Hydrology 3

UNIVERSITÄT STUTTGART INSTITUTE OF HYDRAULIC ENGINEERING CHAIR OF HYDROLOGY AND GEOHYDROLOGY

Prof. Dr. rer. nat. Dr.-Ing. András Bárdossy

Hydrology III October 2003

Institute of Hydraulic Engineering, Universität Stuttgart, Germany Pfaffenwaldring 61 * D-70550 Stuttgart Phone: 0711/685-4679 * Fax: 0711/685-4681 * e-mail: [email protected]

UNIVERSITÄT STUTTGART INSTITUTE OF HYDRAULIC ENGINEERING CHAIR OF HYDROLOGY AND GEOHYDROLOGY

Prof. Dr. rer. nat. Dr.-Ing. András Bárdossy

Hydrology III - I

Table of contents 1

FUNDAMENTAL PRINCIPLES OF RIVER BASIN MODELING .......................... 1 1.1

Scope ................................................................................................................................................ 1

1.2 Principal methods of river basin modeling ..................................................................................... 2 1.2.1 Statistical methods ..................................................................................................................... 2 1.2.2 Deterministic methods ............................................................................................................... 2 1.2.3 Combined methods..................................................................................................................... 3 1.3 Structure of river basin models ....................................................................................................... 3 1.3.1 Partial models ............................................................................................................................ 3 1.3.2 Drainage basin models ............................................................................................................... 4 1.3.3 Streamflow models..................................................................................................................... 6 1.3.4 Complex river basin model......................................................................................................... 8 1.4

Model approaches............................................................................................................................ 8

1.5 Analysis and synthesis ................................................................................................................... 11 1.5.1 Calibration of the model against observed in- and output values............................................... 11 1.5.2 Synthetic streamflow hydrographs............................................................................................ 11

2

3

STRUCTURE OF DRAINAGE BASIN MODELS................................................ 13 2.1

Formation of runoff and runoff concentration.............................................................................. 13

2.2

Formation of outflow and runoff concentration in simple drainage basin models....................... 15

2.3

Base flow ........................................................................................................................................ 16

MODELS OF RUNOFF FORMATION ................................................................ 17 3.1 Runoff coefficient........................................................................................................................... 17 3.1.1 Overall runoff coefficient for single rainfall-runoff events ........................................................ 17 3.1.2 Antecedent precipitation index and coaxial graphical plot ........................................................ 18 3.1.3 The SCS approach .......................................................................................................................... 21 3.2 Models to compute effective rainfall ............................................................................................. 24 3.2.1 Model requirements ................................................................................................................. 24 3.2.2 Runoff coefficient method............................................................................................................... 26 3.2.3 Index approaches, Φ-index............................................................................................................. 26

4

BASIS AND METHODS OF SYSTEMS HYDROLOGY ..................................... 28 4.1 Definition of system properties ........................................................................................................... 28 4.2

Unit hydrograph ............................................................................................................................ 31

4.3

Analysis and synthesis of the unit hydrograph by the black box method..................................... 32

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5

Hydrology III - II

4.4

System operation and instantaneous unit hydrograph.................................................................. 36

4.5

Computation of the unit hydrograph from the ∆t-weighting function ......................................... 41

CONCEPTS OF HYDROLOGIC MODELS ........................................................ 43 5.1

Translation and retention.............................................................................................................. 43

5.2 Translation models......................................................................................................................... 43 5.2.1 Linear translation, linear channel............................................................................................. 43 5.2.2 Time of concentration .............................................................................................................. 45 5.2.3 Floodplan method .................................................................................................................... 47 5.2.4 Time-area diagram................................................................................................................... 51 5.3 Reservoir routing models............................................................................................................... 55 5.3.1 Linear reservoir........................................................................................................................ 55 5.3.2 Non-linear, exponential reservoir ............................................................................................. 58 5.3.3 Linear reservoir cascade........................................................................................................... 58 5.4 Parameter estimation for simple conceptual models..................................................................... 61 5.4.1 Moment method for linear model concepts ............................................................................... 61 5.4.2 Storage-outflow relation of single reservoir models .................................................................. 65 5.4.3 Outflow recession curve of the linear, single reservoir .............................................................. 67

6

COMBINATION OF MODEL CONCEPTS IN DRAINAGE BASIN MODELS ..... 70 6.1 One-component models for direct runoff ...................................................................................... 70 6.1.1 Clark model ............................................................................................................................. 70 6.1.2 Two-reservoir-model (Singh's model)....................................................................................... 71 6.1.3 Influence of precipitation on the model concept........................................................................ 71 6.2

7

Multi-component models, parallel reservoir cascades .................................................................. 71

FLOOD ROUTING MODELS ............................................................................. 74 7.1

Flood routing.................................................................................................................................. 74

7.2 Simple flood forecasting methods .................................................................................................. 77 7.2.1 Gage relation curve .................................................................................................................. 77 7.2.2 Travel time curve ..................................................................................................................... 79 7.2.3 Prediction of discharge changes ............................................................................................... 80 7.3

Hydraulic approaches to instationary flow ................................................................................... 82 Continuity equation ............................................................................................................................. 82

7.4 Hydrologic flood routing concepts................................................................................................. 88 7.4.1 Basic principles of hydrologic flood routing ............................................................................. 88 7.4.2 Muskingum-model ................................................................................................................... 88 7.4.3 Kalinin-Miljukov method, basic principles............................................................................... 93 7.4.4 Kalinin-Miljukov method, linear reservoir cascade................................................................. 103

8

PHYSICALLY BASED HYDROLOGICAL MODELS........................................ 106

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Hydrology III - III

Fundamental principles ............................................................................................................... 106

8.2 Spatial model extension ............................................................................................................... 107 8.2.1 One-dimensional models........................................................................................................ 107 8.2.2 Two-dimensional models ....................................................................................................... 108 8.2.3 Three-dimensional models ..................................................................................................... 109 8.3 Temporal and spatial model resolution ....................................................................................... 110 8.3.1 Temporal resolution ............................................................................................................... 110 8.3.2 Spatial resolution ................................................................................................................... 111 8.4 Modeling of single processes........................................................................................................ 113 8.4.1 Infiltration .................................................................................................................................... 113 8.4.2 Evaporation............................................................................................................................ 114 8.5 Model parameters........................................................................................................................ 115 8.5.1 Parameter estimation.............................................................................................................. 115 8.5.2 Parameter variability.............................................................................................................. 116 8.5.2.1 Temporal variability........................................................................................................... 116

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Prof. Dr. rer. nat. Dr.-Ing. András Bárdossy

Hydrology III - 1

1

Fundamental principles of river basin modeling

1.1

Scope

The basis of rational water use and -management is understanding the temporal and spatial characteristics of water flow. Since transport- and transposition processes take place in the water, and large-scale input of man-made substances occurs, the description of water quality is closely related to the description of water flow. According to the actual problem different statements are required that origin from either statistical or deterministic approaches. The main tasks are: •

Computation of large-scale balances (provides basic information about the water regime and its spatial variations applying long time means and statistical approaches.).

•

Design of water management structures (e. g. flood protection, river development, flood-control reservoirs, carryover storage) usually statistical approaches (e. g. HQ100, NQ10, MQ).

•

Real-time forecasting (e. g. inflow for storage management, flood-forecast service, storage operation) statistical and/or deterministic approach (e. g. forecast by statistical time-series models, prediction of time and height of peak flow computations.

•

Design and assessment of management measures and evaluation of alternatives (e. g. water body development, flood retention, storage operation) usually deterministic approaches (e. g. computation of retention effect of a flood-control reservoir.

•

Process studies for better comprehension of complex hydrologic processes, predominantly deterministic approach.

Another possible subdivision derives from the examination time period: • • •

short-term medium-term long-term

minutes, hours, days (e. g. flood events) weeks, months (e. g. low water, storage management) years (e. g. mean water, sizing of water power plants)

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1.2 1.2.1

Hydrology III - 2

Principal methods of river basin modeling Statistical methods

Statistical methods are exclusively based on the description of observed values without consideration of the underlying causes. Statistical methods are used for: •

theoretical probability distributions (distribution functions) of observed values at a certain river cross-section (e. g. extreme-value statistics of floods and low water),

•

time-series analysis and -synthesis of outflow hydrographs,

•

regionalization, Transformation of e. g. outflow characteristics applying regression/geostatistics based on typical features of the gaged and ungaged sites (e. g. size of drained area, inclination, geologic and morphologic features).

The meaningfulness of statistical investigations is dependent on the density of the gage network and the duration of observation. For more detailed information, see lecture “Hydrological simulation techniques“.

1.2.2

Deterministic methods

Deterministic methods investigate the correlation between cause and effect. It is essential to be able to quantify and mathematically describe the causes and the structure of the affiliated effects. These interrelations may derive from physical laws or from the analysis of short-term observations. Runoff may be attributed to various causes, therefore several different mathematical formulations and mathematical models may be applied. All natural outflow is primarily dependent on precipitation. •

Precipitation (e. g. rainfall-runoff models, drainage basin model); (see Chapter 2).

Additionally secondary effects of precipitation may be regarded as causes for surface runoff. •

Volume of groundwater storage (e. g. low-water models),

•

Melting of snow and glaciers (e. g. equations of snow-melt),

•

Discharge of the upper courses (e. g. streamflow models); (see Chapter 7),

•

Operation and management (e. g. storage, discharge, withdrawal).

The results of the methods are as follows: •

simple outflow characteristics (e. g. time and magnitude of peak flow),

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•

continuous outflow hydrographs (river basin model with temporal and areal resolution).

1.2.3

Combined methods

The combination of statistical and deterministic model calculations leads to combined methods. Often the natural impacts on a hydrologic system can only be described with statistics, whereas the effects of the system can be derived from physical laws. An example of this is the computation of extreme floods for an area that features only short or no outflow measurements. Therefore an extrapolation to determine rare peak discharges is impossible. Assuming that precipitation observations of sufficient duration at one or several representative gage sites are available, extreme flow may be computed from precipitation. This is accomplished by application of a mathematical discharge model which derives from short- term discharge measurements or physical approaches. The relevant input can be obtained from the statistical distribution of precipitation. In the case of a direct dependence the probability of the effect (discharge) equals the probability of the cause (precipitation). Prerequisite for this method is that the effect for the hydrologic system and the probability distribution of the input are known. The method provides the probability distribution of the output (see Figure 1.1).

Statistical technique (distribution)

Variable propability p1

Deterministic model

e.g. Precipitation

Figure 1.1:

1.3 1.3.1

Result propability p1

e.g. Discharge

Combination of statistical and deterministic methods

Structure of river basin models Partial models

A hydrologic river basin model generates outflow according to the relevant hydrologic processes by transforming input (precipitation, meltwater supply, evaporation) into output (discharge at the outlet cross-section of the basin). Consequently the model describes the movement and the storage of the precipitated water on the land surface, subsurface and in the stream itself by partial models. The purpose of a mathematical river basin model is therefore the spatial and temporal reproduction of waterflow within a river basin. Thereby the basin is

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Hydrology III - 4

broken down in the horizontal and vertical direction, however, the selection of the appropriate subdivision is always dependent on the actual problem. In principle a river basin model contains at least models to compute the soil moisture regime, groundwater and evaporation and reproduces the formation of outflow in the drainage basin, runoff concentration in the water-body system and the temporal course of discharge in the streams of the basin. The following three basic elements that contain the previously mentioned subdivisions are applied: •

precipitation drainage basin rainfall-runoff model, drainage basin model (see Chapter 2-6),

•

river course streamflow models, flood-routing (see Chapter 5 and 7),

•

natural or artificial storage structures (storage operation model).

The application of partial models based on physics is recommended if river basin models are applied to rarely gaged or ungaged areas, to assess human impact on the water cycle of an area or if models are coupled to evaluate water quality.

1.3.2

Drainage basin models

The drainage basin model serves the determination of discharge/streamflow caused by precipitation within the basin. The computation is related to a single cross-section of the receiving stream which can be considered the outlet cross-section for the drainage basin above it (see Figure 1.2). The actual size of the area is defined by the borders of the drainage basin. Subsequently precipitation is only considered in liquid matter, which means as rain. The conversion of snow melting is reproduced by a suitable model.

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precipitation iN

Prof. Dr. rer. nat. Dr.-Ing. András Bárdossy

iN(t)

time t areal precipitation iN

discharge Q

discharge Q

Q(t)

drainage basin time t

Figure 1.2:

Principle of a drainage basin model, determination of streamflow from areal precipitation

First of all, the model must contain a method to convert the punctual precipitation measurements at gage sites to areal precipitation. •

areal precipitation temporal and spatial distribution of precipitation from the local data of the gage network (see lecture Hydrology I, Chapter 2.5).

The transformation of areal precipitation to streamflow takes place in two phases. •

Formation of outflow Transformation of precipitation considering evapotranspiration and the retention effect of the basin. Formation of runoff takes place at each point of the drainage basin. However, only a portion of precipitation is transformed into runoff.

