49Surface Hydrology

Arbeitsbereich Wasserbau Prof. Dr.-Ing. E. Pasche

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Master Program Environmental Engineering

SURFACE HYDROLOGY

Summer semester 2002/2003

Chapter 1- Introduction

1.1 Importance of Freshwater

The quantity of water available to the Earth is constant, water is only flowing throughdifferent phases of the hydrologic cycle and its physical conditions are changing. Thefollowing changes are to be considered:

- Arial and temporal changes of water quantity - Arial and temporal changes of water quality

On the other side, the water demand is growing as a consequence of the followingtrends:- continuous population growth (app. 6 billion people)- increase of the specific water consumption due to the higher life quality- increase of consumption by progressing global industrialisation.

The consequence: from 1900 till now the domestic water consumption has grownmore than 10 times. The water consumption for industry has almost doubled. Besidethe withdrawal due to consumption, the water losses have been increased due to theirrigation and higher evaporation form water storage (1.800 km³/a)

As the water demand and its availability are not distributed evenly over time andarea, it is necessary to transfer water and provide it for the following purposes.

Surface Hydrology-Introduction

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- Nourishment- Industry- Food Industry- Energy source

1.2 Water cycle and Water balance

Water cycle (also knows as the hydrologic cycle) is the idealised form of watermovement and its recycling on the earth.

Water is constantly being cycled between the atmosphere, the ocean and land. Thepower of the sun together with the gravitation is driving this cycle and is responsiblefor continuous changes of the physical conditions of water as well as . The watercycle is composed of the following processes:- Evaporation from oceans and surface- Advection of the water vapour- Condensation and precipitation- Back flow of the precipitation as surface and underground discharge

Water balance equation is expressed as:

N = A + V +/- �S

Where is:

N: Mean precipitation height of the catchment area [mm]A: Mean discharge [mm]V: Evaporation [mm]�S: Storage change in the catchment [mm]

Hydrologic cycle is illustrated in Figure 1.1.


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Figure 1.1 Hydrologic Cycle (source: Michael E. Ritter, University of Wisconsin)

1.3 Tasks of the Hydrology and Water Management

Definition Hydrology is considered as a natural science because it is concernedwith the class of natural phenomena governed by general laws.Hydrology is science that treats the waters on the Earth, theiroccurrence above, on and beneath the surface, circulation anddistribution, their chemical and physical properties and their reactionwith the environment including their relation to living things.It is of essential importance to understand water circulationprocesses and interrelations between them, as well as our ownimpact on water use in order to achieve a sustainable balancebetween protecting ecosystems and meeting human needs Ashydrology deals with those processes, is of the great importance forWater Resources Engineering and sustainable management.

Definition Water Resources Engineering (Water Management) is thedetermined order of all human effects on the aboveground andunderground water regarding quality, quantity and biology (DIN4049). Water Resources Engineering deals with the water use byhumans as well as with the water protection. Further, it can bedivided into Qualitative and Quantitative WRE. This coursepredominantly deals with the Quantitative WER. The followingtasks are to be considered in the Water Management:


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- monitoring of quantity and quality of the water- yearbooks- water quality reports

- setting up statistics- derivation of calculation parameters- estimation of frequency

- forecasting of discharge water level and quality to the purpose of the floodprotection and the risk estimation

- by regression analyses - by process modelling

- Management of Reservoirs for the water supply to the flood protection and forpower production

- Assessment of the available water- Design of the reservoirs- Presentation of case scenarios- Operation and monitoring the of the systems using graphical tools

(cumulative line method or mathematical models e.g. rainfall runoff models)

- Storm water management „on spot“ and streets runoff

- determination of the discharges parameters- setting up drainage concepts- monitoring and operation of drainage mechanisms

Integral approach in river basin management. Additionally, setting up water savingconcepts, in which the water use is regulated, e.g.:

- quantity and quality of the wastewater discharge- water withdrawals - flood protection measures- definition of official and natural flood areas (boundaries)- agricultural use of water depending on type and extent of use - usage of pesticide and nitrates in agriculture- irrigation- setting up the concepts for erosion protection of surfaces and water courses

Generally, different managerial tools are used for setting up the concepts in riverbasin management (e.g. river basin models, material transfer models).


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1.4 Importance of modelling in Water Management

In order to accomplish the tasks of Water Management described in the section 1.3,mathematical models are widely used.One differentiates between stochastic models (inductive) ,that only take into accountthe random aspect of natural phenomena and deterministic models (deductive) thatassume that all physical, chemical, and biological parameters of a watershed areknown. They can be further subdivided according to their characteristics(STÖDTER):

Block model: input and output values are determined only by measuring data andexpressed in mathematical algorithms.

Detailed model: based on the physical interrelations between initial conditions andoutput values.(e.g. rainfall runoff models).

Unstructured model: Catchment are is not divided into subcatchments.

Structured model: Catchment area is divided into subcatchments. Output parametersare calculated separately for each of those subcatchment.

Short term model: the simulation of rainfall runoff processes is limited to floodsimulation.

Long term simulation: Rainfall runoff model are conceptualised for the calculationover longer period (water balance models, low water models)

The above mentioned models are used in Water Management in order to solve thefollowing problems:

a) Rainfall runoff model (short term simulation)- design of dams and flood control storage (retention basins)

b) Operational models (block model, water balance model)- Intake for potable water from retention basins- Intake from rivers- Optimisation of operation of a dam

c) Long term simulation model (continuity models, water balance models)those models describe the overall water balance of a selected catchment. They aresuitable for the following tasks:

- low-water analysis- high- water analysis- change of the water regime by groundwater use


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- change of the water regime by agriculture (sealing of the surfaces)- effects of the retention and climate changes on the discharge in the waters

Beside the quantitative models, the mass transfer models are also used. They quantifythe mass flux through the water body.

One differentiates:

- erosion models: calculate the planar soil erosion, e.g. onagricultural area.

- mass transport models: calculate diffusive nitrate and phosphorus masstransport in the catchment area

- hydromorphologicalmodels:

calculate sediment transport in a water body.

Those models are used in order to quantify the impact of the diffuse mass transport tothe waters and to assess the effectiveness of the applied sanitation measures.

This course predominantly deals with the rainfall runoff models, their scope ofapplication, and especially their contribution to efficient flood management as well astheir applicability for the low water calculations.Regarding flood management, the following tasks are to be considered by WaterResources Authorities:

Securing Flood Protection in Urban Areas- No increase of peak flood flow- Preservation o flow capacity- Determination of inundation areas- Resistance to flooding

Modern flood management implies Landuse and River Management and favoursnatural retention against technical flood protectionIn order to achieve high level of flood protection, it is necessary to have reliable andthorough hydrologic data.

