
(8.1b)
4
R generation: 8.2 Runoff coefficient
Runoff coefficient Ψ is the ratio of the direct runoff volume to the
total rainfall volume  portion of excess rainfall:
 
VD Peff ,tot

VP
Ptot
VD
VP
Peff,tot
Ptot
0   1
(8.2)
- direct runoff volume [m3]
- total precipitation volume [m3]
- effective rainfall of the event [mm]
- total rainfall of the event [mm]
 Can be calculated using observed rainfall runoff events
comparing volumes of direct runoff and total rainfall
 Runoff coefficient depends on many factors as listed above
 So, for each event a different runoff coefficient is likely
5
Estimation of effective precipitation time series Peff(t):
1. Separation of base flow:
QD  ti   Q  t i   QB  ti 
(8.3)
QD, Q, QB – direct flow, total flow and base flow [m3/s]
k 1
VD  t  3600   QD (t i )
2. Direct flow volume:
t – time interval [h],
[m3 ]
(8.4)
i 1
3600 – dimensioning factor [s/h]
3. Rainfall volume:
AE – drainage area [km2]
Ptot – total rainfall sum [mm]
4. Runoff coefficient acc. to
Eq. (8.2) and effective
rainfall time series
VP  1000  AE  Ptot
[m3 ]
(8.5)
1000 – dimensioning factor
[m3/(km2mm)]
Peff (t i )    P (ti )
(8.6)
6
Some remarks for application of runoff coefficients:




Simple approaches assume time constant runoff coefficients
It makes more sense to differentiate runoff coefficients according to
soil type, rainfall volume, initial soil moisture conditions (e.g. Fig. 8.2)
If rainfall-runoff events are not available for estimation of the
coefficients often standard values are assumed (e.g. Tab. 8.1)
Another approach is regionalisation of runoff coefficients
Peff
Tab. 8.1: Runoff coefficients for urban
areas after Imhoff (1993)
Type of urban land
Dense development
Standard development
Open development
Suburbs with gardens
Open areas
Parks
Low initial
moisture
High initial
moisture
Ψ [-]
0.7 – 0.9
0.5 – 0.7
0.3 – 0.5
0.2 – 0.3
0.1 – 0.2
0.0 – 0.1
Ptot
Fig. 8.2: Dependence of effective rainfall
on rainfall depth and antecedent
moisture conditions
7
R generation: 8.3 SCS-method


Estimation of runoff coefficients ψ from physical basin characteristics
without direct analysis of rainfall-runoff events
Derived from large investigation of many events for catchments in the
middle west of the U.S.A.  regionalisation approach
Basic assumption
of SCS method:
Continuity:
Empirical estimation:
Peff
P
Ia
F
S
a
 
Peff ! F

P  Ia S
F  P  Ia  Peff
Ia  a  S
(8.7)
(8.8)
(8.9)
- effective rainfall sum of event [mm]
- total rainfall sum of event [mm]
- initial abstraction [mm]
- actual retention (loss) of water in the basin [mm]
- maximum retention (loss) of water in the basin [mm]
- portion of S for initial abstraction [-]
8
Principal equation derived
from Eq. (8.7-8.9):
P  a  S 

P  1  a   S
2
Peff
[mm] (8.10)
a = 0.2 according to original SCS-method, see e.g. Rawls et al. (1993)
a = 0.05 for German catchments according to Maniak (1997)
Relation between maximum
retention and basin properties:
CN






S
25400
 254
CN
[mm] (8.11)
- U.S. Soil Conservation curve number
0 < CN ≤ 100
As larger CN, as smaller max. basin retention S, as larger Peff
For CN = 100  S = 0  Peff = P
CN = f (soil type, land use, antecedent soil moisture conditions)
CN = 100 for impervious and water surfaces
CN < 100 for natural surfaces
9
Soil groups:
Group A
Very permeable soils e.g. deep sand, deep loess,
aggregated silts
Group B
Fairly permeable soils e.g. shallow loess, sandy loam
Group C
Soils with low permeability e.g. clay loams, shallow sandy
loam, soils low in organic content, soils usually high in clay
Group D
Soils with very low permeability e.g. soils that swell
significantly when wet, heavy plastic clays, certain saline
soils
10
Tab. 8.1: CN-values depending on soil group and land use for soil moisture
class II (selection *1)
Land use
CN for soil group
Uncultivated land (without vegetation)
Open spaces, lawns, parks, etc.
good condition: grass cover 75% or more
fair condition: grass cover 50% to 75%
Vine (terraces)
Cereals
Meadow (good condition)
Wood or forest land
thin stand, poor cover, no mulch
good cover
Industrial districts
Roads, parking lots, etc.
Impervious areas
*1
A
B
C
D
77
86
91
94
39
49
64
64
30
61
69
73
76
58
74
79
79
84
71
80
84
82
88
78
45
25
81
98
100
66
55
88
98
100
77
70
91
98
100
83
77
93
98
100
detailed tables see e.g. Rawles et al. (1993)
11
Tab. 8.2: Classification of antecedent moisture classes (AMC)
AMC group
I) dry conditions
II) normal conditions
III) wet conditions
Total five day antecedent rainfall [mm]
Growing season
Dormant season
< 30
30 – 50
> 50
< 15
15 – 30
> 30
 Usually only CN-values for normal antecedent moisture conditions
are tabulated (AMC II)
 For dry (AMC I) and wet conditions (AMC III) a transformation of
CN values according to the following equations is required :
CNI 
CNII
2.334  0.01334  CNII
CNIII 
CNII
0.4036  0.0059  CNII
(8.12)
(8.13)
12
Estimation of Peff(t) using SCS-method:
1. Assessment of catchment characteristics: a) soils, b) land use, c)
antecedent moisture conditions (Tab. 8.2)
2. Estimation of the CN-value (e.g. Tab. 8.1, Eq. 8.12, 8.13)
3. Estimation of effective rainfall for the event Peff (Eq. 8.11 + 8.10)
 