•

Concentration of outflow/ streamflow Concentration of the runoff in the outlet cross-section. In this regard it is important to determine the temporal distribution of outflow.

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point model of areal precipitation

model of discharge formation

precipitation

Figure 1.3:

model of discharge concentration

discharge at catchment outlet

Basic elements of a drainage basin model

Usually drainage basin models are considered basic units that are not subdivided any further. Therefore each rainfall event must be spread evenly throughout the basin (block rain). Significant variations or partial rainfall is not permitted. Since natural precipitation may only on small-scale areas be considered evenly distributed, this provides the upper limit of the size of the model basin size. If the block rain assumption does not provide sufficient precision for the model, other vertical divisions must be found. Some areas always feature a typical areal rainfall distribution (e. g. mountain rims) that can replace block rain. Subdividing the area by hydrologic characteristics (hydrotopes) is often useful and easily applicable. However, since the complexity of the model increases with the horizontal division it is useful only up to a certain degree. The size of the drainage basin is a decisive factor for the design of the model. The respective topographic and orographic characteristics must be taken into consideration. For river basins in southgerman low mountain ranges models covering an area of up to 500 km² are applied.

1.3.3

Streamflow models

Streamflow models reproduce the flow of flood waves in the streams which means instationary open channel flow. The river bed and its piedmonts constitute a retention space which holds the flood wave temporarily back. Continuous retention leads to a flattened flood wave ( wave distortion, see Chapter 7). The streamflow models usually applied in hydrology do not compute the streamflow all the way along the stream, the results are limited to a single control cross-section (outlet crosssection of the examined river section).

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time t

river course

discharge QA

discharge QZ

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time t

QZ inflow

Figure 1.4:

QA outflow

Principle of a streamflow model, one tributary, (floodrouting)

Three tasks can be distinguished: •

Flow of a flood wave in stream without lateral inflow or withdrawal; this means one inflow QZ(t) and one outflow Q A(t) and therefore identical water volume (see Figure 1.4).

•

Confluence of several flood waves from different streams; this means several inflows QZ,i(t) and one outflow Q A(t) (see Figure 1.5).

•

Flow of a flood wave in an open channel with punctual or continuous lateral inflow from the traversed intermediate drainage basin. Combination of streamflow model in the open channel and rainfall-runoff model in the traversed drainage basin (see Figure 1.6).

Figure 1.5:

Principle of a streamflow model with several tributaries (flood forecasting)

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precipitation i

N

Prof. Dr. rer. nat. Dr.-Ing. András Bárdossy

discharge Q Z

time t

areal precipitation i

intermediate catchment

Figure 1.6:

discharge Q A

QZ inflow

time t

1.3.4

N

QA outflow

time t

Principle of a streamflow model with intermediate drainage basin

Complex river basin model

From the previously introduced single components, a complex river basin model including storage spaces may be established. The interfaces of the model components must be selected in a way such that the structure of the model matches the natural formation of outflow (see Figure 1.8). • • • •

Streamflow gages, water-level gages Confluence of tributaries Points of limited discharge capacity, control sites (bridges, villages, etc.) Points that offer management possibilities (e. g. barrages)

1.4

Model approaches

The character of the individual model is selected with the previous knowledge of the hydrologic system. According to the state of knowledge about the physical laws and the extend of required data the model is selected. •

Hydraulic mathematical models are based on physical laws (e. g. conservation of mass and energy, model with previous physical knowledge). The model is developed using detailed geometric and hydraulic measurements (e. g. channel cross-sections, bed slopes, roughness coefficients). To describe the complex spatial flow characteristics of a drainage basin hydraulic models are unsuited. The plurality of essential measurement values and the considerable amount of

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calculations limit their application in hydrologic study. Therefore, hydraulic models are applied merely to compute instationary open channel flow. •

Model concepts are based on simplified physical concepts (e. g. continuity- or storage relations, translation). The complex physical transformation mechanisms are replaced by coarsened model assumptions. The model is defined by a number of parameters (as few as possible) that mostly are derived from only a few and not necessarily very precise geometric and/or hydraulic data or from calibrations against observed values. In this case there is no correspondence between the natural system and the model parameters, just a relation. The application of systemhydrologic models is therefore limited in the case of combined discharge-, transport- and quality analysis.

•

Black-box models contain a merely mathematical description of the transformation characteristics according to systemtheoretical methods (input-output models, models without previous knowledge). Physical principles are completely disregarded. The model is defined by empirical system parameters (see Chapter 4). After the model has been defined the parameters are calibrated against observed outflow values (observed in- and output).

Models that use previous knowledge explain the underlying physical processes, models that do not use previous knowledge only model the processes.

time t

time t

intersection

time t

water flow model

time t

Figure 1.7:

Interfaces of a river basin model

rainfall - runoff model

time t

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rainfall-runoff-model catchment model Si intersections

Figure 1.8:

water flow model

water flow mo including catch (lateral inflow)

Ki gage site (discharge gage)

Structure of a complex river basin model

Many model concepts can be described by methods of system theory. The advantage is that both approaches are based on the same mathematical foundations. Furthermore, due to the connection a direct comparison of the transformation characteristics is possible. Another aspect for the arrangement of model approaches is the relation to the size and shape of the hydrologic system. •

Models that consider the size of the hydrologic system Defining the model considers the areal extension of flow. The parameters are assigned to spatial-, areal- or linear gridpoints (hydraulic- and some conceptual models).

•

Concentrated models, block models A limited space is considered a hydrologic unit. Flow is therefore artificially concentrated at one point (black box- and most conceptual models).

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1.5

Analysis and synthesis

1.5.1

Calibration of the model against observed in- and output values

Models that either use very little or no previous knowledge at all (black box) obtain their transformation characteristics only from analysis of an output based on a known input. Consequently in- and output data such as precipitation-, in- and outflow hydrographs are required. However, the parameters that derive from characteristic values of the hydrologic system are subject to substantial uncertainties. Calibration against in- and outflow data on the other hand provides a means to suit the model better to the respective aim. This is also valid for hydraulic model approaches. The analysis compares the transformation characteristics of the natural hydrologic system and the model. For the same input the respective outputs should match as closely as possible. By specific optimization the model can be suited to the hydrologic system (see Figure 1.9). Thereby either the values of the hydrographs or characteristic hydrograph values such as the moments are compared. The data flow in the course of analysis and synthesis is displayed in Figure 1.10.

1.5.2

Synthetic streamflow hydrographs

To compute synthetic streamflow hydrographs the input values and the transformation characteristics of the model must be known (cause-transformation-effect). For drainage basin models the input is precipitation, for streamflow models it is inflow. observed input

observed output natural hydrological system

e.g. discharge, precipitation

e.g. discharge

calibration of comparison parameters

m athem atical model (parameters)

Figure 1.9:

generated output

Calibration of a mathematical hydrologic model

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The synthesis of streamflow hydrographs can be split up into three tasks. •

Check of historic events If only the input (precipitation or inflow) or output (outflow) of a historic event are known, the unknown values may be found by applying a model. Thus short-term observation data and gaps in observation time-series may be augmented.

•

Estimation of extreme flows For design reasons rare outflow magnitudes of small exceedence probability or probability that outflow falls below the respective value are required. Therefore the input must be connected to a corresponding statistical statement (e. g. 100-year exceedence precipitation as input for a drainage basin model). Methods to determine design precipitation are discussed in the lectures "Hydrology I, Chapter 2.6" and "Hydrologic simulation methods".

•

Prediction of effects of water management projects The effect of water management structures (e. g. storage) can only be assessed applying model calculations. The model simulation is based on historic and/or synthetic outflows.

analysis known input

model, analysis of parameters

known output

synthesis known input

model, known parameters

Figure 1.10: Data flow in the course of analysis and synthesis

synthesis of output

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2

Structure of drainage basin models

2.1

Formation of runoff and runoff concentration

Only a fraction of the precipitation that falls above a drainage basin eventually appears as runoff. Already through the course of the precipitation event evaporation returns a fraction of the water back to the atmosphere. The portion of precipitation that later appears as runoff (effective rainfall) infiltrates dependent on intensity and duration of the precipitation event into different stratums of the drainage basin. Usually flow is separated into three components of roughly uniform character (DIN 4049, Part 1, see Figures 2.1 and 2.2). •

Surface runoff The portion of flow that moves into the receiving stream on the surface.

•

Interflow The portion of flow that flows through the subsurface towards the receiving stream. Interflow may be further subdivided into delayed and fast interflow (unsaturated soil zone).

•

Groundwater flow The portion of flow that flows delayed towards the receiving stream from the groundwater body (saturated soil zone). The total of surface runoff and fast interflow is termed direct runoff. Base flow is formed from groundwater flow and delayed interflow. The formation of runoff is reproduced in the model as a two-phase process.

•

Separation of precipitation into two parts: The first part, called net precipitation or effective rainfall contributes directly to the surface runoff. The other part is composed of losses to interception, evaporation, depression storage, and regional storage.

•

Distribution of the effective rainfall into the three components of flow.

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outflow Q [m3/s]

Prof. Dr. rer. nat. Dr.-Ing. András Bárdossy

flood increase

flood decrease vertex

flood hydrograph Q

surface runoff QO interflow QI

direct outflow Q

base flow assumed drought outflow hydrograph

time t [h]

Figure 2.1:

Separation of flow components of a flood wave

evaporation transpiration interception surface runoff QO areal precipitation iN

effective precipitation iNe

direct outflow QD

overall outflow Q

interflow QI

infiltration

formation of outflow

Figure 2.2:

base flow QB

concentration of outflow

Separation of areal precipitation in the course of flow formation components of runoff concentration

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2.2

Formation of outflow and runoff concentration in simple drainage basin models

Simple drainage basin models only consider two flow components (see Figure 2.3). •

Direct runoff, QD Portion of the flood wave that arises directly and quickly from a precipitation event.

•

Base flow, QB Portion of the outflow that is not directly concerned with the flood event. Base flow is a constant flow that changes only slowly.

Both components are connected to their causes and are treated separately in the setup of the model. The actual precipitation event causes direct runoff, whereas base flow is dependent on the regional soil moisture and the groundwater volume and pertains to the long-term precipitation history (previous precipitation). Simple models compute outflow by separating it into two components (see Figure 2.3). • effective rainfall or net precipitation, iNe Precipitation that eventually appears as direct flow hydrograph at the basin outlet. • losses, iV Combination of all components that are not included in the direct runoff (evaporation, regional storage, etc.). formation of outflow

concentration of outflow

precipitation losses iNV

areal precipitation iN

effective precipitation iNE

direct outflow QD overall outflow Q

precipitation history, initial soil moisture

Figure 2.3:

base flow QB

Flow formation and runoff concentration in simple drainage basin models

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2.3

Base flow

Base flow is the portion of outflow that is not directly associated with the precipitation event. In comparison to direct runoff base flow shows only small magnitudes. Its effects on the flood wave, especially on the peak flow are only marginal. Assuming the base flow hydrograph as a straight line provides sufficient precision, it may even be regarded as constant. The separation is carried out graphically by a horizontal or slightly inclined straight line from the starting point of the flood wave (see Figure 2.4). The starting point is indicated by a recognizable increase of flow. Subtraction of the base flow QB from the overall flow Q provides the direct runoff QD: at the beginning and at the end the direct runoff hydrograph has a value of zero. (2.1)

start

discharge Q [m3/s]

QD ( ti ) = Q ( ti ) − QB ( ti )

time t [h]

Figure 2.4:

Separation of base flow by a a) horizontal or b) slightly inclined straight line

Computation of synthetic outflow is conducted separately for base flow and direct runoff. Adding the two components provides the overall outflow hydrograph. Considering accuracy, the base flow is only of minor importance. Assigning the mean outflow at dry-weather conditions to base flow is of sufficient precision. (For further investigations the analysis of the coaxial graphical plot is recommended, see Chapter 3.1.2).

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3

Models of runoff formation

3.1

Runoff coefficient

3.1.1

Hydrology III - 17

Overall runoff coefficient for single rainfall-runoff events

The overall runoff coefficient is the volumetric ratio of direct runoff to areal precipitation. It is the fraction of a precipitation event that contributes to runoff.