- Determination of flood probability- Determination of flow from storm water network- Determination of flood stages- Determination of inundation areas

Better understanding of flood management measures for:


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- Efficiency of retention measures in watershed area- Unpavement of sealed urban areas- Retention by detention ponds- Extensive use of arable land

- Efficiency of retention measures along the river- Restoration of natural rivers- Restoration of natural flood plains- Flood retention reservoirs


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SURFACE HYDROLOGY


Chapter 2- Hydrologic Cycle

2.1 General considerations

The hydrologic cycle is the most fundamental principle of hydrology. It is theidealised form of water circulation and its recycling on earth.

Water cycle is continuous process by which water is transported from the oceans tothe atmosphere to land and back to the sea. Because the total quantity of wateravailable to the earth is finite, hydrologic system can be looked upon as closed. Inthat system, the processes, such as, evaporation, advection of the water vapour,formation of precipitation and back flow as surface and groundwater discharge, are inequilibrium. This is expressed in the following balance equation:

N = A + V +/- �S

Where is:

N: average height of precipitation in the catchment [mm]A: average height of runoff (discharge) [mm]V: Evaporation [mm]�S: changes of storage in the catchment [mm]

Hydrologic cycle is illustrated in Figure 2.1.

Surface Hydrology-Hydrologic Cycle

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Figure 2.1 Hydrologic cycle (source: Michael E. Ritter, University of Wisconsin)

The concept, that the water cycle is a closed system, is the simplification of the realprocesses in the nature. The hydrologic cycle is made up of many different factors, asa result, it can become quite complicated when trying to analyse the relationshipsbetween those factors. A complete mathematical description of the hydrologicalprocesses is one of the most difficult tasks in engineering and natural sciences andhas not been completely solved yet. It comprises the setting up of a integral modelsystem, in which the components such as global climate model, surface runoff modeland groundwater model of saturated and unsaturated soil layers are coupled together.Considering the fact, that in each of those model components, there are stillunexplained physical processes and lots of natural processes are the small-scale ones,this simplification of the hydrologic cycle becomes reasonable.

Further, a closed mathematical model that describes the water cycle is set and it iscomposed of relevant physical processes in the nature. The processes over the seesurface are not considered. Furthermore, processes like vapour advection orcondensation are also excluded. Those phenomena are meteorological and theirsimulation in Hydrology is not of great importance. But, the important for thissimulation is temporal and arial distribution of the precipitation over the surface. Thisinformation is provided by the weather service or any other meteorologicalinstitution.Finally, considering those modifications and simplification of the real system, onecan distinguish the following processes relevant for the simulation in hydrology:

They are listed in Table 1.


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Table 1. Processes of the Hydrologic Cycle

Rainfall formation

Snow storage/ ablation, -

Interception

Vertical processesSurfacelayer

Surface discharge Horizontal process

Infiltration

Transpiration and Evaporation

Soil/ groundwater storage

Interflow/groundwaterseepage

Unsaturated soillayer(zone of aeration)

Percolation (Groundwaterrecharge)

Vertical processes

Groundwater dischargeSaturated soil layer

Basic flowHorizontal process

Water course Flood wave Horizontal process

As it is shown in the Table above, one can distinguish two types of processes:horizontal and vertical.

2.2 Precipitation (rainfall)

During the precipitation event, 2 types of processes are predominantly occurring overthe surface. Concerning the state of water one can distinguish the following forms ofprecipitation. They are given in Table 2.


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Table 2. Different forms of Precipitation

FormType of formation

liquid solid

Direct condensation or sublimation in theatmosphere ( precipitation from clouds,rainfall)

Drizzle(drops cca 0,5mm)

Rainfall

SnowSnow grainsSnow pelletsGraupelHail (1-6 mm)Hail (ice balls size5-50mm)Iceneedles(1.5mm)

Indirect condensation or sublimation of watervapour close to the surface ( precipitation as condensation)

Dew Frost HoarGlazed frost

In comparison to the other meteorological parameters such as temperature or solarradiation that are relatively constant, precipitation often varies in space and time.

Convective precipitation is the most heterogeneous and is typical of the tropics. It isformed when the air is heated near ground which then expands and rises. It cools asrises and becomes saturated. An example is a summer thunderstorm, that is veryintensive and limited in space (they appear locally).

Cyclonic precipitation is considered as long-lasting precipitation It is formed whencold air mass meets warm air mass. Warm air is less dense and is forced upwardresulting in cooling and precipitation. Cyclonic precipitation can be classified asfrontal and non frontal.Cold advancing fronts move fast bringing intense localised storms. Warm advancingfronts move more slowly creating disperse and less intense precipitation.

Those kind of precipitation are evenly distributed in space. In case of the cold front,the area under the precipitation event reaches 150 km and in case of warm front 650km.

Orographic precipitation, is formed by mechanical lifting of moist air over naturalbarriers such as mountain ranges. It has different intensity and prolongation,depending on the changes in topography. One can distinguish the following intensitygrades and respectively the type of the precipitation:


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Intensity in Description in < 2,5 mm/h Poor rain

2,5 < in < 7,5 Moderate rain

7,5< in Heavy rain

According to the weather service Offenbach, one rain is considered to be heavy if thefollowing condition is fulfilled:

2)24/t(t5N ��

Where is:N = rainfall height in mmt = duration of the rainfall event in min

There is a very important dependence between the intensity and duration of aprecipitation event. The shorter duration of the precipitation event, the more intensiveit is. Contrary, long lasting rainfall has very low intensity.

In order to determine the spatial and temporal distribution of rainfall, the network ofgauging station is established. However, they provide only information about theheight of precipitation at fixed points. In order to achieve accurate estimation of thespatial distribution of rainfall, it is necessary to use interpolation methods, forexample, the Thiessen* method that is considered as the most important inengineering praxis. THIESSEN POLYGON METHOD assigns weight of stationproportional to representative polygon area analogically to Delauny- Triangolation.

The method implies the following steps:1. points are plotted on map2. adjoining stations are connected with the lines3. perpendicular bisectors on lines are constructed4. polygon formed by bisectors gives area (planimeter) are associated with central

location5. rainfall value for gauge is multiplied by area6. all values from (5) are summed and divided by total basin area

An example of spatial precipitation distribution according to Theissen method can beappreciated in Figure 2.1.