4. Estimation of the runoff coefficient:
Peff
P  Ia
(8.14)
5. Estimation of the effective rainfall time series Peff(t):
Peff (t i )    P (t i )
i
for  P (t i )  Ia
otherwise
j 1
Peff (t i )  0
(8.15)
For catchments made up of several land uses and soil types a
composite CN can be calculated weighting the different parts of the
basin according to their areas; or each part can be calculated
separately weighting finally the resulting effective rainfall.
13
P [mm/h]
Initial loss Ia
7
Effective rainfall Peff
6
Retention in basin P - Peff
5
4
3
2
1
1
2
3
4
5
6
7
Time [h]
Fig. 8.3: Scheme for effective rainfall estimation using the SCS method
Advantages SCS:
- Worldwide applications
- Only 1 parameter, the CN
values is required
- Simple application
Disadvantages SCS:
- Quite empirical
- Runoff coefficient independent of
rainfall intensity and soil moisture
- Transferability to conditions other
than middle west of U.S.A.
without calibration questionable
14
R generation: 8.4 Φ-index method
The -index is the constant rate of abstractions PV in mm/h
that will yield an effective rainfall hyetograph with a total depth
equal to the direct runoff of the catchment.
PV (t )  const .  

(8.16)
[mm/h] constant loss rate, Φ-Index
The effective rainfall is calculated for each time step as excess
rainfall greater then the Φ-index:
P (t )   for P (t i )  
Peff (t i )   i
for P (t i )  
0
(8.17)
15
Precipitation P [mm/h]
P(t)
Peff(t)
Φ- index
f(t)
∆t
Time t [h]
Fig. 8.4: Scheme for effective
rainfall estimation using Φindex method
 Estimation of the Φ-index such that the runoff coefficient of the
event is preserved  iterative approach (see Fig. 8.5)
 First the runoff coefficient Ψ needs to be calculated according
to the procedure outlined in chapter 8.2
16
Start
Calculation of the Φ-Index with
known runoff coefficient ψ
Initial value for Φ0
Φ0=0
Next iteration Φ0
Φ0=Φ1
1
n*
n
 n

   Peff (t i )     P (t i ) 
 i 1

i 1
P (t )   0
Peff (t i )   i
0
with
and
Test
P (t i )   0
yes
1   0 
no
Improved Φ-Index
Final value Φ
Φ=Φ1
Φ1=Φ0 ?
P (t i )   0
Stop
n *  number of intervals for
P (t i )   0
Fig. 8.5: Flow chart for iterative estimation of the Φ-index given a known runoff
coefficient
17
R generation: 8.5 Horton’s method
 Approach considers soil moisture accumulation during event
 Assumption: Direct runoff = surface runoff
 Exponential decline of infiltration capacity fv with time:
fV (t )  fc   f0  fc   e  kt
fv
f0
fc
k
(8.18)
- infiltration capacity (maximum rate) [mm/h]
- initial value (max) of infiltration capacity [mm/h]
- final value (min) of infiltration capacity [mm/h]
- parameter [-]
Calculation of actual infiltration rate f(t) and effective rainfall
Peff(t) depending on time and rainfall intensity P(t):
P (t ) if
f (t )  
fV (t ) if
P (t )  fV (t )
P (t )  fV (t )
(10.19)
Peff (t )  P (t )  f (t )
(8.20)
18
P, fV in mm/h
fo
Effective rainfall Peff
Current
infiltration f
Infiltration capacity fV
fc
Time in h
1
2
3
4
5
6
7
8
9 10 11
Fig. 8.6: Scheme for effective rainfall estimation using Horton’s method
19
R generation: 8.6 Comparison of methods
b) Φ
 – -Index
Index-- Verfahren
Method
a) Abfluss
Beiwert - -Verfahren
Runoff -coefficient
Method
f(t)
Peff(t)
∆t
Zeit
Timet [h]
t [h]
- Runoff coefficient or
SCS method
- Simple approach
- SCS considers
regionalisation without
using data from
observed events
Peff(t)
P(t)
Φ
f(t)
∆t
Time
Zeit t [h]
[h]
- Simple approach
- Can be used with and
without pre-estimation
of runoff coefficient
- The latter requires
calibration
Precipitation P
Niederschlag
P [mm/h]
[mm/h]
P(t)
Precipitation P
Niederschlag
P [mm/h]
[mm/h]
Precipitation P
Niederschlag
P [mm/h]
[mm/h]
P(t)
c) zeitl.
Verlust
Timevariabler
dependent
loss
fv(t)
Peff(t)
f(t)
∆t
Zeit t t[h]
Time
[h]
- E.g. Horton
- Simple approach
- Requires calibration of 3
parameters
- Considers time variant
infiltration capacity
Fig. 8.7: Schematic comparison of simple runoff generation methods
20
R generation: 8.7 References




Dyck, S. und Peschke, G. (1995): Grundlagen der Hydrologie. Berlin: Verl. f.
Bauwesen
Maniak, U. (1997): Hydrologie und Wasserwirtschaft. Springer, Berlin, 650 pp.
Imhoff, K.R. (1993): Taschenbuch der Stadtentwässerung. München, Wien:
Oldenbourg.
Rawls, W.J., Ahuja, L.R., Brakensiek, D.L. and Shirmohammadi, A., (1993):
Infiltration and soil water movement. In: D.R. Maidment (Editor), Handbook of
hydrology. MacGRAW-HILL, New York, pp. 5.1 - 5.51.
21
Chapter 9: Theory of hydrologic systems
9.1 Introduction
9.2 System characteristics
9.3 Unit hydrograph (UH)
9.4 Analysis and synthesis of the UH
9.5 Standard functions
9.6 References
1
Systems: 9.1 Introduction
Background:
 Rainfall runoff transformation in nature is a highly complex, nonlinear and dynamic process
 Can be described deterministically using first principles about
conservation of mass and energy
 Possible often only at the homogeneous micro scale (1cm – 10
m), at larger scales (catchments) simplifications are required
 One classical approach is the application of the systems theory
in hydrology using a black-box description of the processes
 This simplified approach is sufficient for many practical
applications where physical processes don’t need to be
considered directly (e.g. real time forecasts of floods)
 However, cannot be applied directly if systems characteristics
change (e.g. land use change)
2
Systems: 9.2 System characteristics
System definition:
 A system is a structure surrounded by a boundary, that accepts
inputs, operates on them internally and produces outputs
 The Inputs ui of the system are the causes for the effects which
are shown in the Outputs vi (Fig. 9.1)
 The system operator φ translates inputs into outputs
Fig. 9.1: A system having several inputs and outputs (input vector ui(t) and
output vector vi(t))
3
In the simplest case the system has only one input u und one output
v (Fig. 9.2). Then it is:
v (t )   u (t )
u(t)
v(t)