ψ=

ψ VD VN hNe hN

VD hNe = VN hN

(3.1)

[-] [m3] [m3] [mm] [mm]

overall runoff coefficient volume of direct runoff volume of precipitation overall effective depth of precipitation of the event overall depth of precipitation of the event

Usually the direct runoff hydrograph QD(ti) is plotted as a succession of linear interpolations between discrete values. The first (i = 0) and the last (i = k) always equals zero. Consequently the discrete integration is reduced to the trapezoidal algorithm. k −1

VD = ∆t ⋅ 3600 ⋅ ∑ QD ( ti )

(3.2)

i =1

QD 3600

[m3/s] [s/h]

direct runoff conversion factor

The volume of the observed areal precipitation is the product of the overall depth of precipitation and the size of the drainage basin. VN = 1000 ⋅ AE ⋅ hN AE hN 1000

[km2] size of the drainage basin [mm] overall depth of precipitation 3 2 [m /(km ⋅mm)] conversion factor

(3.3)

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3.1.2

Hydrology III - 18

Antecedent precipitation index and coaxial graphical plot

The overall runoff coefficient of a rainfall-runoff event is related to the duration of the precipitation event, the overall depth of precipitation, the soil moisture and the season. A measure for the initial soil moisture is the Antecedent Precipitation Index (API) hVN. The seasonal variation of evapotranspiration and the detention storage may be considered by a continuous array of numbers nW assigned to each week. hNe = hN − hNe = f ( hN , TN , hVN , nw )

ψ=

hNe hN

hNv hN

[mm] [mm]

hNe TN hVN nW

[mm] [h] [mm] [-]

(3.4)

(3.5) precipitation losses overall depth of precipitation of a single event (e. g. design precipitation) effective depth of precipitation of a single event duration of the precipitation event antecedent precipitation index week number (of the year)

The antecedent precipitation index is based on the assumption that the soil moisture after a precipitation event decreases exponentially. The more time elapsed between precipitation events, the smaller is its impact and vice versa. The weighted daily depths of precipitation of a limited period of time preceding the actual event are taken into consideration. The individual weights α are always smaller than 1.

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Hydrology III - 19

precipitation hN [mm]

Prof. Dr. rer. nat. Dr.-Ing. András Bárdossy

t0-i

time t [d]

antecedent prec. index hVN [mm]

t0-n

hVN (t0)

t0-n

Figure 3.1:

t0-i

time t [d]

Antecedent precipitation index

The antecedent precipitation index is computed as: n

hVN ( t0 ) = ∑ hN ( t0−i ) ⋅ αi

(3.6)

i =1

t0 hVN hN n α

[d,h] [mm] [mm] [-] [-]

start of precipitation event, antecedent precipitation index, daily depth of precipitation, number of days preceding the event, empirical weighting factor α < 1.

Usually the impact is limited to a time of ≈ 30 days. Preceding precipitation is not considered. Empirical investigations have suggested a weighing factor of α = 0.9. The precipitation losses hNV are displayed in coaxial graphical form (see Figure 3.2).

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depth of prec. losses hNV [mm] (observed)

antecedent precipitation index hVN [mm]

week number

depth of prec. losses hNV [mm] (calculated)

duration of precipitation TN [h]

Figure 3.2:

depth of precipitation hN [h]

Coaxial graphical plot of precipitation losses hNV for a given drainage basin, reading example

The interdependences may be found applying multiple non-linear regression or graphically by trial and error. The equation displayed below indicates a possible non-linear regression to represent the coaxial diagram as a formula, however, here instead of the antecedent precipitation index hVN the base specific discharge qB is used, and instead of the week number nW the month M is used to compute the precipitation losses hNV.

hNV

hNV qB M TN

 π  hN ⋅ eC ⋅qB ⋅ e D⋅TN ⋅  A + B ⋅ sin  ⋅ ( M − 4 )   6   =  π  hN + hN ⋅ e E ⋅hN ⋅ eC ⋅qB ⋅ e D⋅TN ⋅  A + B ⋅ sin  ⋅ ( M − 4 )   6   [mm] 2 [l/s/km ] [-] [h]

precipitation losses (regional storage) base specific discharge at the beginning of the event month duration of precipitation

(3.7)

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A...E

[-]

Hydrology III - 21

parameters

3.1.3 The SCS approach According to the DVWK (1984), for the estimation of effective or net precipitation in the case of rain storm events and small drainage basins the application of the SCS approach developed by the U.S. Soil Conservation Service is recommended. This method considers effective rainfall hNe as a function of the depth of precipitation hN and a curve number CN dependent on the drainage basin: 2

5080   + 50.8   hN − CN  hNe =  20320 − 203.2 hN + CN

(3.8)

The CN value again is a function of the soil type, land cover, cropping practice and the antecedent moisture condition, which is dependent on the antecedent precipitation of the preceding 5 days and the season. Table 3.1 displays CN-values for various soil types and land cover/ cropping practice for antecedent moisture condition II. From Table 3.2 the current antecedent moisture condition may be taken. In case it deviates from II, the final CN-value may be determined applying Figure 3.3.

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Table 3.1:

CN-Values for antecendent soil moisture condition II

Land use

CN for hydrologic soil group A

B

C

D

Bare soil

77

86

91

94

Root crops, wine

70

80

87

90

Wine (terraced)

64

73

79

82

Corn, forage plants

64

76

84

88

Pasture (normal)

49

69

79

84

(barren)

68

79

86

89

Meadow

30

58

71

78

Forest (open)

45

66

77

83

(medium)

36

60

73

79

(dense)

25

55

70

77

100

100

100

100

impervious areas Hydrologic soil group A:

Soils with great infiltration potential, even after antecedent wetting (e. g. thick sand and gravel stratums)

Hydrologic soil group B:

Soils with medium infiltration potential, thick and moderately thick stratums, fine or moderately coarse texture (e. g. moderately thick sand stratums, loess, loamy sands)

Hydrologic soil group C:

Soils with low infiltration potential, sorts of fine or moderately coarse texture or with impervious layers (e. g. thin sand stratums, sandy loams)

Hydrologic soil group D:

Soils with considerably low infiltration potential, clay, thin soil stratums overlying impervious layers, soils with constantly high groundwater stage.

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Table 3.2:

Current antecedent moisture condition

antecedent moisture condition

accumulated depth of precipitation within the preceding 5 days in unit [mm] other

I

< 30

< 15

II

30 - 50

15 - 30

III

> 50

> 30

CN for soil moisture class I, III

vegetation period

CN for soil moisture class II

Figure 3.3:

CN for antecedent moisture condition I and III cross-linked to antecedent moisture condition II

This method should only be applied for rain storm events. Experiences in the past have shown that for depth of precipitation lower than 50 mm the method underestimates effective rainfall. Modifications of equation (3.8) try to compensate this.

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3.2 3.2.1

Hydrology III - 24

Models to compute effective rainfall Model requirements

The current formation of outflow is dependent on the intensity iN, the duration tN and the variations (e. g. breaks) of precipitation. Due to depression storage and wetting the initial losses (or after breaks) are larger than during periods of intensive precipitation. In hydrologic practice usually two methods are applied. Runoff coefficient method In the course of a precipitation event only a portion of precipitation is transformed into direct runoff. The runoff coefficient ψ is the ratio of effective or net precipitation iNe(t) to the observed precipitation iN(t). iNe ( t ) = ψ ⋅ iN ( t )

(3.9)

In principle the runoff coefficient is a function of precipitation intensity and -duration.

ψ = f ( iN ( t ) , t )

(3.10)

However, simple models consider the runoff coefficient as constant throughout the whole precipitation event (see Chapter 3.2.2). Index approaches Only the precipitation that is equal to or more than a certain infiltration capacity iv ( = losses) contributes to direct runoff.

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for iN ( t ) > iNv ( t )

const. coeff. of discharge

Figure 3.4:

time t [h]

phi - index method

time t [h]

prec. intensity i [mm/h]

for iN ( t ) ≤ iNv ( t ) prec. intensity i [mm/h]

prec. intensity i [mm/h]

i ( t ) − iNv iNe ( t ) =  N  0

loss variable with time

time t [h]

Models to determine effective rainfall a) constant runoff coefficient, runoff coefficient method b) constant loss ratio, Φ-index method c) loss ratio decreasing exponentially

base flow Q B (t)

overall outflow Q(t)

separation of base flow

direct outflow Q D (t)

areal precipitation iN (t)

formulation for effective prec.

effective precipitation iNe (t)

precipitation losses iNv (t)

Figure 3.5:

Determination of outflow formation for simple drainage basin models, sequence and data flow

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Hydrology III - 26

The initial rate is always larger than at the end of a precipitation event. Usually the temporal development of losses can be represented by an exponentially decreasing function (see Figure 3.4c).

iNv ( t ) = f ( iN ( t ) , t )

(3.12)

The simplest model features a constant loss rate, which is only a very rough approximation (see Chapter 3.2.3). The analysis of rainfall-runoff events is carried out in established steps (see Figure 3.5). • • •

Separation of base flow (linear course) Computation of overall runoff coefficient from the volumes of direct runoff and areal precipitation Computation of effective rainfall (runoff coefficient method or index approaches)

3.2.2 Runoff coefficient method The runoff coefficient ψ remains constant throughout the entire course of the precipitation event (see Figure 3.4a). It corresponds to the overall of discharge (see Chapter 3.1). Consequently computation of effective rainfall is reduced to the simple formula displayed below. iNe ( ti ) = ψ ⋅ iN ( ti )

(3.13)

iNe(ti) iN(ti)

[mm/h] [mm/h]

effective rainfall intensity in time interval t observed precipitation intensity in time interval t

ψ

[1]

overall runoff coefficient

3.2.3 Index approaches, Φ-index A constant loss rate in the course of a precipitation event is referred to as Φ-index. iV ( t ) = const. = Φ

Φ

[mm/h]

(3.14) constant loss rate, Φ-index

The portion of precipitation that exceeds the Φ-index is the effective rainfall.

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i ( t ) − Φ iNe ( t ) =  N  0

Hydrology III - 27

for iN ( t ) > Φ

for iN ( t ) ≤ Φ

The Φ-index must be determined by step-by-step iterations since negative precipitation is impossible. The iteration provides a constantly increasing Φ-index; the procedure is repeated until it equals the known effective rainfall.

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4

Basis and hydrology

methods

of

systems

4.1 Definition of system properties A system is a distinguished arrangement of interrelated structures (DIN 19226). Each system features an entrance where the cause (input) ui affects the system, and an exit where the effect (output, system answer) vi occurs (see Figure 4.1). The interrelations of these values describe the system. The system-operation establishes a definite relation between input and output. input

output

system load

Figure 4.1:

result

System with several in- and output variables (input vector ui(t) and output vector vi(t))

The simplest case is the definite relation between one output magnitude v and one input magnitude u (see Figure 4.2), e. g. effective rainfall - direct runoff. The mathematical relation between in- and output can be represented by

v ( t ) = ϕ {u ( t )}

(4.1)

where, u(t) v(t) ϕ

time-dependent input signal, time-dependent output signal, system operator.

system input variate

Figure 4.2:

output variate

System with one input magnitude u(t) and one output magnitude v(t)

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Hydrology III - 29

The system operation may be defined by certain regularities that allow a classification of the model systems. As an example these regularities are applied to the precipitation-runoffrelation in drainage basins. The input magnitude is the areal effective rainfall expressed as intensity in unit [mm/h], the output magnitude is the outflow at the basin outlet in unit [m3/s] (see Figure 4.3). It is essential to estimate effective rainfall correctly, as the system input affects the quality of the computed unit hydrograph. •

The drainage basin is an open, dynamic system.

A system is termed dynamic if at any time t1 the output signal v(t1) is not merely dependent on the input signal ui (t) at the same time t1 but also from preceding input signals u(t) for t < t1. In physical regard this feature corresponds to a temporal storage of the input magnitude which can be regarded as a system memory. Drainage basins, open channels and storage structures can be considered dynamic systems, because they temporarily store outflow and deliver it later and damped (retention). Figure 4.3a displays how a precipitation event of short duration TN is discharged as a flood wave of much longer duration Tb. • Theory of proportionality Any input signal multiplied by a constant C produces an output signal multiplied by the same constant.

ϕ {C ⋅ u ( t )} = C ⋅ ϕ {u ( t )}

(4.2)

The effective rainfall displayed in Figure 4.3b is double the amount as in Figure 4.3a and produces a doubled outflow hydrograph. The duration Tb of the outflow hydrograph remains constant. • Theory of superposition The system answer to accumulated input signals equals the total of the single output signals.

ϕ {u1 ( t ) + u2 ( t )} = ϕ {u1 ( t )} + ϕ {u2 ( t )}

(4.3)

• Theory of linearity The combination of the theory of proportionality and superposition provides the theory of linearity.

ϕ {C1 ⋅ u1 ( t ) + C2 ⋅ u2 ( t )} = C1 ⋅ ϕ {u1 ( t )} + C2 ⋅ ϕ {u2 ( t )}

(4.4)

• Theory of time invariance The system operation is not time-dependent. Shifting an input signal by the time interval T results in an output signal shifted by the same time interval without changing the signal itself (see Figure 4.4c). Tb is preserved.