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Figure 2.2 Tyssen method

Another example of Thiesssen method is given in Figure 2.2

Figure 2.3 example of Thiessen method


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Although widely used in engineering praxis, this method has its shortcomings. In order to overcome them, more accurate methods are used, in which the whole areis “rastered” and each of those rasters is calculated according to the quadrant method.For each of the four quadrants is the nearest gauging station determined and theprecipitation at the raster point is weighted according to the following relation:

��

��

4

1iNiijNj hwh

where is:ijw = weight of the gauge station regarding the raster point j

Njh = precipitation height at the raster point j

Nih = measured precipitation heights at the gauging station i

In order to achieve a homogenous field, it is necessary to calculate the weight ijw asspatial distance according to the following formula:

��

�� 4

1ii

iij

d

d1w

If the weather conditions differ considerably between gauging station and raster point(e.g. river, mountains) or there are significant topographic changes between them, itis necessary to consider those differences using correction parameters and applyingone of the convenient methods (e.g. Kringing-method).

Still in the testing phase, are the methods that measure arial and temporalprecipitation continuum e.g. precipitation radar.Finally, DWD weather forecast service of Germany and Europe provide with theinformation about the area with the rainfall for 48 hours in advance.The quantitative distribution of this forecast quality value is, however, interestingonly for water management purposes.

2.3 Interception

Precipitation that is falling on the surface , is partly retained by the canopy. Thisstorage of water above the ground surface, mostly in vegetation is called interception.

First of all, the leaves are becoming wet as fine drops are collecting on their surfaceand in the end, the intercepted water is drained from the leaves. The intercepted watercan be drained in different ways. It can drip down from the leaves or can be drainedalong the stem of the plant. Parallel to this drainage process, part of the intercepted


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water is returning back to the atmosphere (interception evaporation). In the beginningthis process is not intensive.Before the precipitation really reaches the ground, it is further partly retained by nearground plants and leaves. This process is similar to the interception of biggervegetation species.

Interception plays an important role in the calculation of water balances in hydrology.Almost one third of the precipitation can be retained as interception and thanevaporate.The storage capacity and consequently the interception capacity, heavily depends onthe vegetation cover. The most important vegetation parameters that shape theinterception are specific area under vegetation cover, lifestage of the plants and thecharacteristics of the land. Interception is highly seasonal dependent (seasonallanduse).

For the maximal interception capacity, the following values are considered :

Deciduous without leaves till 1mmwith leaves till 2mm

Conifers till 9mm

For the calculation purposes of accurate mathematical models, three main

components of canopy interception can be identified

1. throughfall

2. stemflow

3. canopy storage

that part of the precipitation that doesn't reach the ground, because it evaporates fromthe canopy (canopy interception loss- e.g. MEUSER, 1989) and from near-groundplants and leaf (interception loss) or, to a lesser extent, is absorbed by plants.

Under still conditions throughfall and stemflow do not begin until the storagecapacity of the canopy is completely full. Once storage capacity is exceeded, thenadditional water made available cannot be retained, and gravitational effects prevail.Under windy conditions the branches are shaken and drip (throughfall) is enhanced.This renders the mass of water storage variable. After the gust of wind there is someadditional storage capacity available, so the drip rate decreases temporarily.The main processes related to interception are given in Figure 2.4.


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Figure 2.4 Downward losses

Where Pa Actual precipitationTF ThroughfallSF - StemflowI - InterceptionE - Evaporation

Since the necessary data for this differential approach is usually not available,Interception loss is usually calculated according to the treshhold value method. In thismethod, the whole vegetation cover is considered as a storage and the followingcontinuity equation is obtained:

)t(i)t(ET)t(idt

)t(dIna

'n

c��

Where is:)t(Ic = actual content of the interception storage mm

)t(i'n = intensity of the precipitation on the soil mm/h)t(ETa = actual evapotranspiration rate from the interception storage mm/h

)t(in = intensity of the rainfall that reaches soil mm/h

2.4 Snow-hydrological processes

The snow is a form of precipitation and it is formed if the air in a cloud is belowfreezing. The longer temperature stays below zero (expressed as mean dailytemperature), the thicker the snow cover is(accumulation phase).If the temperature rises over freezing point, warm air compresses the snow cover andincreases its compactness. It is about the same effect that is caused by rain.


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Maximal compactness is achieved (cca 45%), if it is not possible to increase densityof the snowpack any more. The excessive amount of water, in form of thawed snow,penetrates the ground (ablation).When the temperatures are above zero and there is no snow cover, the rainfall reachesthe groundwater storage without time delay. Those snow melt processes can bemathematically described by the Snow-Compaction-Method. This method is basedon the physical processes during the snow melting, while the water loss through theevaporation is neglected as the order of those values is too small (0,1 -0,8 mm/d). Inthis method, the influence of wind speed, humidity and temperature corrections,depending on the elevation is considered through averaged parameters.

2.4.1 Accumulation

The increase in the snow depth �l occurs when the precipitation occurs by thetemperatures below zero. As temperature values, the mean daily temperatures areconsidered. The snow depth resulting from the precipitation is calculated by the absolute amountof precipitation and the water content of snow according to the following equation:

100/W)t(P)t(l A��

where is:P(t) : precipitation [mm] and WA : Initial water content in snow [%].

2.4.2 Compression

The compression of the snowpack occurs when the temperatures are above zero.It is assumed that the melt rate and free water from precipitation cause thecompression (i.e. reduction of the snow height) and higher compactness.Procentual height reduction of the snowpack can be calculated according to thefollowing empirical equation (BERTLE, 1966):

WD P4774,04,147P ��

where is:PD : snow height in % of the initial height [%] and

PW : accumulated water of the water equivalent of the dry snow (initial watercontent) [%].

Considering the above mentioned relation, the compaction �h is calculated asfollowing:

)P474,04,147(100/1h100/Phh WADA ��


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where is:

hA : initial snow height [mm] and�h : compression of the snow pack [mm].

The indication of accumulated water PW requires the estimation of the potential snowmelt rate. It can be quantified according to the Temperature-Factor-Method. In thismethod, the snow melt rate is divided into the temperature dependent and thetemperature independent part.The temperature dependent part is proportional to air temperature and the temperatureindependent part is represented by a constant. This constant is to be estimated basedon the average rates of the absorption of the snow pack ,wind speed, relativehumidity and the cloudiness. Further, it is necessary to consider the heat flux that reaches the snow cover byprecipitation. This dependencies is shown in the following equation(MEUSER 1989)

)t(T)t(i0125,0)t(Taa)t(i PTus ��

where is:

is : potential snow melt rate [mm/h],au : temperature independent melt rate [mm/h],aT : temperature dependent melt rate [mm/h/oC], with the plausibility area

[mm/h/oC],T : positive air temperature (Consideration: precipitation temperature is

equivalent to the air temperature) [oC] andiP(t) : precipitation intensity [mm/h].