(9.1)
time dependent input,
time dependent output,
system operator (transfer function)
Fig. 9.2: System having one
input u(t) and one output
v(t)
Application of systems theory in hydrology for e.g.:
 Catchment: calculation of direct runoff from effective rainfall
 River section: calculation of flood routing
 Reservoir: calculation of retention
4
1. Dynamic system
behaviour:
Precipitation
Prec. P
For any time t with v(t)>0 the
output v(t) depends not only on
the input u(t) for that time t, but
also on the antecedent
hydrologic conditions:
Runoff QD
Runoff
v (t i )  f u (t i ), u (t i 1 ), , u (t i  n ) (9.2)
The input is stored temporarily.
The system shows persistence in
time. This feature applies for
catchments, river sections,
reservoirs etc..
Time
Fig. 9.3: Dynamical system: the duration
of precipitation TN is smaller than the
duration of the direct runoff TQD
5
2. Principle of proportionality:
If the input is multiplied by a constant C the new output can be
obtained by multiplying the original output with the same constant C:
 C  u (t )  C   u (t )
(9.3)
Precipitation
Prec. P
Prec. P
Precipitation
Runoff
Runoff QD
Runoff QD
Runoff
Time
Fig. 9.4: Principle of proportionality
Time
6
3. Principle of superposition:
Precipitation
Prec. P
The response of the sum of two input
signals is equal to the sum of the two
specific output signals:
 u1(t )  u2 (t )   u1(t )   u2 (t ) (9.4)
Runoff QD
Runoff
4. Principle of linearity:
Combination of superposition and
proportionality:
 C1  u1(t )  C2  u2 (t )  
C1   u1(t )  C2   u2 (t )
Time
(9.5)
Fig. 9.5: Principle of superposition
7
5. Principle of time
invariance:
 u (t - T )  v (t - T )
(9.6)
Runoff
Runoff QD
 The response behaviour of the
system is independent of the
time.
 If the input is shifted by the time
span T it results in an output
which is shifted by the same time
span T (without change in form).
Prec. P
Precipitation
Time
Fig. 9.6: Principle of time invariance
8
Summary of system characteristics:
1.
2.
3.
4.
5.
Dynamic behaviour
Principle of proportionality
Principle of superposition
Linearity: combination of 2. and 3.
Time invariance
 Characteristics are precondition for the modular calculation
of output signals (e.g. direct runoff) from input signals (e.g.
effective rainfall) using the systems theory in hydrology.
 For instance, hyetographs can be broken up into discrete
rainfall signals, for which specific runoff responses can be
calculated separately with subsequent superposition of
responses to obtain the total flood hydrograph.
9
Systems: 9.3 Unit hydrograph
Fig. 9.7: For the application of
the unit hydrograph a
catchment is considered as a
linear, dynamic, time invariant
system with effective rainfall as
input and direct runoff as output
 First a system identification is required i.e. the derivation of the
system operator φ  Analysis
 If the system operator φ is once known output can be calculated
for any input  Synthesis
 To transform effective rainfall into direct runoff the Unit
hydrograph (UH) is used as system operator
 Thus the UH method can be considered as simple runoff
concentration/ transformation model
10
Definition of the unit hydrograph (UH):
 The UH depends on the catchment
characteristics; each catchment has
it’s own specific UH
 The UH is used as system operator
for the transformation Peff  QD
 A more universal system operator is
the pulse response function (see
Chap. 9.5)
Prec. P
Precipitation
Runoff
Runoff QD
The unit hydrograph is the direct
runoff hydrograph of a watershed
resulting from 1 mm effective
rainfall occurring uniformly over
the drainage area at a constant
rate for a specific duration.
Time
Fig. 9.8: Unit hydrograph gE(∆t,ti)
11
Systems: 9.4 Analysis and synthesis
 The derivation of the unit hydrograph, i.e. the system
identification, is possible directly by analysing observed rainfallrunoff events (analysis).
 Since no physical laws are considered this way, this kind of
model is termed black-box-approach.
General procedure:
1)
2)
3)
4)
5)
Selection of rainfall-runoff event(s), definition of ∆t
Separation of base flow: Q  QD (see exercise)
Calculation of effective rainfall: P  Peff (see Chap. 8)
Analysis: derivation of the system operator φ from QD and Peff
Synthesis: prognosis of direct runoff from any given effective
rainfall QD = φ (Peff)
12
Synthesis: convolution operation
Calculation of the system response using the given system operator:
1. Multiplying of each input pulse P(i) by a weighing function G(j) (e.g.
unit hydrograph)
2. Superposition of the single pulse responses to obtain the total
response Q(i)
Discrete convolution using the unit hydrograph:
i
QD  t i    g E  t , t k   Peff  t i  k 1 
for i  1,..., o
(9.7)
k 1
Peff(ti)
gE(∆t,ti)
QD(ti)
o
Effective rainfall during interval ti-1 – ti [mm]
Unit hydrograph at time ti [m3·s-1·mm-1]
Direct runoff response at time ti [m3·s-1]
Number of ordinates for the direct runoff hydrograph
13
Example for discrete convolution equation (after Eq. 9.7):
Pi = Peff(ti)
Gi = gE(∆t,ti)
Qi = QD(ti)
Effective rainfall during interval ti-1 – ti [mm]
Unit hydrograph at time ti [m3·s-1·mm-1]
Direct runoff response at time ti [m3·s-1]
n=3 pulses of P
m=5 ordinates of G ≠ 0
 o=7 ordinates of Q ≠ 0
Q1 
Q2 
P1  G1
P1  G2
P2  G1
Q3 
P1  G3
P2  G2
P3  G1
Q4 
P1  G4
P2  G3
P3  G2
Q5 
Q6 
P1  G5
P2  G4
P2  G5
P3  G3
P3  G4
Q7 
o  m  n - 1 or
m  o  n 1
(9.8)
(9.9)
P3  G5
14
P
P2
P3
P1 t
t0
G
t1
Effective rainfall, n = 3
t
t2 t3
G2
G3
G4
G1
G
t
Unit hydrograph, m = 5
G5
G2
Unit hydrograph
G1
G
Unit hydrograph
Q
Q(ti) total response, o = 7
Response to P1
Response to P2
Response to P3
0
1
2
3
4
5
6
7
Fig. 9.9: Convolution principle
15
Zeit t
8
Analysis: system identification:
 Derivation of the system operator, i.e. here calculation of the unit
hydrograph ordinates G(i) from given input P(i) and given output
Q(i) using the convolution equation system
 Considering m + n - 1 equations with m unknowns the equation
system is (n-1)-times overdetermined
 The unknown G(i)’s can be estimated using the method of least
squares with the objective to minimise the squared differences
between calculated flows QD,cal and observed flows QD,obs:
m  n 1
 Q
i 1
D,cal
 ti   QD,obs  ti  
2
 Minimum!
(9.10)
 For details on solution of Eq. (9.10) see text books
 For 1 or 2 – times overdetermined systems it is possible to find
the solution manually by trying
16
For the sum of UH ordinates
the following should hold:
AE
gE(∆t,ti)
3.6
3.6  t
AE
m
 g  t, t   1
i 1
E
i
(9.11)
Catchment area [km2]
Unit hydrograph [m3·s-1·mm-1]
Dimensioning factor [mm·km2·s·h-1·m-3] with ∆t in [h]
 A more robust UH is obtained if several rainfall runoff events are
used for it’s derivation, e.g. by averaging the UH ordinates after
separate estimation considering the constraint in Eq. (9.11)
 If no observed rainfall runoff events are available, synthetic unit
hydrographs can be used, which are defined analytically based
on certain parameters like base time, lag time, time to peak,
peak discharge, etc. (e.g. triangular UH)
 These parameters of the synthetic UH can be estimated using
regionalisation from observed basins relating the parameters to
physiographic catchment characteristics.
17
Systems: 9.5 Standard functions
 It is useful to define different standard input signals and
corresponding standard response functions, for a more
generalised description of the system behaviour
 This allows especially an analytical derivation of the response
(e.g. a synthetic unit hydrograph) for simple conceptual
hydrological models (see Chap. 10 and 11)
Table 9.1: Standard input signals and response functions
Input
Output
1
Unit step input ε(t)
Step response function h(t)
(S-hydrograph, S-curve)
2
Unit pulse input u(t)
(rectangular pulse)
Pulse response function g(∆t,t)
(corresponding to UH: gE(∆t,t)
3
Unit impulse input δ(t)
(Dirac function, delta function)
Impulse response function g(t)
(corresponds to Instantaneous UH)
18
Unit step input ε(t) [-]:
 0 for t  0
 1 for t  0
 (t )  
(9.12)
Input u(t)
Corresponds to switching on the
input at time t = 0
Step response function h(t) [-]:
  (t )  h(t )
with h(t )  0 for t  0
(9.13)
System responds with attenuated
rising of the output to the step input
Output v(t)
19
Unit pulse input [1/∆t]:
u (t ) 
1
 (t )   (t  t )
t
(9.14)
Input u(t)
• Superposition of two unit step inputs
shifted by ∆t having opposite signs
• Volume = 1, Intensity = 1/∆t
Pulse response function g(∆t,t) [1/∆t]:
g  t , t i  
1
 h(t )  h(t - t )
t
(9.15)
• Superposition of two step response
functions shifted by ∆t having
opposite signs
• Volume = 1
• Corresponds to the unit hydrograph
Output v(t)
20
Unit impulse input δ(t) [1/∆t]:
0 for