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ϕ {u ( t − T )} = v ( t − T )

(4.5)

unit hydrograph of discharge precipitation

intensity iNe [mm/h]

intensity iNe [mm/h]

Applying the theory of time-invariance and proportionality, consecutive input signals of different intensity can separately be assigned to individual output signals (see Figure 4.4d). The theory of superposition allows to overlay the individual signals to one.

time

Figure 4.3:

precipitation

discharge discharge QD [m3/s]

discharge QD [m3/s]

discharge

principle of linearity

time

a) dynamic system operation, unit streamflow hydrograph b) theory of proportionality

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principle of time invariance

intensity iNe [mm/h]

intensity iNe [mm/h]

Prof. Dr. rer. nat. Dr.-Ing. András Bárdossy

precipitation

principle of superposition precipitation

discharge

discharge QD [m3/s]

discharge QD [m3/s]

discharge

time

time

Figure 4.4: a) Theory of time-invariance b) Theory of superposition

4.2

Unit hydrograph

The unit hydrograph method is a linear, time-invariant model to determine outflow.

effective precipitation iNe*Ae input

Figure 4.5:

drainage basin gE system (linear, dynamic, time-invariant)

direct outflow QD output

Runoff concentration model as linear, dynamic, time-invariant system

To describe the system operation it is sufficient if the output function that pertains to one constant input signal is known. Using the theory of linearity and superposition the system answer to any series of discrete input signals can be determined. It is useful to relate the characteristic output function to a constant unit input of the duration ∆t and the magnitude one (see Figure 4.3a). This function is referred to as discrete weighting function g(∆t,ti) with

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Hydrology III - 32

reference interval ∆t. Note that not the intensity, but the volume of the unit input possesses the value 1 and therefore the intensity is 1/∆t. The reference time interval ∆τ is a defining feature of the weighting function g(∆t,t)i. For a linear and time-invariant drainage basin model the weighting function g(∆t,ti) is replaced by the unit hydrograph gE(∆t,ti) that considers the different dimensions of precipitation and outflow and the size of the drainage basin. The unit hydrograph describes the system operation of effective rainfall to direct runoff (see Figure 4.5). The unit hydrograph is a characteristic outflow hydrograph of a surface drainage basin that develops from constant effective rainfall of uniform distribution of 1 mm in depth and defined length (DIN 4049 Part 1). Thereby, effective rainfall is expressed by the depth, not by the intensity of precipitation.

4.3

Analysis and synthesis of the unit hydrograph by the black box method

The determination of the unit hydrograph for a system can be achieved directly by the analysis of the observed rainfall-runoff events. This approach ignores the physical structure of the system and applies only the system properties and is referred to as black-box. First, a reference time interval ∆t is chosen that is valid for the discretion of all time-related data. When applying the computer program, a time interval that splits the flood hydrograph into 30-50 units is recommended. Separation of the base flow ( see Chapter 2.3) provides the system input, the direct runoff QD(ti). Subsequently the overall runoff coefficient and the effective rainfall hNe(ti) is computed (see Chapter.3.2). The flood hydrograph is composed of the hydrographs of the individual precipitation intervals (see Figure 4.6). Each time interval ti provides (assuming that the unit hydrograph is given) the direct runoff (synthesis). The overall outflow function that results from the overall inflow function may be developed by superposition of the hydrographs that pertain to the individual precipitation events. This procedure is referred to as superposition. Subsequently the equation system for a simple example enclosing n = 3 effective rainfall ordinates and m = 5 ordinates of the unit hydrograph is displayed:

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Hydrology III - 33

Q1 = N1 ⋅ G1 Q2 = N1 ⋅ G2 + N 2 ⋅ G1 Q3 = N1 ⋅ G3 + N 2 ⋅ G2 + N 3 ⋅ G1 Q4 = N1 ⋅ G4 + N 2 ⋅ G3 + N 3 ⋅ G2

(4.6)

Q5 = N1 ⋅ G5 + N 2 ⋅ G4 + N 3 ⋅ G3 Q6 =

N 2 ⋅ G5 + N 3 ⋅ G4

Q7 =

N 3 ⋅ G5

where [m3/s] [mm] [m3/(s⋅mm)]

Qi = QD(ti) Ni = hNe(ti) Gi = gE(∆t,ti)

direct runoff with time ti, effective rainfall in the interval between ti-1 and ti, unit hydrograph at time ti.

The equation system may be reduced to a differential equation, the so-called discrete equation of superposition. For a given time ti: i

QD ( ti ) = ∑ g E ( ∆t , tk ) ⋅ hNe ( ti − k +1 ) k =1

(4.7)

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Hydrology III - 34

precipitation intensity

areal and effective precipitation

time t [h]

direct runoff

principle of superposition

time t [h Figure 4.6:

Theory of superposition of the unit hydrograph method

Since the initial and terminal value of the direct runoff hydrograph and the unit hydrograph always equal zero, these times are not considered. Therefore the number of discrete values and intervals are

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Hydrology III - 35

o = m + n −1

(4.8)

m = o − n +1

o

[1]

number of discrete values of the unit hydrograph method QD ≠ 0,

n

[1]

number of effective rainfall intervals hNe, precipitation duration/∆t,

m

[1]

number of discrete values of the unit hydrograph gE ≠ 0.

For the analysis the ordinates of the unit hydrograph must be computed from the linear equation system. For m + n - 1 equation and m unknown values the equation system is n - 1 times overdefined. The optimum solution provides computed runoff QD,ber as close to the observed runoff QD,gem as possible. This is accomplished by applying the method of the lowest square error. m + n −1

∑ (Q

D ,ber

i =1

( ti ) − QD, gem ( ti ) )

2

= Minimum !

(4.9)

The solution of an overdefined equation system by the method of the lowest square error is available in closed form (BRONSTEIN-SEMENDJAJEV p.513-514). For a lower number of unknown values the solution may be found by trial and error. A test of plausibility derives from the definition of the unit hydrograph as the direct runoff hydrograph caused by the effective rainfall of 1 mm in depth within a time interval ∆t. Therefore

3.6 ⋅ ∆t m ⋅ ∑ g E ( ∆t , ti ) = 1 AE i =1 AE 3.6

∆t

2

[km ]

mm ⋅ km 2 ⋅ s ] h ⋅ m3 [h] [

(4.10) size of drainage basin, conversion factor, time interval.

The runoff concentration within a drainage basin may only approximately be represented by a linear system. In the individual case the unit hydrograph method provides satisfactory results even though it features a certain variation when analyzing several rainfall-runoff events of same duration ∆t. The generally applicable unit hydrograph is the mean of all obtained unit hydrographs while still considering the test of plausibility.

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Hydrology III - 36

Oftentimes no suitable runoff- or precipitation measurements for the drainage basin are available. A number of methods exist to compute the unit hydrograph if measurements are unavailable; two of them are commonly used: Direct application of the unit hydrograph of a similar drainage basin. Selection according to the drainage basin properties. From similar, well-observed drainage basins the dimensionless system may be taken. The reference drainage basin is selected according to size, geology, slope, soil type, characteristic values of the receiving stream and land use (Literature: DVWK-Merkblätter 1982, 1988 catalogue of system operations). Formulation of a synthetic unit hydrograph by regionalization of the drainage basin properties: Relations between the parameters of a system operation and the properties of a drainage basin can be established. However, it is essential to examine a plurality of drainage basins and to apply weighting functions that can be described analytically (e. g. triangular hydrographs, hydrograph of constant rise and exponential recession or the gamma-function). Important parameters of the system operation are the time of rise, the peak and various recession parameters etc. The correlation of unit hydrograph and basin parameters can be described by e. g. regressions. The Geomorphologic Unit Hydrograph (GUH) is a physically based model. It makes use of stream network characteristics to determine the probability of occurrence of individual water particles at the basin outlet (Literature: SIVAPALAN et al. 1990).

4.4

System operation and instantaneous unit hydrograph

The system operation of linear, time-invariant systems, as outlined in Chapter 4.2, can be represented by a characteristic answer to a constant input signal of duration ∆t. Other typical input signals exist that define the system by their affiliated output function. For several linear, time-invariant conceptual models (see Chapter 5) The output functions may be determined analytically. A special system input is the unit step function ε(t) (see Figure 4.7): 0 ε (t ) =  1

ε(t)

[1]

for t < 0 for t ≥ 0

(4.11)

unit step function

The unit step function is equivalent to an on-switch at time t = 0. The affiliated system answer is termed s-curve h(t). For a dynamic system that features a storage effect, the system operation transforms the input to an s-shaped, lagged and dampened outflow hydrograph (see Figure 4.7). In practice, this may occur if a river or channel weir is suddenly opened. For

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most hydrologic systems the system answer has the value zero at the time t = 0 which simplifies the subsequent formulas.

ϕ {ε ( t )} = h ( t ) with h ( t ) = 0 h(t)

[1]

for t ≤ 0

(4.12)

system operation

u Input u (t):

E (t) 1

unit jump E (t) 0 for t < 0 E (t) = 1 for t > 0

0

t v Output v (t):

h (t) 1

system operation h (t)

h (t) = 0

Figure 4.7:

0 for t < 0 0 - 1 for 0 Tc) the whole basin AE dewaters and a constant outflow rate is acquired (see Figure 5.4). On the rising limb (t ≤ Tc) the area At(t) that contributes to the outflow is constantly increasing with time. t   AE ⋅ TC At ( t ) =   AE  At(t) AE

[km2] [km2]

for 0 ≤ t ≤ TC for t > TC area contributing to outflow, size of the drainage basin.

effective precipitation

drainage basin

Figure 5.3:

Floodplan-model, area At(Tt) that contributes to outflow at time Tt

(5.10)

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Hydrology III - 48 outflow

time t [h]

Figure 5.4:

contributing area At (t) [km2]

normalized contributing area

precipitation direct runoff [m3/s]

eff. prec. iNe [mm/h]

Prof. Dr. rer. nat. Dr.-Ing. András Bárdossy

time t [h]

System operation of the floodplan for precipitation in form of a jump function, temporal course of the area contributing to outflow At(t), system operation function

Since the floodplan method is based on linear translation it is a linear, time-invariant system. The function of the ratio of the contributing area to the size of the drainage basin At(t)/AE represents the system operation function h(t) of the floodplan method (see Figure 5.4). t for 0 ≤ t ≤ TC At ( t )  = h ( t ) =  TC AE  1 for t > T C  h(t)

[1]

(5.11)

system operation function of the floodplan method

From the system operation function the unit hydrograph of the floodplan method can be determined (see Chapter 4.4). However, it is only representative if the concentration time Tc is a integer multiple of the time interval ∆t. 1 for 0 ≤ t ≤ TC d  g ( 0, t ) = ⋅ h ( t ) =  TC dt  0 for t > T C  g ( ∆ t , ti ) =

with h ( +0 ) = 0

1 ⋅ ( h ( ti ) − h ( ti − ∆ t ) ) ∆t

1 1 ⋅ AE ⋅ ⋅ ( h ( ti ) − h ( ti − ∆t ) ) 3.6 ∆t 1 = ⋅ AE ⋅ g ( ∆t , ti ) 3.6

(5.12)

(5.13)

g E ( ∆ t , ti ) =

g(0,t)

[1/h]

weighting function of the floodplan,

(5.14)

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g(∆t,ti) [1/h] gE(∆t,t) [

Hydrology III - 49

∆t-weighting function of the floodplan,

3

m ] s⋅mm

unit hydrograph of the floodplan.

For a constant time interval ∆t the hydrograph of the direct runoff QD(ti) is computed using the effective rainfall hNe(t) in the equation of superposition. For the time interval ∆t an integer fraction of the concentration time Tc is recommended. i

QD ( ti ) = ∑ g E ( ∆t , tk ) ⋅ hNe ( ti − k +1 )

(5.15)

k =1

If several time intervals ∆ti of different duration are applied, for each precipitation interval the assigned hydrograph can be computed by adding all partial hydrographs at the discrete values. For the direct runoff QD that pertains to one precipitation interval of the duration ∆t and the effective depth of precipitation hNe (intensity iNe = hNe/∆t):

hNe 1 ⋅ ⋅ AE ⋅ ( h ( t ) − h ( t − ∆t ) ) ∆t 3.6 1 = iNe ⋅ ⋅ AE ⋅ ( h ( t ) − h ( t − ∆t ) ) 3.6

QD ( t ) =

(5.16)

The ratio (∆t/Tc) of the interval ∆t to the concentration time TC governs the course of the weighting function and the partial outflow hydrographs. Figure 5.5 displays the construction of the ∆t-weighting function for different time intervals. In both cases the depth of precipitation equals one. For a precipitation interval ∆t shorter than the concentration time Tc (Figure 5.5d) only a fraction of the drainage basin contributes to outflow.

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dt pulse input

normalized area

dt pulse input

and normalized contributing area

time t [h] dt pulse function = dt weighting function

time t [h]

normalized area

dt weighting function

and normalized contributing area

time t [h]

Figure 5.5:

time t [h]

Effect of the time interval on the course of the ∆t-weighting function (the pulse volume equals in both cases one a) ∆t > TC b) ∆t < TC

Figure 5.6 contains the governing characteristic values for the variations of ∆t/ TC.