Accumulated snow water content PW is dependent on the snow melt rate, according tothe following equation:

)()()()(

tWtPtWtP

rW

��

where is:W(t) = absolute water content in the snow pack [mm],�t = time between the simulations [h] and

t)t(i)t(W)t(W sr ��

Wr(t) = water content reduced by portion of melted snow


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2.4.3 Ablation

Snow ablation usually refers to removal by melting.Water removal out of a snowpack occurs if compactness Pw(t) is higher that maximalcompactness Pmax of a snowpack. This water removal can be calculated as following:

t/)100/Ph)t(W()t(i maxA �� , falls maxPh)t(W ��

where is:iA : water removal in [mm/h],Pmax : maximal snow compactness [%] (i.d.R. 45%) andH : snow depth [mm].

This amount of water that is removed from the snowpack is used as output load forthe soil water. For example, if the snow depth is 0 and the temperature is above 0,there is no storage in “snow storage”.

2.5 Evapotranspiration

The physics of the evaporation process is based on the process of providing sufficientenergy for breaking the bonds between water molecules. Supplying the system withheat, causes water molecules to become increasingly moveable which results inincrease in distance between them. The higher temperature is ,the more watermolecules escape the surface into the lower layers of the air. The physical cause forthis phenomenon is Brownian movement.

Evaporation can be defined as the process where liquid water is transformed into agaseous state.For changing the aggregate state from liquid to gaseous, it is necessary to providehigher amount of energy (2450 J/g) then for the melting process (340 J/g).There are two processes considered for evapotranspiration: evaporation andtranspiration. Evaporation is loss of water from a wet surface through its conversationinto its gaseous state. Surface can be bare soil (soil evaporation), open water(including river, lakes and oceans) or intercepted water held upon plant surfaces(interception evaporation).Transpiration comprises water taken up by plants from soil which is then moved upto leaves and lost through biological processes in vegetation (DIN 4049,1994).

The sum of soil, interception and open water evaporation and transpiration is calledevapotranspiration.


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One should distinguish between maximal(potential) and actual evapotranspiration.

Maximal (potential) evapotranspiration ETp is by definition the rate ofevapotranspiration from an extended surface of an 8 - 15 cm tall green grass cover,actually growing, completely shading the ground and not short of water (DIN4049,1994). This maximal evapotranspiation is also called referenceevapotranspiation.

The actual evapotranspiration ETa is by definition “the rate of the evapotranspirationfrom a surface under field conditions and with limited water supply.” Under thoseconditions, the actual evapotranspiration is much less then the potential, especially ifthe soil water storage capacity is limited.

In order to illustrate the ratio and the contribution of the above mentioned processes,an example for Germany is given. The evaporation of free water bodies is 2.2%,Interception 16.0% and transpiration 72,6% of the total evapotranspiration.According to KELLER,1979 the amount of total evapotranspiration to the amount ofprecipitation is 64%.At the global scale, evapotranspiraton and precipitation are principal elements of thehydrologic cycle.Rate of evapotranspiration depends on the different parameters (arial and temporalvariable). Those parameters are defined considering atmosphere, soil and type ofvegetation cover.While it is possible to assess accurately those parameters for open water andatmosphere, it is very difficult to quantify forces in soil and vegetation cover.In soil, for example, they depend on radius in capillaries and in vegetation, thisprocess is regulated through small pores on the leaves (stomata).Soil, plant and atmosphere form parts of a continuos flow in which water moves atvarying rates. In soil, water moves under the influence of moisture gradient towardthe roots of the plant. It is then absorbed and travels up the plant stem to the plantleaves from where it is finally vaporised.Plants can control transpiration by varying the opening of stomata. In that way theycan prevent dehydration.

2.5.1 Methods to determine evapotranspiration

Despite the crucial importance of evapotranspiration in hydrological cycle, it is verydifficult to measure it and quantify.In order to assess evapotranspiration, the following methods are applied:- direct methods- indirect methods


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- computational methods

Direct measuring of evapotranspiration is not possible, as such a method has tomanage turbulent mass transport of water vapour, which technically has not beensolved yet. Measuring techniques, such as evaporimeter, are although considered asdirect methods, can give only reference values. In this method, evapotranspiration iscalculated from water balance equation, where the other elements (variables) of theequation are measured.

Indirect methods are based on the interrelation between directly measuredmeteorological parameters and water vapour transport or heat flow of the evaporationprocess in air layer close to the surface.

As the measuring methods that are mentioned above are very complex, computationalmethods, are developed. Computational methods calculate evapotranspiration usingenergy budget equation and aerodynamic principles.

2.5.1.1 Direct Measuring

The results obtained by the instruments for direct measuring of evapotranspiraton(Evaporimeter) do not coincide with the real potential or actual values as those resultsdepend on the type of the instrument and local conditions (place the equipment islocated). It is however possible to calculate reference values using correction factorsand formulas.There are two types of instruments - atmometer- atmograph

Atmometer: with daily readout and atmograph with continual registration areequipped with standardised glass cylinder (PICHE atmometer) with inner diameter of11mm and outer of 14mm. This atmometer uses a filter paper disc or fleece paper asthe evaporating element. It is pressed against the bottom opening on the device with aspring clip. The amount of water evaporated through the paper is read at thegraduated tube reservoir (in ml) and this value is to be corrected as a function of thefilter paper size.

Due to the different physical conditions the rate of evaporation of the PICHEevaporimeter is higher than actual evaporation of free water or bare soil surface.Therefore this conversion is necessary.

Evaporation pans

Evaporation pans are open vessels, that are filled with water. Because of its simplicityit is probably the instrument used most widely to estimate potential evaporation. The


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values for water loss obtained for an evaporation pan are used as reference values forwater bodies or potential evapotranspiration (ET0).Although this method is very simple and widely used it has shortcomings which areto be considered. The evaporation from a pan can differ significantly from that froman adjacent water body or land covered with vegetation. Therefore it is necessary toaccommodate these differences using empirical pan coefficients. Those coefficientsvary significantly with siting and pan design as well as with climatic factors.