 (t )   for
0 for

t 0
t 0
t 0
(9.16)
Input u(t)
For ∆t -> 0 the unit pulse input
becomes the unit impulse input with
Volume = 1 and Intensity = ∞
Impulse response function g(0,t)= g(t) [1/∆t] :
d
h(t )
dt
g (0, t ) 
(9.17)
t
h(t )   g (0, t ) dt
Output v(t)
0
Can be obtained by the derivation of the
step response function.
Corresponds to the instantaneous unit
hydrograph IUH with Volume = 1
21
Discrete convolution using pulse response function g(∆t,t):
i
v  t i    g  t , t k   u  t i - k 1  t
(9.18)
k 1
v(ti)
g(∆t,ti)
- Output [mm·h-1],
u(ti)
- pulse response fnc. [h-1] ∆t
- Input pulse [mm·h-1]
- time interval [h]
Transformation between UH and pulse response function:
g E ( t , t i )  g ( t , t i ) 
3.6
gE(∆t,ti)
AE
3.6
(9.19)
- Dimensioning factor [mm·km2·s·h-1·m-3]
- unit hydrograph [m3·s-1·mm-1]
AE - drainage area [km2]
Analytical convolution using impulse response function g(t):
t
v (t )   g (t   )  u( ) d
(9.20)
0
System hydrological methods can be used for modelling of runoff
concentration/ transformation and flood routing!
22
Systems: 9.6 References