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Characteristic values of the ∆t-weighting function or the partial outflow

Figure 5.6:

hydrographs for different ratios ∆t/TC

∆ t > TC

∆ t = TC

∆ t < TC

TC

TC

∆t

TC

∆ t - TC

0

TC - ∆ t

Tges

∆ t + TC

2TC = 2∆t

TC + ∆ t

1

1

∆t/TC

1/∆t

1/∆t

1/TC

T1 = T3

X1 for h(t) - h(t-∆t) X1 for g(∆t,t)

5.2.4

Time-area diagram

The term time-area diagram denominates a runoff concentration model that is founded on linear translation, but, in contrast to the floodplan method, also considers the topography of the drainage basin. In an arbitrary shaped topographic drainage basin the overland flow velocity is not constant, but depends on the actual inclination. The travel time from any point within the basin to the outlet is lower in steep parts (high flow velocity) than in less inclined areas (low flow velocity). Equal travel time Tt results in different distances from the outlet. Those distances are displayed as isochrones in a map. Isochrones are lines of equal travel time to an outlet (see Figure 5.7). Applying the area-related Kirpich formula adapts the travel time to the inclination of the drainage basin, therefore a topographic map is needed. In steep areas the isochrones are further apart than in low-angled areas (steep areas = contour lines are close = high flow velocity = short travel time = isochrones far apart). The time-lag between two neighboring isochrones is equated with the time interval of the precipitation- and outflow hydrographs.

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Isochrone plan (lines of equal travel time), areas enclosed by isochrones ∆At,i

Figure 5.7:

∆Tt = ∆t , ti = Tt ,i

∆ Tt ∆t ti Tt,i

[h] [h] [h] [h]

(5.17) lag time of neighboring isochrones time interval of precipitation- and outflow hydrographs discrete hydrograph time interval discrete isochrone time interval

For precipitation in the form of a jump function the areas ∆At,i enclosed by the isochrones contribute successively to outflow (see Figure 5.8a). In-between the isochrones, similar to the floodplan method, a continuously rising participating area is assumed. The function At(t) represents the temporal development of the areal runoff contribution. Its course is, in contrast to the floodplan method, not linear but considers the actual shape of the drainage basin. i

At ( ti ) = ∑ ∆At ,i

for ti ≤ TC

k =1

At ( t ) = AE

∆at,i At(t) Tc

[km2] [km2] [h]

for ti > TC area between two neighboring isochrones, time-area function, time of concentration.

(5.18)

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direct runoff

area contributing to outflow

normalizedarea contributing to outflow

Prof. Dr. rer. nat. Dr.-Ing. András Bárdossy

system answer to jump function

dt weighting function

dt unit hydrograph

system answer to dt weighting function

time t [h]

weighting function g(0,t) time - area diagram

system answer to needle pulse time - area diagram W(t)

time t [h]

Figure 5.8: a) System operation of the time-area diagram in case of jump function precipitation, temporal development of the contributing area At(t), system operation function

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The function At(t)/AE corresponds to the system operation h(t) of the time-area diagram. According to the correlation discussed in Chapter 4, the ∆t-weighting function and the unit hydrograph can be computed. h (t ) =

At ( t )

(5.19)

AE

g E ( ∆ t , ti ) = g E ( ∆ t , ti ) ⋅

AE 1 At ( ti ) = ⋅ 3.6 3.6 ∆t

1 ⋅ ( h ( ti ) − h ( ti − ∆t ) ) ∆t 1 At ( ti ) − At ( ti −1 ) 1 At ( ti ) = ⋅ = ⋅ AE ∆t AE ∆t

(5.20)

g ( ∆ t , ti ) =

h(t) [1] g(∆t,ti) [1/h] gE(∆t,ti) [m3/(s⋅mm)]

(5.21)

system operation of the time-area diagram ∆t-weighting function of the time-area diagram unit hydrograph of the time-area diagram

The unit hydrograph gE(∆t,ti) is a discrete function with linear interpolation between the discrete values (see Figure 5.8b). The weighting function g(0,t) (instantaneous unit hydrograph) is here designated time-area diagram w(t). The differentiation of the system operation function is a histogram-shaped continuous curve (continuous terraced line, see Figure 5.8c). Applying h( + 0) = 0:

w ( t ) = g ( 0, t ) =

d 1 d h (t ) = ⋅ At ( t ) dt AE dt

(5.22)

and from the isochrone plan At (Tt ,i ) − At (Tt ,i −1 ) A d At ( t ) = t ,i = dt ∆Tt ∆Tt

w(t) At(t) AE ∆ Tt

[1/h] [km2] [km2] [h]

(5.23)

time-area diagram (corresponding to the weighting function) area contributing to outflow size of the drainage basin time-lag between two neighboring isochrones (usually ∆Ti = ∆t)

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In contrast to the ∆t-weighting function (discrete values), the time-area diagram w(t) remains constant within the time intervals Tt,i-1 < t < Tt,i (histogram). The floodplan method can be regarded as a simplified version of the time-area diagram with parallel isochrones of equal distance. The term "simplified time-area diagram" is valid for all methods that approximate the drainage basin and the isochrone plan by basic geometric figures such as triangles, trapeziums or circle sectors.

5.3

Reservoir routing models

5.3.1

Linear reservoir

The linear reservoir is a fictitious reservoir, where outflow is proportional to the storage volume.

S = k ⋅ QA [m3] [m3/s] [s]

S QA k

(5.24) storage volume outflow storage or retention constant

QZ

v=0 S QA

Figure 5.9:

Reservoir routing model, single reservoir and reservoir retention

The storage constant k usually obtains the same dimensions as the elapsing time t (for floods in unit hours).

S = 3600 ⋅ k ⋅ QA k 3600

[h] [s/h]

(5.25) storage constant, conversion factor.

For stationary flow the constant k equals the detention period of a water particle in the reservoir and consequently the time through which the whole storage volume has been

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exchanged once. The change in storage volume is computed with the methods of reservoir retention (see Hydrology II, Chapter 4.2).

dS d = k ⋅ QA ( t ) = QZ ( t ) − QA ( t ) dt dt d k ⋅ QA ( t ) + QA ( t ) = QZ ( t ) dt [m3/s]

QZ

(5.26)

inflow

The inhomogeneous differential equation of the linear reservoir can be rearranged to determine the outflow QA(t):

QA ( t ) = QA ( t0 ) ⋅ e

−

( t −t0 ) k

− 1 t + ⋅ ∫ QZ ( t ′ ) ⋅ e k t0

( t −t′ ) k

⋅ dt ′

(5.27)

applying the initial condition:

Q A ( t0 ) = t QA(t0) S(t0) e

S ( t0 ) 3600 ⋅ k

[h] [m3/s] [m3] [2.7183]

start, t > t0 outflow at time t0, initial condition storage volume at time t0 (initial storage volume S0) Eulers constant

If at the time t0 the reservoir is empty (S(t0) = 0) and therefore the outflow equals zero also (QA(t0) = 0), the solution of the differential equation is simplified and acquires the form of the integral of superposition (see Chapter 4.4). t

1 − QA ( t ) = ∫ ⋅e k 0

( t −t ′) k

⋅ QZ ( t ′ ) ⋅ dt ′

(5.28)

where t0 = 0, QA(t0) = 0 and QZ(t) = 0 for t < 0. The linear reservoir is a linear, time-invariant system that provides the weighting function g(0,t) (instantaneous unit hydrograph) as g ( 0, t ) =

g(0,t)

1 −tk ⋅e k

[1/h]

(5.29) weighting function of the linear reservoir.

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Applying the system operation function h(t) provides the ∆t-weighting function of the linear reservoir g(∆t,ti) (see Chapter 4.4). t

h ( t ) = ∫ g ( 0, t ′ ) dt ′ = 1 − e

−t

k

(5.30)

0

1 ⋅ ( h ( ti ) − h ( ti − ∆ t ) ) ∆t − ( ti −∆t ) 1  − ti k  = ⋅  −e k + e  ∆t  

g ( ∆ t , ti ) =

(5.31)

Determination of the outflow hydrograph by application of the discrete theory of superposition is only possible if the inflow hydrograph is available as a histogram (with constant values within each interval). If the reservoir inflow only exists as discrete values with linear interpolation, the reservoir retention may be resolved by the finite difference method.

∆S ( ti ) ∆t

=k⋅

∆ Q A ( ti ) ∆t

=k⋅

QA ( ti ) − QA ( ti −1 ) ∆t

1 1 k QZ ( ti −1 ) + QZ ( ti ) ) = ( QA ( ti −1 ) + QA ( ti ) ) + ⋅ ( QA ( ti ) − QA ( ti −1 ) ) ( ∆t 2 2

(5.32)

(5.33)

Hence the working equation of the linear reservoir in finite difference form is established (QZ and QA in discrete value form) as:

QA ( ti ) = C1′ ⋅ ( QZ ( ti ) + QZ ( ti −1 ) ) + C2′ ⋅ QA ( ti −1 )

(5.34)

with the coefficients C’1 and C’2:

C1′ =

and

∆t

2 ∆ k+ t

∆t >> k =

, 2

C2′ =

k − ∆t k + ∆t

∆S ∆QA

In rearranged form and with C1 and C2:

2 2

(5.35)

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QA ( ti ) = QA ( ti −1 ) + C1 ⋅ ( QZ ( ti −1 ) − QA ( ti −1 ) ) + C2 ⋅ ( QZ ( ti ) − QZ ( ti −1 ) ) C1 =

∆t k + ∆t

,

C2 = 1 − C1 ⋅ k

∆t

2

(5.36)

(5.37)

If we assume the interpolation between the discrete values is linear when integrating the differential equation, an exact solution for C1 and C2 is obtained. C1 = 1 − e

5.3.2

−∆t

k

,

C2 = 1 − C1 ⋅ k

∆t

(5.38)

Non-linear, exponential reservoir

A cylindrical reservoir that contains an outlet or overfall shows a storage-outflow relation that can be expressed as an exponential equation. S = k ⋅ QAm

(5.39)

parameter of the storage-outflow relation storage constant, dependent on m

m k

The affiliated inhomogeneous differential equation of the reservoir retention cannot be resolved analytically. Therefore an approximation is determined applying the finite difference method. Integration is only possible in the case of a drainage process (with known initial storage volume at time t0, no inflow after t0, QZ(t) = 0 for t ≥ t0 homogeneous differential equation, m ≠ 1) 1

m − 1 t  m −1 m −1  QA ( t ) =  QA ( t0 ) − ⋅  m k  t0 QA(t0)

[h] 3 [m /s]

for m ≠ 1

(5.40)

start, t ≥ t0, outflow at time t0, initial condition.

For m = 1 we obtain the single, linear reservoir (see Chapter. 5.3.1).

5.3.3

Linear reservoir cascade

The reservoir cascade is a series connection of reservoirs. The simplest form is the linear cascade, a series connection of n equal linear reservoirs, each with the same storage constant k (see Figure 5.10). For the concentration of outflow, the linear reservoir cascade is also known as Nash-cascade (named after the English hydrologist Nash). The Sowjet scientists Kalinin

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and Miljukov applied this concept to a streamflow model (in the general form the series connection of non-linear reservoirs, see Chapter 7.4.3). Outflow may be computed step by step by the use of the working equation of the finite difference method (linear single reservoir , see Chapter5.3.1).

1st linear reservoir time t [h] 2nd linear reservoir time t [h] 3d linear reservoir time t [h]

nth linear reservoir time t [h]

Figure 5.10: Linear reservoir cascade, development of the equilibrium equation in case of increasing number of reservoirs A more advanced method is based on the systemtheoretical approach. The series connection of linear, time-invariant partial systems leads to a single combined, linear, time-invariant system (see Chapter 5.3.1), the linear reservoir cascade features the same properties. Therefore a weighting function (instantaneous unit hydrograph) g(0,t) can be determined.

1 t g ( 0, t ) = ⋅  k ⋅ ( n − 1) !  k  g(0,t) k n e

[1/h] [h] [1] [2.7183]

Note that (0)! equals 1.

n −1

⋅e

−t

k

weighting function of the linear reservoir cascade, storage constant, number of reservoirs, (n = 1 equals an single linear storage), Eulers constant.

(5.41)

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The weighting function g(0,t) of a linear reservoir acquires its peak at the time t = 0 and then falls exponentially (see Figure 5.10). With an increasing number of reservoirs the retention effect increases. The peak flow decreases and shifts further to the right. Since the procedure can be regarded as reservoir retention, the vertex or peak lies on the recession limb of the inflow hydrograph. Integration leads to a superposition system operation function and the ∆tweighting function g(∆t,ti) (see Chapter 4.4). −

ti −∆t k

k i ⋅ ( ti − ∆ t ) e g ( ∆ t , ti ) = n ⋅∑ k ⋅ ∆t i =1 ( n − 1)! g(∆t,ti) [1/h]

n

n −1

−

ti k

n −1

k i ⋅ ti − n ⋅∑ k ⋅ ∆t i =1 ( n − 1) ! e

n

(5.42)

∆t-weighting function of the linear reservoir cascade

The concepts of factorial, which initially was defined for integer, positive figures n, can be generalized for any real figure x by means of the gamma function.