The most widely used instrument is Class-A-Pan introduced by U.S. Weather Bureau.It consists of the following parts:- cylindrical container with inner diameter of 1.207mm and 255mm depth placed on

a slatted wooden frame so that the top rim is about 10cm from the ground.- mechanism for reading off the water level- swimming minimum/maximum thermometer- instrument for wind speed measuring (anemometer), 50cm over the ground- rain gage

Raft evaporation tanks

The methods that are explained so far, give us reference or approximate values ofevaporation over land.The measurement of the evaporation over free water is done using raft constructions,which serve as carriers for one or more evaporation tanks immersed into the water.Those tanks are of different size, between 0.2 m² and 3 m²in area, and 40-60cm deep.As material brass tank with copper bottom, white-enamelled or with silver, bronzepainted sheet iron, is used.

The results obtained during those measurements should also been corrected. Thiscorrection is between 3-5%

Lysimeter

Lysimeter is an instrument for determination of the water balance of a soil volume,with known dimensions, characteristics and vegetation growing on it. Onedifferentiates between weighing and non weighing lysimeters. In case of weighinglysimeters, which measure the weight in regular time intervals (e.g. daily), the changein water storage is determined by weight difference:

∆W = (WE - WA)

Further, the equation for the actual evaporation (Eta) in function of drainage from thesoil volume (SW) and precipitation (p in 1m height) is derived from water balanceequation:


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tWSWPETa

�

��

Not weighing lysimeter(drainage gauges-measurers) are suitable for determination ofthe average values of evaporation over a longer period of years, calculated as adifference between precipitation and drainage, as in this case the water content of thesoil is balanced and it is not necessary to measure it.

When using big lysimeters e.g. drainage gaugers in the forest area, it is necessary toprovide the natural conditions in the vessel. It is achieved by filling the lysimeter withsoil in layers as they are stratified in the natural compression.

Determination of water content in the soil

Evaporation of bare soil, if the groundwater is far enough from the sample area, canbe determined by termin measurements of water content in soil W. There is a varietyof practical methods for those measurements of soil moisture, from which thefollowing ones are most widely in use:

1. gravimertical methods:soil sample is extracted and in the water content is determinated in the lab bymeasuring and drying the sample. When converting the water content into soilmoisture value, the compaction of the soil layers is to be considered. Shortcoming ofthis method is contained in the fact that the natural soil profile is disturbed(disrupted). The following methods are overcoming this deficiency.

2. tensiometerthis device measures the pressure potential or matrix potential of the at variousdepths. This is the force with which water is held in the soil. Pressure potential ortension (which is the positive value of pressure potential) is a measure of this force. It is then possible to express the soil moisture based on the curve water pressure-water content (pF-Curve).

3. neutron scatteringin this method, the gradient of deceleration of neutrons is measured. This decelerationof neutrons in proportional to the soil moisture.

4. Time-Domain-Reflectometry (TDR)- MethodTDR measures the propagation of an electromagnetic pulse along the transmissionlines (wave guides). By measuring the travel time, the velocity and hence theapparent dielectric constant of the soil can be estimated. This then allows the watercontent of the soil to be determined. This method is limited only to loose rocks. In


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order to determinate the evaporation of the soil, it is necessary to consider followingcases:

If the horizon between increasing water percolation and drainage in deeper soil layers(watershed) is below the soil root layer, the evaporation can be determined accordingto the following equation:

tWPETa

�

��

If the water shed is in the root layer, the plants take up a certain amount of water aswell as a part of ground water dischargeTherefore, total evaporation can be calculated according to the following formula:

� �t

SWWWPET uoa

�

��

where: ∆Wo: change in water content above the watershed∆Wu: change in water content below the watershed

In order to determine SW it is necessary to determine permeability of an unsaturatedsoil layer and afterwards the drainage(infiltration) according to Darcy law.

5. Turbulence-Correlation method (Eddy Flux)

Near the surface, turbulent eddies within the body of the moving air causemovements of the evaporated water. Vertical movement of water vapour can becalculated from the temporal averaging of the products of the fluctuations.For determination of the fluctuations, different measuring instruments are used (e.g.ultrasonic anemometer for wind velocity, LYMAN alpha or infrared hygrometer forhumidity.

2.5.1.2 Indirect methods

- gradient measurements in the near the surface air layer

The assumption for this method is that evaporation causes horizontal and verticalturbulent exchange. It causes the flow of water vapour, heat and impuls and can becalculated applying diffusion equation.

6. Energy Balance (Budget) Method

According to this method, it is possible to calculate evaporation as


- 18 -

-LE = Rn + G + Hwhere:

LE = latent heat flux outRn = Solar energyG = sensible heat flux into ground (QW)H = sensible heat flux

Evaporation height(E) can be calculated as :

E = LE / L

L = specific heat flux (245 J/cm²/mm at 20° C atmosf. temperature)

The individual components of the energy balance equation are determined by thefollowing measuring methods:

7. Solar radiationIt is measured directly by special radiation balance measures with range of 1,5-2,0m. Those devices measure the difference between upwards and downwardsoriented radiation from short and long wave radiation. Alternatively, radiation canbe determined as a difference between other wave lengths.

Measurements with pyranometer (Global radiation)Shortwave solar radiation and sky (diffuse) radiation is detected on the horisontalsensor surface with the termosensitive element.Alternatively, it is possible to measure relative sunshine duration S/S0, where thesunshine duration is registered by the instrument called sushine autograph. Itconsists of a glass bowl with special effect, so that the sunshine is registered onthe adhesive tape.

- Heat Flux GIt is determined with so called ” Heat Flux Plates ” under closed conditions, or bythe gradients of the soil temperature applying the heat conduction equation.

- - perceptible heat flow with application of the fluctuation method, wind and temperature componentsunder dissolution of their turbulent fluctuation sizes are measured and set incorrelation to the perceptible heat flow.