Chow, V.T., D.R. Maidment & L.W. Mais (1988): Applied Hydrology. McGraw-Hill,
Available online at http://www.knovel.com/knovel2/Toc.jsp?BookID=136.
Dooge, J.C.I. (2003): Linear theory of hydrologic systems. EGU reprint series 1.
EGU, Katlenburg-Lindau.
Dyck, S. und Peschke, G. (1995): Grundlagen der Hydrologie. Berlin: Verl. f.
Bauwesen.
Dyck, S. (1980): Angewandte Hydrologie, Teil 2. VEB Verlag für Bauwesen,
Berlin, 544 pp.
23
Chapter 10: Conceptual Models
10.1 Introduction
10.2 Translation
10.2.1 Linear translation
10.2.2 Time of concentration
10.2.3 Rational method
10.3 Retention
10.3.1 Linear reservoir
10.3.2 Linear reservoirs in series
10.4 References
1
Conceptual models: 10.1 Introduction
 Simulation of complex hydrological processes using simple
conceptual models or model concepts; first step to consider
physical properties in a simple way (no black box anymore)
 The conceptual models considered here can be used for
modelling runoff transformation and flood routing
 Two basic processes in catchments and rivers are considered:
1) Translation  output is obtained by shifting the input in time
without changing it in form (Chap. 10.2)
2) Retention  output is obtained by changing the input in form
without offset in time“ (Chap. 10.3)
 In the following the basic modules of conceptual models are
presented; in hydrological catchment models several modules
need to be combined to simulate the rainfall-runoff processes
2
Translation: 10.2.1 Linear Translation
 Translation is a pure temporal shift of the signal (e.g. the
hydrograph) in time without change in form
 The input signal u(t) occurs again as output signal v(t) only
shifted by the translation time Tt
 If the translation time Tt is constant, independent of the input
signal and time invariant, then it is called linear translation
v (t )  u(t - Tt )
u(t)
v(t)
Tt
[div.]
[div.]
[h]
(10.1)
time dependent input
time dependent output
translation time
3
time t
Fig. 10.1: Input and output of a system with linear translation
The output v at time t is equal to the input u at time t-T
4
Linear Channel:
Most simple fictitious flow routing model, where flow velocity v is
constant in time (steady flow) and space (uniform flow)
The flow time TF is the time, which is required for a water particle
to travel a certain distance LF  for the linear channel the flow
time is assumed equal to the translation time Tt :
TF  Tt 
1 LF

3.6 v
(12.3)
Qout (t )  Qin  t  Tt 
-
(12.4)
LF
v
3.6
[km]
travel distance
[m/s]
flow velocity
[km∙s∙m-1∙h-1] dim. factor
Qin,
Qout
[m3/s]
inflow, outflow
Most simple approximation of reality
Retention is not considered
Can be applied only as sub-module within a hydrological model
5
Translation: 10.2.2 Time of concentration
 The time of concentration Tc is the maximum flow time of the
direct runoff in a catchment
 It is the time a water particle needs to travel from the farthest
point on the watershed to the outlet of the catchment
Time of concentration Tc after Kirpich:
 L 
Tc  0.06625  
 I
Tc
hF
LF
L
I
[h]
[m]
[km]
[km]
[1]
0,77
with
I
hF
LF  103
(10.5)
time of concentration
specific elevation difference
specific flow distance
real flow distance from outlet to watershed devide
average slope for maximum elongation of the catchment
For the estimation of hF, LF and I see Fig. 10.2!
6
hF
LF
Lmax
[m]
[km]
[km]
specific elevation difference
specific flow distance
maximum elongation of the catchment
watershed
outlet
Lmax
elevation
watershed
Elevation at
watershed
I
hF
LF  103
for A1  A2
outlet
Elevation at
outlet
LF
Lmax
distance
Fig. 10.2: Estimation of the average slope for calculation of the time of
concentration by area balancing at longitudinal section of the catchment
7
Translation: 10.2.3 Rational method (Flutplan)




Flow transformation on the surface of a plane rectangular area
Homogeneous areal effective rainfall with constant intensity
Constant flow velocity on the surface in space and time
Often applied for sewer design in urban hydrology
Flow contributing area:
Effective
areal
Effektiver
Gebietsniederschlag
rainfall
t

 AE
At (t )   Tc
A
 E
I
(T)
AA
t(Ttt)
AE
Einzugsgebiet AE
X=v
mꞏ
Tt
Q(t)
Q
At(t)
AE
Vm
[km2]
[km2]
[mꞏs-1]
für 0  t  Tc
(10.6)
für t  Tc
contributing area
catchment area
flow velocity
Fig. 10.3: Scheme of the rational method (Flutplan) or simplified time-area
method with parallel isochrones (lines of constant travel time)
8
The rational method can be seen in the framework of system theory:
 Input  effective precipitation as unit step input ε(t)
 Output  direct runoff as step response function h(t)
Time [h]
Contributing area
Runoff
Direct runoff QD
Effective Prec. P
Precipitation
Fig. 10.4: Input and
output of the rational
method (Flutplan)
Time [h]
t
At (t ) 
h(t ) 
 Tc
AE
1