( x − 1)! = Γ ( x )

(5.43)

Employing the gamma function in the weighting function of the linear reservoir cascade provides a model that allows non-integer numbers of reservoirs.

1 t g ( 0, t ) = ⋅  k ⋅ Γ ( n)  k  n g(0,t)

[1] [1/h]

n −1

⋅e

−t

k

(5.44)

number of reservoirs, weighting function of the linear reservoir cascade.

In this form the weighting function of the linear reservoir cascade corresponds to the density function of two-parametric gamma-distribution (see Lecture "Hydrologic simulation methods"). Again, integration to determine the superposition system operation function and the ∆t-weighting function is not possible. For this reason, either tables or computer routines of the "unfinished gamma-function" p(x) (1-parametric form) must be developed with substitution to a 2-parametric form. Substitution: x = t

h (t ) = ∫ 0

t k

1  t′  ⋅  k ⋅ Γ (n)  k 

n −1

⋅e

where p(x) selected from tables:

−

t′ k

⋅ dt ′ = p ( x )

(5.45)

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p ( x) = ∫ 0

x′n−1 ⋅ e − x′ dx′ Γ (n)

(5.46)

integral of the unfinished gamma-function (probability, that x is not equaled or exceeded).

p(x)

5.4

Hydrology III - 61

Parameter estimation for simple conceptual models

5.4.1

Moment method for linear model concepts

Moments are characteristic values of functions that are especially suitable for the description of time-dependent functions (see Lecture "Hydrologic simulation methods"). In the case of linear systems, certain parameters of the in- and output variables of the moments for the weighting function exist and may be used to estimate the model parameters. For the rth moment µ’r,f of a time-dependent function f(t) related to zero (t0 = 0) (initial moment) the indefinite integral ∞

µ′r , f = ∫ f ( t ) ⋅ t r ⋅ dt

(5.47)

0

where

µ’r,f f(t)

rth moment of the function f(t) related to zero, any time-dependent function

holds true. The functions are limited to these for which: f(t) = 0 for t < 0 and f(t) →€0 for t →€∞. The hydrographs of effective rainfall and direct runoff meet these requirements. The initial moment µ’0,f of the function f(t) represents the area below the function (e. g. the volume of effective rainfall or of the direct runoff). ∞

∞

µ′0, f = ∫ f ( t ) ⋅ t ⋅ dt = ∫ f ( t ) ⋅ dt 0

0

(5.48)

0

The center of mass ts,f of the function f(t) equals the ratio of the first and the initial moment.

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ts , f =

µ1,′ f µ′0, f

[s,h,d]

ts,f

(5.49) temporal center of mass of the function f(t)

These moments are related to the center of mass ts = µ’1 /µ’0 and are referred to as central moments µr. The first central moment µ1 is equal to zero by definition. ∞

µr, f = ∫ 0

µr,f

 µ1,′ f f (t ) ⋅  t −  µ′ 0, f 

r

  ⋅ dt 

µ1,′ f = 0

(5.50)

central moment of the rth order of the function f(t) (related to the temporal center of mass)

The second central moment µ2f derives from the initial moments. µ 2, f = µ′2, f − µ′21, f

(5.51)

For linear system the following relations are valid. For r = 1:

(relation of the initial moments)

µ1,′ v µ1,′ g µ1,′ u = + µ′0,v µ′0, g µ′0,u

(5.52)

µ1,′ g µ1,′ v µ1,′ u = + µ′0, g µ′0,v µ′0,u

(5.53)

ts , g = ts , v − ts ,u

(5.54)

th

where the r initial moments of

µ'r,u µ'r,v µ'r,g

the input function u(t), the output function v(t) and the weighting function g(0,t).

The temporal centers of mass of ts,u ts,v ts,g

[h] [h] [h]

the input function u(t), the output function v(t) and the weighting function g(0,t).

The center of mass of the weighting function corresponds to the difference of the centers of mass for the in- and output functions.

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The area below the weighting function i. e. the moment of the 0 order has the value one. Since the in- and output volumes are assumed equal, the results may be simplified further. Where: µ´0,g = 1 and µ´0,u = µ´0,v

µ1,′ g =

1 ⋅ ( µ1,′ v − µ1,′ u ) µ′0,v

(5.55)

For r = 2: (relation between the central moments) µ′2,v µ′2,u µ′2, g = + µ′0, v µ′0,u µ′0, g

(5.56)

Estimation of parameters according to the moment method assumes that the model is suited to the hydrologic system as soon as the empirical moments mr,g of the model correspond to the theoretical moments µrg of the model. Moments are termed empirical if they derive from observed values. µ′r , g = mr′, g

µr,g mr,g

or

µ r , g = mr , g

(5.57)

theoretical moment of the model weighting function of rth order, empirical moment of the weighting function of rth order, computed from the moments of observed in- and output variables.

The number of required moment conditions corresponds to the number of model parameters. The theoretical moments of model concepts derive from the respective weighting functions. Therefore a relation between the parameters of the weighting function and the moments can be established. An essential disadvantage of the moment method is the sensitivity of the high order moments. In the equation that determines the moments, the term f(t)⋅tr can be regarded as the weighting of the actual value of the function f(t) with the time-factor tr. As the time t elapses, the weighting of the function is constantly increasing, i. e. that the last part of the hydrograph is overemphasized in comparison to the beginning. This effect increases as the order of the moment increases. The recession limb of an outflow hydrograph, for example, has larger effect than the crest, whereas the aim of the model is more focused on the peak flow. Marginal variations of the observed values may cause substantial variations in the moments. This disadvantage of the moment method occurs from the second order on. However, this is managed quickly and simple, which suggests it is good as a rough parameter estimator. The method provides good results when only the first moments are required for the model.

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The computation of empirical moments is dependent on the form of the discrete hydrograph, either in the form of a polygonal line or as a histogram. The subsequent approximations are usually of sufficient accuracy. •

Hydrograph in the form of polygonal lines with discrete values, the first and the last value equals zero (e. g. direct runoff hydrograph Qd(ti)).

f ( t0 ) = f ( tn ) = 0 n −1

mr′, f = ∑ f ( ti ) ⋅ tir ⋅ ∆t

(5.58)

or

i =1

n −1

mr , f = ∑ i =1

•

 m′ f ( ti ) ⋅  ti − 1, f  m0,′ f 

r

  ⋅ ∆t 

Histogram (e. g. effective rainfall)

(

n

mr′, f = ∑ f ( ti ) ⋅ ti − ∆t i =1 n

mr , f = ∑ i =1

mr,f f(ti) ∆t n

) ⋅ ∆t r

2

 m′ f ( ti ) ⋅  ti − ∆t − 1, f 2 m′  0, f 

[s,h,d] [1]

or r

  ⋅ ∆t 

(5.59

rth empirical moment of the discrete function f(ti), discrete, time-dependent function, time interval, number of intervals of the discrete function.

For the more important model concepts the relation between moments and parameters have been determined.

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•

linear translation (see Chapter 5.2.1) ′ = Tt µ1,g

•

(5.60)

linear single reservoir (see Chapter 5.3.1) ′ =k µ1,g

•

Hydrology III - 65

(5.61)

linear reservoir cascade (see Chapter 5.3.3)

µ1,′ g = n ⋅ k

1st central moment

µ′2, g = n ⋅ k 2

2nd central moment

5.4.2

(5.62)

Storage-outflow relation of single reservoir models

If for an single reservoir the in- and outflow hydrographs QZ(t) and QA(t) are known, the storage hydrograph S(t) can be computed. t

t

t0

t0

S ( t ) = ∫ QZ ( t ′ ) ⋅ dt ′ − ∫ QA ( t ′ ) ⋅ dt ′ + S ( t0 )

(5.63)

= VZ ( t ) − VA ( t ) + S ( t0 ) [m3] [m3/s] [h] [m3]

S QZ, QA t0 V Z, V A

storage volume, in- and outflow, start time, in- and outflow volumes.

If the outflow hydrograph is available in discrete form, the outflow volume V at the points of time ti may be determined. •

histogram (e. g. effective rainfall), i

V ( ti ) = ∑ Q ( t k ) ⋅ ∆ t

(5.64)

k =1

•

discrete values, trapezoidal approach (e. g. outflow hydrograph), i

Q ( tk −1 ) + Q ( tk )

k =1

2

V ( ti ) = ∑

⋅ ∆t

(5.65)

For a reservoir with an initial condition of S(t0) = 0 the discrete storage hydrograph S(ti) is:

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S ( ti ) = VZ ( ti ) − VA ( ti )

(5.66)

storage content S [m3]

At any point of time ti the storage S t(t) can be plotted versus the reservoir outflow QA(ti) (see Figure 5.11).

outflow QA [m3/s]

Figure 5.11: Determination of the storage constant k of a linear reservoir from the storageoutflow relation at the points of time ti If the mean variation is only marginal, the storage-outflow relation can be represented by an equalizing curve. For a linear reservoir, the relation is of a straight-line form and has the slope k. S ( t i ) = k ⋅ QA ( t i )

(5.67)

The fitting can either be accomplished graphically or by means of regression. This method to determine the parameters is best be applied for exponential reservoirs (see Chapter 5.3.2). In this case, there is a straight-line relation between the logarithmic values of in- and outflow (non-linear regression).

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S = k ⋅ QAm

or

Hydrology III - 67

ln S = ln k + ln QA ⋅ m

ln ( S ( ti ) ) = ln k + ln ( QA ( ti ) ) ⋅ m

5.4.3

(5.68)

Outflow recession curve of the linear, single reservoir

If the inflow ends at a given time t0, a drainage process results. With this type of reservoir the outflow features a specific hydrograph (recession curve). For the linear reservoir, the recession curve (see Chapter 5.3.1) is of an exponential form.

QA ( t ) = QA ( t0 ) ⋅ e t0 t k

[h,d] [h,d] [h,d]

−( t − t0 )

k

(5.69)

time when inflow ends, time variable, in this case only time intervals, no inflow, t > t0 storage constant of the linear reservoir

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precipitation intensity iN [mm/h]

areal precipitation iN (t)

outflow Q [m3/s]

outflow Q (t)

time t [h]

recession curve 1 recession curve 2

time t [h]

outflow Q [m3/s]

logarithmic outflow ln Q

outflow in logarithmic scale

time t [h]

Figure 5.12: Determination of the storage constant of a linear reservoir from the logarithmic outflow curve. Here: Example of a rainfall-runoff system, hydrographs of the precipitation intensity, the outflow and the logarithmic outflow The logarithmic form provides a straight-line relation. − ( t −t 0 )   ( t − t0 ) k ln QA ( t ) = ln  QA ( t0 ) ⋅ e  = ln QA ( t0 ) − k  

(5.70)

Consequently the outflow recession curve of a linear reservoir plotted on logarithmic paper is a straight line (see Figure 5.12). This characteristic may be used to determine the storage

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constant. The time intervals where no inflow occurs must be known. The storage constant k derives from the ratio

k=

t − t0 ln QA ( t0 ) − ln QA ( t )

(5.71)

(see Figure 5.12) For a rainfall-runoff system the precipitation intensity, the outflow hydrograph and the logarithmic outflow hydrograph are plotted (see Figure 5.12). Only if the logarithmic recession curve is of a straight line, relations can be established for parts of the hydrograph. Since the storage constant is determined by the recession curve, this method to estimate parameters is especially suitable for low-flow models that represent the development of flow during periods of drought.

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6

Combination of model concepts in drainage basin models

6.1

One-component models for direct runoff

6.1.1

Clark model

Clark put the time-area diagram and the single linear reservoir in series. The outflow hydrograph of the time-area diagram (see Chapter 5.2.4) is the inflow hydrograph for the single linear reservoir (see Chapter 5.3.1). The model considers the translation - and the retention effect of a drainage basin (see Figure 6.1).

effective precipitation iNe * AE Figure 6.1:

time time-area - area diagram diagram

linear reservoir

direct runoff QD

Clark model, series connection of the time-area diagram and linear reservoir

The weighting function of the Clark model g(0,t) derives from the superposition of the timearea diagram and the weighting function of the single linear reservoir. Integration is possible because w(t) remains constant through the time intervals (see Figure 5.8c). g ( 0, t ) =

w(t) k g(0,t)

t

−t ′ 1 −t k ⋅ e ⋅ ∫ w ( t ′ ) ⋅ e k ⋅ dt ′ k 0

[l/h] [h] [1/h]

(6.1)

time-area diagram storage constant weighting function of the Clark model

The Clark model contains two parameters, the travel time Tt (expressed in the isochrone plan) (see Figure 5.7) and the storage constant k. In addition to single linear reservoirs, non-linear reservoirs or reservoir cascades may be used.