2.5.1.3 Calculation of evaporation

The following methods are considered:


- 19 -

- Aerodynamic method (Dalton-process)- Energy balance methods- Combination of the above mentioned processes

2.5.2 Determination of potential evapotranspiration of an area with vegetation cover

In most cases, only total evaporation from an area plus transpiration (fromvegetation) is of real interest. Penman- Monteith Model is a method that combinesenergy budget and mass transport for determining the evapotranspiration. It assumesthat the heat flux into and out of the soil is small enough to be neglected and that theplants are taking the whole available energy for evapotranspiration.Evapotranspiration is limited by two antagonistic conditions: regarding small poreson leaves-stomata and aerodinamic conditions.If those conditions are known, it is possible to determine potential evapotranspirationET0 according to DVWK, booklet 238 as grass reference evapotranspirationaccording to FAO-Standard:

� �

� �2

s2N

0 V34,01s

e)T(eV273T75,3

LGRs

ET��

��

��

�

with the following terms (according to DVWK-bulletin 238/1996):

ET0 mm Grass-/Reference evapotranspiration (FAO-Standard)s hPa/K Slope of the saturation vapour pressure curve for waterRN

2mW Net solar radiation

G2m

W Heat flux between soil and atmosphere

L Heat required for evaporation of 1mm waterT °C Air temperatureV2

sm Wind speed, at 2m above the ground

�

KhPa Psychrometric constant (= 0,65 hPa/K)

With the following consideration:

L = (249,8 - 0,242 �T) [J / (cm² ( mm)]


- 20 -

In this equation, G can be neglected and � is a constant (� = 0,655 [hPa/K]).Therefore, the equation for saturation vapour pressure eS(T) gives:

��

��

�

�

�

��T273T48,7

s 1011,6)T(e

Further, it allows the slope of the saturation vapour pressure curve for water to becalculated as follows:

� �2S

T12,2434284)T(es

�

��

Net solar radiation RN can be calculated as follows:

� � � � � �5,0

0

4N e044,034,0

SS9,01,015,273TRG1R ��

�

��

��

where is �= 0,49 ( 10-6 [(J : cm² � k4)]�

42KmJoule Stefan-Bolzmann-Constant

� diml. Albedo for grass and agricultural used landS h Sunshine durationS0 h Astronomic possible sunshine duration RG

2mW Global solar radiation

R02m

W Extraterrestrial radiation

� diml. Function considering the selected day in year� Grad Latitude

Where the global radiation is calculated in the following way:

� �5,0

00G e044,034,0.

SS55,019,0RR ��

�

��

��

If the measurements of solar radiation are available, it is possible to use it directly inthe following way:


- 21 -

��

��

��

00G S

S55,019,0RR

Extraterrestrial radiation can be calculated as follows:

R0 = 245 � [9,9 + 7,08 � sin� + 0,18 � (�-51) � (sin�-1)]

Where � is function that considers the day in year JT = � = 0,0172 � JT - 1,39

If there is no available measuring data for sunshine duration, it is possible to calculateit according to the following formula:

��

��

�� 35,0

RR82,1SS

0

G0 with S = 0 for S < 0

Finally, it is necessary to determine the astronomic possible duration of sunshine. Itcan be done according to the following equation:

��

��

� ��

6513,4sin3,12S0

This theoretical analysis clearly shows that , just by measuring air temperature,sunshine duration, wind speed and relative moisture as daily values it is possible todetermine evapotranspiration. The calculated value for reference evapotranspirationET0 is corrected with the correction factor Kc and converted into canopy specificpotential (reference) evapotranspiration ETpfl.

ETpfl = Kc � ET0

Factor Kc can be found in Table 6.2 in DVWK-bulletin 238/1996. In order toconsider forest conditions additional correction factor for higher values ofinterception evaporation of 1,1 is considered. (see DVWK-bulletin 238/1996).Additionally, it is necessary to set the rooting depth as deep or shallow, depending onthe age and type of forest.The important difference between the evaporation from open water and fromvegetation covers is that in open water it is not determinated by the water regime ofthe plants. However, the heat transport occurs also in water body and has to beconsidered for the balance equation.

The evaporation from open water can be determinated according to Dalton’s law.Basic formula for this procedure is:


- 22 -

EW = f(v) � (es(Tw0) - e) in [mm/d]

Where is TW0 [°C] the temperature of water surface and f(v) wind functionexpressed in the following way:

f(v) = a + b � vc

Constants a and b (according to Richter, see DVWK-bulletin 238/1996) arecalculated depending on the wind speed. Temperature of the water surface is rarelymeasured. It is usually calculated as a function of the of the measured mean airtemperatures T [°C] in the following way:

TW0 = p � T* + q (for the period without ice April - December)

Where is:

Tmn

1Tmn

1��

��

�

� (n+m)

and T* [°C] mean at temperature of the day n and number of days before the day n-m, that is calculated as follows:

m = 1,04 � z + 4,0

and depends on the water depth z [m]

For winter months January till March it can be calculated as:

TW0 = 0,98 � T* - 0,12

where m is set to constant = 10

Finally, T* is given as follows:

T111T

111

1��

�� (11)

Empirical constants p and q can be found in the Table 5.2 of DVWK-bulletin238/1996

2.5.3 Determination of the actual evapotranspiration

The potential evaporation of the vegetation cover ETpfl and the soil water tension (reduction of soil moisture in root layer) is to be considered. WENDLING proposesthe following balance equation


- 23 -

EVA = P(t) - (P(t) - ETpfl) � Ri

Where is:P(t) mm PrecipitationRi diml. Reduction function

Reduction factor Ri is calculated as follows:

pfli ET

ETP1,0

WEFKWEWP1

)t(BFWEWP1

R ��

�

��

��

�

Where is:WP % Mean wilting point in effective root layerWE dm Effective root depthBF(t) mm Soil moisture in the effective root layer in period t FK % Mean field capacity in the effective root layer ETP mm/d Mean long-term value of ETpfl in the vegetation period

Further, according to WENDLING (DVWK-bulletin283/196) it gives:Ri = 1, P(t) > ETpflRi = 1, Ri > 1,0Ri = 0, Ri > 0,0


- 1 -


SURFACE HYDROLOGY


Chapter 3- Modelling of the soil moisture regime

3.1 Hydrologic processes in the unsaturated soil layer

Interception splits precipitation into that delivered to the land and water surfaces andthat caught on the canopy and returned to the atmosphere by evaporation.Water delivered to the land surface may run off directly, as overland flow or infiltratethe soil. To which extent the infiltration in the soil occurs, depends on the physicalcharacteristics of the upper soil layer as well as on the actual soil moisture.Reliable quantification of the process of water distribution in the soil, requires soilmoisture modelling. It implies calculation of the actual soil moisture BF(t) usingcontinuity equation for the unsaturated soil layer.

Elements of the vertical water flow can be appreciated in Figure 3.1. In Figure 3.2,different types of soil layer are given.