Step response function of
the rational method h(t) [-]
Pulse response function of
the rational method g(∆t,t) [h-1]
g  t , t i  
for
0  t  Tc
for
t  Tc
(10.7)
1
 h(ti )  h(ti - t )
t
(10.8)
9
g(Δt,t)
g(Δt,t)
Tc
∆t
∆t>Tc
g(Δt,t)
Tc
∆t=Tc
∆t
Tc
∆t<Tc
∆t
1//∆t
1/∆t
1/T
/Tc
T1
t0
T2
T3
t
t0
T1
t
T3
T1
t0
T2
t
T3
∆t > Tc
∆t = Tc
∆t < Tc
Tc
Tc
∆t
T2
∆t - Tc
0
Tc - ∆t
TTges
tot
∆t + Tc
2Tc = 2∆t
Tc + ∆t
h(t0)=t0/Tc
1
1
∆t/Tc
h(t0-∆t)
0
0
0
g(∆t,t0)
1/∆t
1/∆t
1/Tc
T1 = T3
see Eq. 10.7
see Eq. 10.8
Fig. 10.5: Characteristic shapes and properties of the pulse response function
Resulting peak (see Eq. 9.18): 𝑄
“Rational Formula” with runoff
coefficient C included 0≤C≤1
𝑄
𝑡
𝑃
⋅ Δ𝑡 ⋅ 𝑔 Δ𝑡, 𝑡
𝑡
𝑃
⋅ Δ𝑡 ⋅ 𝑔 Δ𝑡, 𝑡
mm ⋅ ℎ
·𝐶
mm ⋅ ℎ
10
Retention: 10.3.1 Linear reservoir
Most simple retention model is a linear reservoir, for which the
output is proportional to the current reservoir storage:
Linear reservoir:
Qout 
dS
 Qin  Qout
dt
Continuity:
S
Qout
k
Qin
Qout
1
S
k
 S  k  Qout
 Qin  Qout 
[m3]
[m3/s]
[s]
[m3/s]
dS
dt
(10.9)
(10.10)
reservoir storage
outflow
storage constant
inflow
Fig. 10.6: Scheme of a linear reservoir
11
Substituting S in Eq. (10.10) by Eq. (10.9) yields :
Differential equation of the linear reservoir:
Qin  Qout  k 
dQout
dt
(10.11)
General solution of linear reservoir:
t
1
Qout (t )  Qout (t0 )  e   t  t0  / k   Qin ( )   e   t   / k d
k
t0
Drainage term:
Pure drainage of the reservoir,
starting from time t0, when inflow
has stopped and outflow is Qout(t0)
Impulse response function
(response term):
(10.12)
Response term:
Outflow response to certain inflow Qin(t),
corresponds to the analytical convolution
Eq. (11.20)
g (t ) 
1 t / k
e
k
(10.13)
12
Outflow as response to step input (from Eq. 10.12):
Qout (t ) 
Qin,const
k
t
 e  t / k  e / k d  Qin,const  1  e  t / k 
(10.14)
0
with step response function:
h(t )  1  e  t / k
(10.15)
Outflow as response to pulse input (Can only be given in two parts):
Qout (1)  Qin,const ( t )  1  e t / k 
Rising limb of the hydrograph
(from Eq. 10.14):
Qout (t )  Qout (1)  e ( t t ) / k
Falling limb of the hydrograph
(1st term from Eq. 10.12):
(10.16)
(10.17)
13
Q(t)
Qout(1)
Qin,const(Δt)
0
t
Fig. 10.7: Outflow of a linear
reservoir as response to one input
pulse
2
1
t
3
Q(t)
5
Qin,2
k=1
4
3
Qin,1
2
Fig. 10.8: Outflow of a linear
reservoir as response to two input
pulses
1
0
0
t
1
2
3
t
14
As alternative to using the differential equation of the linear
reservoir the method of differences can be applied:
Discretisation of reservoir differential equation (Eq. 10.11) into:
1
1
k
Qin (ti 1)  Qin (ti )  Qout (ti 1)  Qout (ti )  Qout (ti )  Qout (ti 1)
t
2
2
(10.18)
Working equation of the linear reservoir with differences:
Qout (t i )  C1  Qin (t i )  Qin (t i 1 )   C2  Qout (t i 1 )
with
C1 
t / 2
k  t / 2
and
C2 
k  t / 2
k  t / 2
(10.19)
(10.20)
where Qin and Qout are given as ordinates
To apply this method of differences
the following condition is required:
t  k
15
Estimation of parameter k for the linear reservoir :
1. Calibration  linear reservoir is often used as module within
rainfall-runoff models for different processes and for several
spatial units; here usually calibration of k is applied with the
objective to fit simulated to observed flows
2. Storage – discharge – relation  k = ΔS/ΔQA  may be
used for flood control reservoirs; seldom applied for runoff
transformation of natural catchments since storage –
discharge – relation is usually not known
3. Hydrograph separation in runoff components  for small
catchments assuming the considered runoff component of the
total catchment can be represented be a linear reservoir
4. Regionalisation  analysing of many observed catchments
and assessing the retention constant k a transfer to ungauged
catchments is possible using the relation of k to physical
catchment characteristics
16
Retention: 10.3.2 Linear reservoirs in series
Qin
1. lin. reservoir
time t [h]
Qout,1=Qin,2
2. lin. reservoir
time t [h]
Qout,2=Qin,3
3. lin. reservoir
time t [h]
Qout,3
Qin,n
n. lin. reservoir
time t [h]
Qout
Fig. 10.9: A watershed represented by a series of n identical linear reservoirs each having
the same storage constant k (Nash cascade); scheme of reservoirs on the left, impulse
17
response functions for each reservoir on the right
 Routing an inflow volume through the n linear reservoirs leads
to the impulse response function
Impulse response
function:
Generalisation for
positive real n :
1
g(0,t) =
k  (n -1)!
g(0,t) =
t 
 
k 
1
t 
 
k  (n )  k 
n 1
 et / k
(10.21)
n 1
 e t / k
(10.22)
Pulse response function g(∆t,t) from Eq. (10.21) :

ti t
n
k i   t i  t 
e k
g ( t , t i )  n

(n  1)!
k  t i 1
-
n 1
ti
n
ek
k i  t in 1
 n

k  t i 1 (n  1)!
(10.23)
Derivation of g(∆t,t) from Eq. (10.22) is not possible
Instead approximation of g(∆t,t) by g(0,t) using ordinates (see Chap. 11)
Parameter estimation n and k similar as for linear reservoir
In addition method of moments for parameter estimation can be applied
(see Chow et al., p. 261)
18
Conceptual models: 10.4 References