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6.1.2

Two-reservoir-model (Singh’s model)

This model consists of two single linear reservoirs in series. The two storage constants k1 and k 2 characterize the detention storage and the retention effect within the receiving streams (see Figure 6.2).

effective precipitation iNe * AE Figure 6.2:

linear reservoir k1

linear reservoir k2

direct runoff QD

Two-reservoir-model, series connection of two different linear reservoirs k1 and k2

Analogous to the linear reservoir cascade (see Chapter 5.3.3) a weighting function g(0,t) can be determined. 1 g ( 0, t ) = k1 − k2

6.1.3

t −  − kt k2 1 ⋅e − e  

   

(6.2)

Influence of precipitation on the model concept

Usually, a model concept that comprises of only one linear flow component provides too rough of an approximation of the hydrologic system. The outflow that pertains to a short and intensive precipitation event is more distinct than in the case of steady rain. The one-component model, with stated parameters, shows the same system operation for each precipitation event. To consider the changing outflow characteristics, the parameters are regarded as variables of precipitation and antecedent soil moisture. So coefficients and travel times may be related to precipitation intensity and -duration.

6.2

Multi-component models, parallel reservoir cascades

As outlined in Chapter 2.1 the runoff of a drainage basin is made up of several components: groundwater flow, interflow and surface runoff (see Figure 2.1). Surface runoff may again be subdivided according to its origins (paved or unpaved areas). A widely used model concept is the parallel connection of linear reservoir cascades. Each cascade represents one of the runoff components.

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The model is linear if precipitation is distributed among the cascades at a constant ratio. Diskin introduced the model of two linear reservoir cascades and a constant distribution factor.

surface runoff

effective precipitation iNe AE

linear distribution alpha = constant

direct

n1 reservoir intermediate flow

runoff QD

n2 reservoir

Figure 6.3:

Diskin model, parallel connection of two linear reservoir cascades and constant distribution factor

The model encloses five parameters n1, k1, n2, k2 that pertain to the linear reservoir cascades and the constant distribution factor α. For α1 = α and α2 = (1 - α) the weighting function is:

g ( 0, t ) = k1,k2 n1,n2 α α1,α2

[h] [1] [1] [1]

n −1

t

1 − α1 α2 t t ⋅   ⋅ e k1 + ⋅  k1 ⋅Γ ( n1 )  k  k2 ⋅ Γ ( n2 )  k 

n2 −1

⋅e

−

t k2

(6.3)

storage constant of the reservoir cascades number of the reservoirs in each cascade constant distribution factor load factor of each reservoir cascade

Generally, the load of each component depends on the intensity and duration of the precipitation event and the antecedent soil moisture content. As a result, the model of runoff formation must precede the model of runoff concentration. Its task is to perform the separation into components in relation to precipitation. It is not necessary to separate direct runoff and base flow since the concept takes all runoff components into consideration. The input variable is therefore simply areal precipitation iN(t), the output variable the streamflow Q(t).

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areal precipitation iN AE

Figure 6.4:

distribution dependent on input

overall outflow Q

Multi-component model, parallel connection of linear reservoir cascades and separation of precipitation according to precipitation

An example of a multi-component model is displayed in Figure 6.5. It is based on a exponentially falling distribution ratio that derives from the loss rate approach (see Figure 3.4c). Consequently the model is non- linear (see Figure 6.4) and weighting functions may only be determined for the single linear reservoir cascades of the individual flow components.

precipitation intensity i [mm/h]

In addition to the parameters of the single cascades ni and ki those of the distribution ratio must be determined. Calibration is only possible by means of non-linear optimization methods.

iN (t)

surface runoff intermediate flow

losses iV (t)

groundwater outflow

time t [h]

Figure 6.5:

Model of outflow formation, exponentially falling distribution ratio, threecomponent model

In general, the drainage basin model that contains linear reservoir cascades in parallel connection and non-linear component formation provides a very close approximation of the natural hydrologic system.

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7

Flood routing models

7.1

Flood routing

As a flood wave passes through a river reach, the peak of the outflow hydrograph is attenuated since the wave front proceeds faster than the recession limb due to the larger inclination of the water surface. The wave is attenuated and extended (see Figures 7.1 and 7.2). The riverbed constitutes a retention space that lags the flow of water by a certain time (river retention).

water depth W [m]

flood recession (low angle)

r iv e r

drop of wave crest be d

long

flow distance [km]

Figure 7.1:

flood rise (high angle)

Flood wave in a river, principal sketch

short

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river kilometer [km]

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time t [d] Isoplethe and line of temporal succession for Q = constant Line of temporal succession for successive max Q Line of temporal succession for successive max h

Figure 7.2:

Lines of equal flow, flood wave of the river Elbe in September 1890

For stationary flow, a straight water stage-outflow relation exists (discharge curve). In a uniform channel the water stage drop ISp corresponds to the riverbed slope ISo. In the case of a flood wave (instationary flow), however, the rising limb exceeds, and the recession limb slope is smaller than the slope of the riverbed (see Figure 7.3). For a given water level h different discharge magnitudes for rising and recession limb occur.

flood recession

Figure 7.3:

flood rise

Surface slope, flow velocities and the rising and recession limb of a flood wave

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•

stationary flow, uniform river cross-section I Sp = I So = I E = I stat

[1] [1] [1] [1]

ISp ISo IE Istat •

Hydrology III - 76

(7.1) surface slope riverbed slope slope of hydraulic gradient slope of hydraulic gradient for stationary flow

instationary flow, rising limb (see Figure 7.3) I Sp > I So

(7.2)

If the surface slope exceeds the bed slope, the flow accelerates.

vinst ( h ) > vstat ( h )

and

Qinst ( h ) > Qstat ( h ) v Q h stat inst •

[m/s] [m3/s] [m] stationary instationary

(7.3)

flow velocity outflow water depth

instationary flow, recession limb (see Figure 7.3) I Sp < I So

(7.4)

At the recession limb the surface slope is lower than the riverbed slope and flow is attenuated.

vinst ( h ) < vstat ( h ) Qinst ( h ) < Qstat ( h )

and

(7.5)

The water stage-outflow relation for instationary flow is no longer definite. Instead of a uniform water stage-outflow curve (stationary flow) a loop is obtained (instationary flow, see Figure 7.4). The shape of the loop is dependent on the gradient of the individual flood wave. The change in outflow dQ/dt also varies from flood to flood. Especially for rising water stage, the difference to stationary flow is distinct. According to Figures 7.2 and 7.4, peak discharge occurs earlier than the maximum water stage.

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outflow Q [m3/s] water depth W [cm]

water depth W [cm]

Prof. Dr. rer. nat. Dr.-Ing. András Bárdossy

flood recession

flood rise stationary outflow curve time t [h]

Figure 7.4:

outflow Q [m3/s]

Water stage-outflow relation in the course of a flood wave

In contrast to reservoir retention, in river retention the peak of the outflow hydrograph is not necessarily located on the recession limb of the inflow hydrograph. Reservoir retention is therefore only a special case of river retention.

7.2

Simple flood forecasting methods

Flood routing in river reaches intends to answer the question: to which extend do flood waves from drainage basins get attenuated by the storage effect of the reach. Therefore, methods must be determined, that allow flood forecasting in a longitudinal cross-section of the river, based on observed or computed discharge values at inflow gage sites. For flood forecasting, the time and the magnitude of the peak flood flow are of special interest. Observations at one or several gage sites upstream provide information and allow conclusions about the critical cross-section. All the methods described subsequently are easy to handle, but suffer the disadvantage of relatively inexact results.

7.2.1

Gage relation curve

This method determines the relation between the peak flow water stages at different river cross-sections. To apply the method, the following assumptions must be made: Only cross-sections at the same river may be compared, and they may not be located too far apart (causal connection). Only comparable water stages may be considered. Since there is no definite water stagedischarge connection for instationary flow, the method is limited to the peak discharge water stages (dW/dt = 0).

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water depth at gage A WA [cm]

time t

Analysis of flood waves to determine a gage relation curve

HW at C MW at C

water depth at gage B WB [cm]

Figure 7.5:

time t

water depth at gage B WB [cm]

water depth at gage B WB [cm]

water depth at gage A WA [cm]

The analysis of numerous flood discharges eventually provides the gage relation curve (see Figure 7.5). For a given flood event, based on the observations made at gage site A predictions may be made for gage site B downstream. Since the method is very simple, it is especially suitable for approximate real time forecasting.

NW at C

WAmin

WAmax

water depth at gage A WA [cm]

Figure 7.6:

Gage relation curve including the water stage at a tributary (gage site C)

The application of this method is limited to smaller distances. If no definite connection between the water stages at gage A and B can be determined, lateral inflow may be the cause. The water stages of a single tributary can be considered by a number of curves (see Figure 7.6). When plotting the chart, one must consider if several rivers contribute to the discharge of the observed river (see Figure 7.7).

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cross arrows on this line

Figure 7.7:

Gage relation nomogram, prediction of peak flow water stage at gage B based on observations on several gage sites A, C, D

7.2.2

Travel time curve

The travel time curve provides the time of the peak discharge of an expected flood event. To determine the curve, representative flood discharges must be analyzed. To identify the travel time between two river cross-sections, the peak flow must be clearly visible. From the analysis of several flood events the average wave travel times are determined and plotted against the water stage of the observed gage site. (see Figures 7.8 and 7.9). The cross-sectional shape of the river strongly influences the shape of the travel time curve (see Figure 7.8). In principle, the travel time will decrease with increasing water stages due to greater flow velocities. If the cross-section is enlarged the storage effect increases, this again leads to smaller wave travel velocities. (see Figure 7.8, area ).

€

A

B

cross-section between A and B

Figure 7.8:

water depth WA [cm]

The travel velocity of flood peaks is higher than the flow velocity since they proceed with wave velocity.

1

travel time Tl [h]

Travel time curve, influence of the river cross-section on the travel time of the flood peak

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water depth WA [cm]

gage reference curve WB (WA)

water depth WB [cm]

Figure 7.9:

travel time Tl [h]

Gage relation curve and travel time curve to predict magnitude and time of the peak water stage at a critical gage

7.2.3

Prediction of discharge changes

The conversion of water stages to discharge using the stationary discharge curve is not correct in the case of instationary flow. However, the error decreases if the discharge change ∆Q is considered rather than the absolute discharge Q. For short-time forecasting and only small discharge changes the error remains within an acceptable range. The initial conditions for this method are similar to those of linear translation: • •

The travel time Tt of the discharge change ∆Q is considered constant between two crosssections. A discharge change ∆Q at an upstream gage occurs in the same magnitude at a downstream gage.

The discharge changes at the critical gage result from the time lag of the discharge change at an observed gage, not from inflow along the reach (see Figure 7.10).

∆Qvor ( t1 + ∆t ) = Qi ( t1 + ∆t − Tt ,i ) − Qi ( t1 − Tt ,i )

∆Qvor Qi t1 ∆t t1+∆t

[m3/s] [m3/s] [h] [h] [h]

discharge change at the critical gage discharge at the observed gage i time of observation forecasting time period forecasting time

(7.6)

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[h]

travel time from observed gage i to the critical gage

outflow at gage A QA [m3/s]

Tt,i

travel time Tt,A

outflow at gage C QC [m3/s]

here

outflow at gage B QB [m3/s]

travel time Tt,C

critical gage

observation time

prediction time

Figure 7.10: Forecasting based on discharge change (see ground plan Figure 7.5), critical gage B, observed gages A and C, lag of the discharge change ∆QA and ∆QC by the travel time Tt,A and Tt,C (here Tt,A = ∆t and Tt,C > ∆t) The more observed gages are taken into consideration, the more accurate the results get. The overall discharge change of the critical gage is the sum of the single discharge changes. k

Qvor ( t1 + ∆t ) = Qvor ( t1 ) + ∑ ∆Qi ( t1 + ∆t ) i =1

Qvor

[m3/s]

discharge at the critical gage

(7.7)

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∆Qi

[m3/s]

k

[1]

7.3

Hydrology III - 82

discharge change at the critical gage as a result of the discharge change at the observed gage i number of observed gages

Hydraulic approaches to instationary flow

From the hydraulic point of view, flood wave deformation is an instationary flow problem. The governing equations derive from two underlying physical laws: • •

mass conservation (continuity equation), energy conservation (energy equation).

The usual approaches consider 1-dimensional open channel flow: • • • • • •

along the river bed only gravity and friction forces are effective, no oscillating waves that transport energy or momentum occur, only mass-transport waves, evenly distributed flow velocity in each cross-section (1-dimensional flow), changes in the water stage as a function of time can be neglected, no vertical acceleration component, crossfall negligible, no acceleration in cross-direction, roughness coefficient kSt for stationary flow may be applied.