Surface Hydrology- Modelling of the soil moisture regime

- 2 -

Figure 3.1 Vertical profile soil-vegetation-atmosphere with water balance components (source DVWK)

Figure 3.2 Distribution of the soil layers

For the purposes of mathematical modelling the soil is divided into layers withhomogeneous structure- lamellas (as shown in Figure 3.3).


- 3 -

Figure 3.3 Functional dependence between Infiltration (Inf), Evapotranspiration (Eva), Percolation (Perk)and Interflow (Intf) to soil moisture

Thus, the quantity equation gives:

)()())()(()()( tCUtEvatIntftPerctInf

dttdBF

iiiii

��

where is:BF : Available water (for plants) [mm],Inf : Infiltration [mm/h],Perk : Percolation [mm/h],Eva : actual Evapotranspitration [mm/h] Intf : Interflow [mm/h].CU : Capillary uprise [mm/h]

Analysing this equation, it becomes clear that the soil moisture can be determinedfrom inflow components Infiltration rate (Inf) and capillary uprise (CU) as well asfrom Percolation (Perc) and Evapotranspiration (Eva).Before each of those components is thoroughly analysed it is necessary to give anintroduction to soil characteristics and soil water.

3.1.1 Soil moisture

Definition: Soil moisture is the water content that can be removed via drying the soilsample at 105° C in the oven.Seasonal differences in water inflows and losses in soil are summarised under theterm „soil moisture regime“.

The amount of rainfall that infiltrates into the soil, percolates only partly togroundwater. The rest remains in the unsaturated soil layer i.e remains above thegroundwater level. In the case of saturated soil layer, water fills all available porespace in that layer.Important parameters considering the extent to which the water is attached to the soilparticles and size of pores are :Wilting point (WP), Field capacity (FC), availablefield capacity (aFC), air capacity (AC), total pore volume (TPV) and maximal watercapacity (BFMAX). They are defined according to the BodenkundlicherKartieranleitung, Arbeitsgruppe Boden, 1982 They are also called “soil waterconstants” and have been introduced to facilitate comparisons between differenthydrological status of different soils.

Field Capacity (FC):It is defined as the amount of water that remains 2-3 days after the saturation ofa soil with water (after gravity movement of water has largely ceased). As a


- 4 -

rule, soil moisture tension of 101.8 and 102.5 mbar occurs in soil. It variesconsiderably depending on the clay and silt content, soil texture, content oforganic matter, humus form, coherency of soil and ion characteristics in soilcolloids.Field capacity rate depending on soil texture is as follows:

Sand>Loam>Silt>Clay>Peat

- Permanent Wilting Point (WP)Permanent wilting point (PWP) is the lowest amount of water that is hold in thesoil and which plants are unable to use. Soil water that is incorporated into soilstructure with higher soil moisture tension than 104.2 mbar is not useful formajority of the plants through root system. The remained soil moisture isconsidered as „dead water“.

- Available Field Capacity (aFC): Available Field Capacity is the difference between Field Capacity (FC) andwilting point of a soil. It is considered as water available for plants to extractfrom the soil moisture zone. According to the definition of WP and FC, thevalues for the soil moisture tension (SMT) are between 101.8 and 104.2 .

- Air Capacity (AC):It specifies the size of the drained macro-pore space. The water percolating inthe soil has flown downwards and is not available for the plants any more. Itcoincides with the size of the pore space that is filled with air at the fieldcapacity conditions.

- Total Pore Volume (TVP): The amount of all cavities in soil expressed in Volume-%

- Maximal water capacity BFMAX: Maximal water content of soil (by total water saturation). If it is expressed inVolume-% it coincides with the total pore volume (TVP).

According to BEAR, the following soil water types are considered in the unsaturatedsoil layer. (see Figure 3.4, BEAR).

- Hygroscopic water

Hygroscopic water is held in the soil between air dry and oven dry. Unavailable waterremains when soil is drier than wilting point. Unavailable water is soil waterheld so firmly to soil particles by adsorptive soil forces that it cannot beextracted by plants. Neither evaporation nor percolation into deeper soil layeris possible.


- 5 -

- Capillary water

Soil moisture in this zone is between Wilting Point (WP) and Field Capacity(FK). This water is available for plants and transpiration also occures in thiszone. The capillar forces are held by formation of menisci at the contact pointswith the mineral particles, so that neither percolation into deeper soil layers nordischarge between different soil layers is possible.

- Gravitational water (excess soil water)

This zone is above the Field Capacity. Gravitational water drains or percolatesreadily by gravitational force. Since drainage takes time, part of the excesswater may be used by plants before it moves out of the root zone.Alternatively, this water flows horizontally forming interflow.

Distribution of those zones depending on the soil moisture can be appreciated inFigure 3.4

Figure 3.4 Porosity according to Bear

Within those soil moisture zones, it is possible to simplify the dependencies of theInfiltration, Pecolation, Transpiration and Interflow rates and present them as linearfunctions of soil moisture. It can be appreciated in Figure 3.5.


- 6 -

Figure 3.5 : Infiltration (Inf), Evapotranspiration (Eva), Percolation (Perk) and Interflow (Intf) expressesas linear function of soil moisture

For setting up this mathematical model, the conditional equations for Infiltration,Percolation, Evapotranspiration and Interflow are obtained.They vary between Wilting Point (WP) and Field Capacity (FK) as well as betweenField Capacity (FK) and maximal soil saturation (BFMAX).

3.1.2 Infiltration

Water that falls on the ground in form of precipitation, can either infiltrate or runs offas overland flow. Infiltration rates mostly depend on the soil type and surfacestructure. In case of sealed areas, almost 100% of the rainfall is converted intooverland flow and no infiltration occurs. In case of natural areas, this rate dependspredominantly on the soil structure. It is further shaped by pore volume in the soil. Itis possible to distinguish between 2 types of pores:- Macro(equivalent radius > 3mm)- Micro (capillary system)Different pore types and their spatial distribution in the soil is illustrated in Figure3.6.

Figure 3.6 Pore distribution in the soil


- 7 -

Macro pores are of importance for the infiltration process as water percolates 100-400 times faster through macro pores than through the micro ones. The macro poresare created by plant roots, channels and paths of the soil fauna (e.g. mice, moles) soilaggregation and the soil cultivation. The most important factor that shapes soilporosity created by fauna are earth worms, that make uniform vertical pore system.Beside that, physical soil parameters, such as granulation, particle shape, soilstructure and soil compaction are also of relevance.