Chow, V.T., D.R. Maidment & L.W. Mais (1988): Applied Hydrology. McGraw-Hill.
Dooge, J.C.I. (2003): Linear theory of hydrologic systems. EGU reprint series 1.
EGU, Katlenburg-Lindau.
Dyck, S. und Peschke, G. (1995): Grundlagen der Hydrologie. Berlin: Verl. f.
Bauwesen
Dyck, S. (1980): Angewandte Hydrologie, Teil 2. VEB Verlag für Bauwesen,
Berlin, 544 pp.
19
Chapter 11: Models for flood routing
11.1 Introduction
11.2 Principle of flood movement
11.3 Simple methods for flood forecasting
11.3.1 Peak rating curve
11.3.2 Travel time curve
11.4 Hydrological flood routing
11.4.1 Muskingum
11.4.2 Kalinin-Miljukov
11.5 Hydraulic flood routing – overview
11.6 References
1
Flood routing: 11.1 Introduction
Objectives:
 How do the flood waves, which have been generated in
catchments, move and changes within the channel network?
 Prognosis of flood waves in rivers for points downstream from
known flood waves upstream.
Gauge A
River
Q
[m3/s]
Gauge B
A, given
B, forecast
Zeit t
Fig 11.1: Flood wave at two gauges at a river
2
Flood routing: 11.2 Principle
 Redistribution of a flood wave when it travels downstream; it
becomes longer and more smooth  „River retention“
 The peak decreases, since the wave front travels faster than
the tail of the wave because of its larger slope
Water level [m]
Wave back
(smaller slope)
Decrease of wave
peak
Wave front
(larger slope)
Larger
distance
Smaller
distance
Travel distance [km]
Fig. 11.2: Principle of wave movement downstream a river reach
3
 Steady flood flow  invariable stage-discharge relationship 
water surface slope ISp = channel bottom slope ISo
 Unsteady flood flow  variable stage-discharge relationship 
different slope for front and back of wave
Wave back
Wave front
Fig. 11.3: Water surface slope (ISp), bed slope (Iso), steady flow velocity (vstat)
and unsteady flow velocity (vinst) at rising and falling limbs of the flood wave
4
For the same water level h  there are different flow velocities v 
and thus different flows Q  that leads to a variable stagedischarge relationship  looped rating curve
h(t)
h(t)
Falling limb of
the flood wave
hmax
Invariable curve
(steady flow)
Qmax
h
Rising limb of
the flood wave
Q
t1
t2
t
Q(t2)
Q(t)
Q(t1)
Fig. 11.4: Stage-discharge-relationship of a gauge for the passage of a flood wave
5
Flood routing: 11.3 Simple methods
For flood forecasting the most important information is:
a) Maximum water level of flood peak Wmax
b) Arrival time of flood peak t(Wmax)




The simple (minimal) methods focus on the prognosis of these
two variables and look for relations between peak water levels
of different cross sections of a river
Then based on observations at upstream cross sections a
simple forecast for downstream cross sections is possible
Required is an empirical analysis of many historic flood events
to establish the relationships
These approaches are very simple in handling but not very
accurate (pure empirical approaches, don’t consider changes
in river and flood plain, don’t consider tributaries, etc.)
6
Methods:
1) Peak rating curve  Wmax,B=f(Wmax,A)
2) Travel time curve  tA,B=f(Wmax,A)
Preconditions:
 Considering only cross sections at the same river with a
reasonable small distance
 Since for unsteady flood flows the rating curve is looped, the
method is restricted to the peak water levels Wmax
 If no clear relationship between peak water levels can be
derived tributary inflow might be considered as a reason
Catchment boundary
Water level WA [cm]
7
Time t [h]
HW at C
NW at C
Water level WA [cm]
Water level WB [cm]
Water level WB [cm]
MW at C
Time t [h]
Fig. 11.5: Analysis of flood waves at two gauges for the derivation of a Peak
rating curve Wmax,B = f (Wmax,A)
8
Water level WA [cm]
Channel geometry between A and B
Travel time t [h]
Water level WA [cm]
Fig. 11.6: Travel time curve tA-B=f(Wmax,A) and channel geometry
Peak rating curve Water level WB(WA)
Travel time curve T(WA)
Fig.11.7: Using peak rating
curve and travel time curve
for forecasting magnitude
and arrival time of the peak
water level downstream
Water level WB [cm]
Travel time t [h]
9
Flood routing: 11.4 Hydrological flood routing
Introduction:
a) Simple methods are often inaccurate
b) Hydraulic methods are often too demanding considering
operational flood forecasting and large river basins
 Alternative are the hydrologic model concepts
Hydrologic flood routing models use the continuity equation in the
following form:
dS
 Qin  Qout
dt
(11.1)
As replacement for the momentum equation an empirical
relationship of the following form is used:
Qout (t )  f  S,Qin 
(11.2)
10
Flood routing: 11.4.1 Muskingum method
Simulates storage volume of flooding in a river reach by
combination of a wedge storage for the unsteady flow part and a
prism storage for the steady flow part
Wedge storage S2=k·x·(Qin-Qout)
Qin
Qout
Qstat= Qout
Channel reach
Prism storage S1=k·Qout
Fig. 11.8: Separation of storage volume of flooding into prism and wedge
storage for application with the Muskingum method
11
Prism storage S1:
Steady discharge part Qin = Qout assuming:
S1
K1
Qout
[m3]
[s]
[m3/s]
S1  k1  Qout
(11.3)
contents of the prism storage
retention constant of the prism storage
outflow of the channel reach
Wedge storage S2:
Non-steady discharge part with Qin > Qout for flood advancing and Qin <
Qout for flood recession:
S2  k2  Qin  Qout 
S2
K2
Qin
[m3]
[s]
[m3/s]
(11.4)
contents of the wedge storage
retention constant of the wedge storage
inflow into the channel reach
12
Total storage:
S  S1  S2  K1  Qout  K 2  Qin  Qout 
(11.5)
Classical notation of the Muskingum model:
S  k   x  Qin  1  x   Qout 
k
x
[s]
[1]
(11.6)
retention constant (time of travel through the channel reach)
weighing factor
Range: 0  x  0.5
Range in natural streams:
0  x  0.3
Special cases: x = 0  linear reservoir; x = 0.5  linear translation
Differential equation with Eq. (11.6) in Eq. (11.1):
Qout  k 1  x 
dQout
dQin
 Qin  k  x
dt
dt
(11.7)
13
Solution of the differential equation:
t
t -
Qout (t 0 ) - t
1
x
Qout (t ) 
Q
(
)