1-dimensional, instationary flow can be represented by two independent parameters, the time t and the flow length x. Dependent parameters are the water depth h(x,t) (or the water stage W(x,t)) and the flow velocity v(x,t) (or the discharge Q(x,t)). To derive the governing equations, we consider a channel sector of infinitesimal length dx (see Figures 7.11 and 7.12). Analyzing this element, the temporal changes d/dt and the changes parallel to the direction of flow d/dx must be taken into consideration. The shape of the channel is assumed to be prismatic Continuity equation The temporal change of volume within the element corresponds to the difference of the inand outflow. dV = QZ − QA dt

(7.8)

If the change of discharge along a longitudinal cross-section of the channel is denoted ∂Q/∂x, for the in- and outflow of an element of length dx (see Figure 7.11a), it is

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QZ = Q

,

QA = Q +

Hydrology III - 83

∂Q ⋅ dx ∂x

(7.9)

Equation (7.9) in equation (7.8) provides:  dV ∂Q  ∂Q = Q −Q + dx  = − dx dt ∂x  ∂x 

(7.10)

Figure 7.11: Principle of continuity at a channel cross-section of the length dx principle sketch a) longitudinal section, discharge change along the channel element b) cross-section in the middle of the channel element and temporal change of cross-section. The volume element dV is determined with the element length dx multiplied by its crosssectional area dA in the middle of the element (see Figure 7.11b): dV = dA ⋅ dx

(7.11)

Due to the temporal change of the cross-sectional area ∂A/∂t, for the temporal volumetric change in an element of length dx

dV ∂ A = dx dt ∂t After canceling down dx and rearranging the formula with A = A(x,t) and Q = Q(x,t) :

(7.12)

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∂ A ∂Q + =0 ∂t ∂ x

(7.13)

Applying the continuity principle, the equation may be rearranged even further: Q ( x, t ) = v ( x, t ) ⋅ A ( x, t ) ∂ A ∂( v ⋅ A) + =0 ∂t ∂x

(7.14)

According to the chain rule of differentiation and with A = A(x,t) and v = v(x,t):

∂A ∂A ∂v +v⋅ + A⋅ =0 ∂t ∂x ∂x A Q V bSp h t v x

[m2] [m3/s] [m3] [m] [m] [s] [m/s] [m]

(7.15)

cross-sectional area flow into the channel element ds volume water level width water depth time flow velocity coordinates in flow direction

In a prismatic channel, the cross-sectional area A and the water level width bSp are directly related to the water depth h: A = A(h)

and

bSp = bSp ( h )

(7.16)

The differential cross-sectional area dA may now be computed from the water level width and the water depth (see Figure 7.11b): dA = bSp ( h ) ⋅ dh

(7.17)

After further rearrangements finally the continuity equation of 1-dimensional instationary flow is obtained. The water depth h(x,t) and the flow velocity v(x,t) are directly related to the flow coordinate x and the time t, the cross-sectional area A(h) and the water level width bSp(h) only indirectly by h.

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∂ h A ∂v ∂h + ⋅ +v⋅ =0 ∂t bSp ∂ x ∂x

(7.18)

Oftentimes it is necessary to apply the water stage W related to a given geodetic datum instead of to the water depth h. W ( x, t ) = h ( x, t ) + z ( x )

ISo W Z

[1] [m] [m]

or

h ( x, t ) = W ( x, t ) − z ( x )

(7.19)

riverbed slope water stage related to any geodetic datum elevation of the river bed

By partial differentiation the continuity equation of the related variables v(x,t) and W(x,t) can be obtained:

∂ h ∂ W d z ∂W = − = + I So and ∂x ∂x d x ∂x ∂W A ∂v ∂ W v⋅ + v ⋅ I So + ⋅ + =0 ∂x bSp ∂ x ∂t

∂h ∂W = ∂t ∂t

(7.20)

Figure 7.12: Principal sketch to determine the energy balance at a river stretch element, length dx Energy balance between two river cross-sections at the distance dx: For free surface flow the principle of energy conservation can be applied (see Figure 7.12)

UNIVERSITÄT STUTTGART INSTITUTE OF HYDRAULIC ENGINEERING CHAIR OF HYDROLOGY AND GEOHYDROLOGY

Hydrology III - 86

Prof. Dr. rer. nat. Dr.-Ing. András Bárdossy

hS + h + hk + ha + hv = const. [m] [m] [m] [m] [m]

hS h hk ha hv

(7.21)

geodetic elevation of the riverbed, water depth, kinetic energy head, acceleration head (deceleration head), head loss due to friction.

I So ⋅ dx + h +

v2 2g

left

v2 ∂h v2 1 ∂v 2g h+ ⋅ dx + + ⋅ dx + ⋅ ⋅ dx + I v ⋅ dx ∂x 2g ∂x g ∂t

side 1 (7.22)

∂

right

side 2

Rearranging the equation under inclusion of the dependent variables v(x,t) und h(x,t) provides

∂v ∂v ∂h +v⋅ +g⋅ = g ⋅ ( I So − I v ) ∂t ∂x ∂x [m] [m/s] [s] [m/s2] [-] [-]

h v t g ISo Iv

(7.23)

water depth, flow velocity, time, gravitational acceleration, riverbed slope, slope of hydraulic gradient.

If the water depth is replaced by the water stage W, the energy equation with the dependent variables v(x,t) and W(x,t) becomes:

∂v ∂v ∂W +v⋅ +g⋅ = − g ⋅ Iv ∂t ∂x ∂x

(7.24)

The slope of the hydraulic gradient can be computed according to Manning’s equation. V and h provide only an indirect dependence.

Iv =

kSt rhy

v⋅ v 4

k St2 ⋅ rhy3

(7.25)

s

[m1/3/s2] [m]

roughness coefficient according to Strickler hydraulic radius

UNIVERSITÄT STUTTGART INSTITUTE OF HYDRAULIC ENGINEERING CHAIR OF HYDROLOGY AND GEOHYDROLOGY

Prof. Dr. rer. nat. Dr.-Ing. András Bárdossy

Hydrology III - 87

The system of partial differential equations from the energy- and continuity principle are known as St. Venant’s equations. The equations are almost linear, inhomogeneous and of the hyperbolic type. The independent variables are x and t, the dependent variables are v(x,t), h(x,t) or W(x,t). For lateral, perpendicular inflow q(t) that remains constant along the river stretch, the equations of St. Venant are transformed to:

v⋅

∂ h A ∂v ∂h q + ⋅ + = ∂ x bSp ∂ x ∂t bSp

(7.26)

∂v ∂v ∂h v + v ⋅ + g ⋅ + ⋅ q = g ⋅ ( I So − I v ) ∂t ∂x ∂x A [m3/(m⋅s)]

lateral, perpendicular flow (mainly inflow), q(x) = const.

outflow Q [m3/s]

q

flo

w

c re di

n tio

x

m [k

(7.27)

time t [h]

]

Figure 7.13: Flow of a flood wave, finite difference grid of the temporal and spatial development, principle sketch The St. Venant equations can only be solved step by step with finite differences. The relevant period of time t and the river stretch x must be subdivided into elements of ∆t and ∆x size (finite difference method according to Lax). The discrete values are computed at the gridpoints of the finite difference grid (see Figure 7.13). This method requires extensive numerical computation and is therefore carried out best with a computer. In addition, exact geometric and hydraulic channel data must be available.

UNIVERSITÄT STUTTGART INSTITUTE OF HYDRAULIC ENGINEERING CHAIR OF HYDROLOGY AND GEOHYDROLOGY

Prof. Dr. rer. nat. Dr.-Ing. András Bárdossy

7.4 7.4.1

Hydrology III - 88

Hydrologic flood routing concepts Basic principles of hydrologic flood routing

Hydrologic concepts usually take advantage of a special case of instationary open channel flow, the reservoir retention. This reduces the conditions that must be satisfied to the equation of continuity. The energy equation is considered in an implicit way by the design of the routing models, the storage function and the outflow retention. Computing instationary flow with the storage equation of reservoir retention offers two major advantages: • •

Similar to stationary flow, a definite relation between discharge and storage capacity exists. The computational steps are relatively easy.

In this chapter, two model concepts will be introduced that differ among other things in their typical field of application. • •

Muskingum-model (mountainous areas and lowlands) Kalinin-Miljukov-method (low mountain ranges and river deltas)

The models are defined by characteristic parameters for each river stretch section. It is assumed that the specific boundary conditions for each event do no influence the parameters. The models are calibrated against observed in- and outflow hydrographs (e. g. Muskingum method) or the parameters are obtained from roughly guessed geometric and hydraulic channel properties (e. g.Kalinin-Miljukov method). For composed channel cross-sections different parameters for the piedmonts and the main flow channel may be defined (different retention effects).

7.4.2

Muskingum-model

The Muskingum model represents the river stretch between two gages by a storage space. This storage space consists of two parts, the prism and the wedge storage (see Figure 7.14).

UNIVERSITÄT STUTTGART INSTITUTE OF HYDRAULIC ENGINEERING CHAIR OF HYDROLOGY AND GEOHYDROLOGY

Hydrology III - 89

Prof. Dr. rer. nat. Dr.-Ing. András Bárdossy

wedge storage S2 = k*x*(QZ - QA)

ch an ne

l s ect ion

channel section prism storage S1 = k * QA

Figure 7.14: Muskingum model, subdivision of a river stretch into prism- and wedge storage • Prism storage S1 For stationary flow (QA = QZ) a definite relation between the water volume stored in the river stretch and outflow exists. The storage space may be approximated by a prism. A linear storage-outflow relation is assumed (linear storage).

S1 = K1 ⋅ QA S1 K1 QA

[m3] [s] [m3/s]

(7.28) prism storage volume prism storage retention constant outflow from a river stretch

• Wedge storage S2 The second storage space represents the momentary difference between in- and outflow. The water level is assumed to be linear in shape. Consequently, a wedge-shaped storage space is formed (see Figure 7.14). The wedge storage overlays the prism storage and acquires either a positive (QZ > QA,, flood rise) or a negative value (QZ < QA, flood recession). Again a linear storage-outflow relation is assumed. S 2 = K 2 ⋅ ( QZ − QA )

S2 K2 QZ

[m3] [s] 3 [m /s]

(7.29) wedge storage volume wedge storage retention constant inflow in river stretch

The overall storage volume S is the total of the two storage volumes.

UNIVERSITÄT STUTTGART INSTITUTE OF HYDRAULIC ENGINEERING CHAIR OF HYDROLOGY AND GEOHYDROLOGY

Hydrology III - 90

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S = S1 + S2 = K1 ⋅ QA + K 2 ⋅ ( QZ − QA )

(7.30)

In the commonly used form of the Muskingum equation, the parameters K1 and K2 are replaced by k and x, since these parameters are more meaningful. K1 = k K2 = x ⋅ k

(7.31)

S = k ⋅ QA + x ⋅ k ⋅ ( QZ − QA ) S = k ⋅ ( x ⋅ QZ + (1 − x ) ⋅ QA ) k x

[s] [1]

retention constant of the Muskingum method weighting factor of the Muskingum method

The weighting factor x ranges from 0 to 1. 0 ≥ x ≥1

(7.32)

For x = 0, the Muskingum-model corresponds to the linear, single reservoir. The case x = 1 must be excluded, since a storage volume S that only depends on the inflow QZ is unrealistic. For practical use, x usually acquires values between 0.2 and 0.4. The value is dependent on the ratio of instationary and stationary outflow depth in the inflow cross-section, or in other words, from the rate of inflow change. For triangular cross-sections, x ≈ 0.3 is proposed. For a given inflow hydrograph QZ(t), the solution of the reservoir retention differential equation is t −t ′

t − Q (t ) − 1 x k ⋅(1− x ) ′ QA ( t ) = A 0 ⋅ e k ⋅(1− x ) + ⋅ Q t ⋅ e ⋅ dt ′ − ⋅ QZ ( t ) 2 ∫ Z ( ) 1− x 1− x k ⋅ (1 − x ) t0 t

(7.33)

The last term in the equation reduces for small values of t the outflow QA(t) to a magnitude lower than the initial value QA(t0). This means that the outflow hydrograph decreases at the beginning of a flood event and constitutes one of the disadvantages of this method. If a discrete inflow hydrograph QZ (ti )is given by discrete values, the finite difference approach is applied. If the in- and outflow hydrographs are known (point values and linear interpolation), the reservoir retention equation in finite difference form is obtained (trapezoidal approach). The time interval ∆t must be substantially smaller than the retention constant k.

UNIVERSITÄT STUTTGART INSTITUTE OF HYDRAULIC ENGINEERING CHAIR OF HYDROLOGY AND GEOHYDROLOGY

Hydrology III - 91

Prof. Dr. rer. nat. Dr.-Ing. András Bárdossy

∆t