 Q in(t )
k
x )  d (1e k (1- x ) 
e
in
2 
1- x
1- x
k (1- x ) t 0
(11.8)
In practice the Muskingum model is applied in discrete form:
Qout (t i )  C1  Qin (t i )  C2  Qin (t i 1 )  C3  Qout (t i 1 )
C1 
 k  x  t / 2
,
k  1  x   t / 2
Test:
C2 
(11.9)
with
k  x  t / 2
k  k  x  t / 2
, C3 
k  1  x   t / 2
k  1  x   t / 2
(11.10)
C1  C2  C3  1!
Where Qin and Qout are given as discrete values
Precondition for discrete application:
t  k
14
Parameter estimation methods:
I. If Muskingum flood routing is part of a more complex rainfall runoff
model often x and k are considered as calibration parameters
II. An alternative procedure for determining the parameters x and k from
channel geometry and flow rate is the Muskingum Cunge method see
text books
III. If observed inflow and outflow hydrographs are available the
parameters x and k can be estimated using a semi-graphical method
Semi-graphical method:
Calculation of “weighted discharge” Q* from Eq. (11.6):
S  t i   k   x  Qin (t i )  1  x   Qout (t i )
S  k  Q*
(11.11)
with Q * called here weighted discharge
15
Calculation of storage from continuity equation: ∆S/∆t=Qin-Qout:
S(t i )  S(t i -1)  0.5  t Q in(t i )  Q in(t i -1)  Q out (t i )  Q out (t i -1)
(11.12)
Equalise Eq. (11.11) and (11.12) allows estimation of k:
k



S S  t i 1   0.5  t  Q in(t i )  Q in(t i -1) - Qout (t i )  Q out (t i -1)

Q*
 x  Qin (ti )  1 x   Qout (ti )
(11.13)
The computed values of S and Q* are plotted for given inflow and
outflow hydrographs of a flood event in chronological order
This usually gives a looped storage discharge function S=f(Q*)
Plotting is repeated for of several different x
16
Storage content S [m3]
Storage content S [m3]
Weighted discharge Q* [m3/s]
Weighted discharge Q* [m3/s]
Weighted discharge Q* [m3/s]
The value of x which produces a loop closest to a single line is taken to
be optimal for the reach (in Fig. 11.9 for x=0.2)
The parameter k equals then the slope of the respective line
For analytical derivation correlation and regression can be used
Alternatively k can be derived from the flood peak travel time in the reach
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Flood routing: 11.4.2 Kalinin-Miljukov method
 Application of the model concept of linear reservoirs in series
(see Chap. 10.3.2 ) for flood routing on river reaches
 Parameters can be estimated from observed inflow and outflow
or from channel geometry (see textbooks)
River reach
Inflow Qin
Outflow Qout
Outflow Qout



Fig. 11.9: Relationship between
storage S(t) and weighted
discharge Q*(t) for different
weighting factors x
Inflow Qin

Storage content S [m3]
Storage content S [m3]
Weighted discharge Q* [m3/s]
Time t
Linear reservoirs
in series
Time t
Qout
Fig. 11.10: Flood routing using linear reservoirs in series
18
Calculation option (1):
Replace input hydrograph by sequence of rectangular pulses
Apply discrete convolution with Eq. (9.18) using pulse response function
from Eq. (10.23) (no fractional number n of reservoirs possible)


Calculation option (2):


Replace input hydrograph by sequence of impulses with distance of ∆t
Approximate pulse response function by multiplying impulse response
function Gl. (10.22) with ∆t:
t
t
 
g ( t,t) = t  g (0, t ) 
k  (n )  k 
*
n 1
 e t / k
[1]
(11.14)
 Then apply discrete convolution with approximate pulse response function:
i
Qout  t i    g *  t , t k   Qin  t i - k 
[m3 / s]
(11.15)
k 0
19
Qin(t)
Fig. 11.11: Replacement options for the
inflow hydrograph
(1) Sequence of rectangular pulses
(2) Sequence of impulses
(2)
(1)
1
2 3
4 5 6
7 8
9 10
t
20
Flood routing: 11.5 Hydraulic flood routing - overview
 Dynamic wave propagation in open channels is a unsteady (δv/dt ≠
0), non-uniform (δv/dx ≠ 0) complex flow process
 Can be described using basic physical principles for conservation
of mass, energy and momentum using distributed parameters
 For flood routing often the one-dimensional description is sufficient
 The respective hydraulic model are the „Saint-Venant-Equations“
 They have various simplified forms as indicated in Eq. 11.16
 For the solution of those equations usually numerical techniques
are required (e.g. finite differences; see module environmental
hydraulics)
 Here, the basic hydraulic method are just mentioned and not
discussed any further
21
Energy
grade line
Energiehorizont
1
2
Energy
Energielinie
v2
2g
h
I E  dx
Friction
Reibung
1 v
 dx
g t
lokale
Locale
Beschleunigung
v2 v  v

 dx
2g g  x
konvektive
Convective
Beschleunigung
acceleration
acceleration
h
dx
x
z
z
 dx
x
h
z
Datum line
Bezugshorizont
dx
Water
Wasserspiegelgefälle
surface slope
SohlChannel
gefälle
bottom slope
Energy:
1 v
g t
locale
Fig. 11.12: Onedimensional unsteady,
non uniform flood flow in
a river reach
Continuity
v v

g x
convective

h
x
IW

z
x
IS

hl
x

0
IE
Q
x

A
t
0
(11.16)
Hydrological methods
acceleration
Kinematic wave
Diffusion wave
Dynamic wave (complete Saint-Venant)
22
Flood routing: 11.6 References


Chow, V.T., D.R. Maidment & L.W. Mais (1988): Applied Hydrology. McGraw-Hill.
Dyck, S. und Peschke, G. (1995): Grundlagen der Hydrologie. Berlin: Verl. f.
Bauwesen
23