As it can be deduced from the figure 3.5, CIN determines the rate of infiltration. Thehigher values for CIN are, the smaller portion of precipitation is converted intooverland flow.Due to the large number of relevant parameters for infiltration ,an accuratedetermination of the infiltration capacity (CIN) is possible only by performing fieldtests.Approximatively, it is possible to estimate the infiltration capacity from soil type,effective compaction according to Bodenkundlicher Kartieranleitung, ArbeitsgruppeBoden, 1982 and hydraulic conductivity obtained for soil saturation conditions kFand presuming loose soil (Ld 1-2)

2/1,FkCIN�

Real infiltration is determined as function of the soil structure, slope inclination andactual water contentIn the zone of Wilting point, this value coincides with the infiltration capacity.It is linear till it reaches the point of maximal soil capacity.

CEX parameter shows the rate of ex filtration capacity (to the layers laying below theobserved one).

Concerning the values for soil moisture portions corrected for Wilting point, potentialInfiltration of the soil Infp, the following conditional equation is obtained:

WPBFMAX)t(BFBFMAXCIN)t(Infp

i

iiii

�

�

��

The actual infiltration Inf depends on the intensity of the rainfall and percolation rateof the soil layers.

for soil layer i=1:)()()()( 1,11 tInfptiiftInfptInf effN ��

)()()()( 1,,1 tInfptiiftitInf effNeffN ��


- 8 -

for soil layer i>1:)()()()( 1 tInfptPerkiftInfptInf iiii ��

�

)()()()( 11 tInfptPerkiftPerktInf iiii ��

3.1.3 Percolation

Percolation is water trickling downward through the cracks and pores in soil andsubsurface material, similar to the infiltration process. However, it reaches maximumat the field capacity conditions. Percolation capacity heavily depends on the soilstructure and texture, air capacity and soil compactness. It can be estimated byhydraulic conductivity and for higher compactness of the soil, this equation gives:

5/4,FkCEX�Considering those assumptions, potential percolation Percp from soil lamella i can becalculated as follows:

��

��

��

�

�

iiFKBFMAXFKtBF

i

iii FCBFifCEX

FCBFiftPercp

ii

ii ,,0

)((

))((

3.1.4 Determination of the actual evapotranspiration

Potential transpiration ETpl should be reduced to actual transpiration ETa where thewater tension (limitation of available soil moisture in effective rooting depth) is to beconsidered. According to WENDLING, this balance equation is expressed as follows:

EVA = P(t) - (P(t) - ETpfl) � RiIt calculates actual transpiration as function of actual soil moisture and vegetationspecific transpiration parameters as daily value continuum.

Where is:

P(t) mm Precipitation Ri diml. Reduction function

Determination of the reduction function Ri can be calculated as follows:

pfli ET

ETP1,0

FKWP1

)t(BFDWTWP1

R ��

�

��

�

Where is:WP % Mean wilting point in the effective root layer


- 9 -

DWT dm Effective root layerBF(t) mm Soil moisture in effective root layer at the tine point t FK % Average field capacity in the effective root layerETP mm/d Average value of ETpfl over longer period of years in

vegetation period

And further, according to Wendling (DVWK-bulletin 283/196) :Ri = 1 for P(t) > ETpflRi = 1 for Ri > 1,0Ri = 0 for Ri > 0,0

3.1.5 Capillary Uprise

Capillary uprise from ground water in the effective root layer close to thegroundwater horizon (level) can be of great importance for water balance assessment.It can be calculated based on the water conductivity and suction power (force) and isexpressed as Capillary Uprise (CU). It varies as a function of the depth of thegroundwater level (GW) below the lower bound of the effective root layer and thesoil characteristic. For sand it the capillary upise reaches 3mm/d if the groundwaterlevel is 5dm below the lower boundary of the effective root zone. If this distance is8dm, capillary uprise is only 0.2mm/d. In order to estimate the capillary uprise rates,die Bodenkundliche Kartieranleitung and DVWK bulletin 238 are used.

3.1.6 Interflow

That part of the precipitation which infiltrates the surface soil and moves laterallythrough the upper soil horizons above the water table toward surface waters. Alsocalled subsurface runoff (see baseflow).

The interflow Intf from the soil lamella i can be calculated as difference betweenpotential percolation Percp and actual Percolation Perc.

)t(Perk)t(Perkp)t(Intf iii ��

In order to consider this lateral movement of water in the soil, this value is multipliedwith the ordinate of the time-area function, that discharges in the time interval ∆t.

)t(IntfA6,3/1)t(QIntf iii ��


- 10 -

where isIntfi : Interflow through the lamella i [mm/h],QIntfi : Interflow through the lamella i [m3/s] andAi: contact surface part [km2], that in the time interval ∆t discharges to the

lamella i

The remaining portion of the interflow contributes to the increase of the soil moistureof the lamellas above the observed lamella i. At the lower boundary it is necessary to correct the value for the soil moisture asthere are no chargeable lamellas bellow it:

��

��

��

iiiiiii

iiiiii BFMAXtIntfAtBFiftIntfAtBF

BFMAXtIntfAtBFifBFMAXtBF

)()1()(),()1()()()1()(,

:)(

If the first condition is fulfilled, then is:

iiii BFMAX)t(Intf)A1()t(BF ��

returned to lamella i-1.

3.2 Mathematical solution to soil water equation

The continuity equation for inhomogeneous, linear differential equation in ist generalform is expressed as:

12 C)t(BFCdt

)t(dBF��

Constants C1 and C2 are calculated in dependence of the Infiltration, Percolation,Evapotranspiration and Interflow.


- 1 -


SURFACE HYDROLOGY


Chapter 4- Processes of discharge concentration

1. General Information

The discharge formation can be mathematically described as predominantly verticalprocess and therefore, the catchment area is divided into individual homogeneousvertical layers (columns). The next step of the rainfall runoff modelling would be themathematical definition of the runoff formation (discharge concentration). For thismodelling, the physical process is considered that transforms the effectiveprecipitation of a catchment area into a discharge hydrograph at the outlet node ofthis catchment area.In one detailed rainfall runoff model, each discharge component is separatelyconsidered and afterwards linearly superpositioned.In principle, it is possible to describe accurately the process of the dischargeconcentration applying the partial differential equations. But the closed solution tothis equation is only rudimental due to the high numerical complexity of the system.For example, in the SHE-Model (Système hydrologique europienne) for theindividual components such as groundwater, interflow and surface runoff the systemof the differential equations is solved and the interaction between those componentsis implemented in form of the boundary conditions.For the surface runoff is, for example, applied the 2 dimensional depth-averagedshallow water equations for quasi laminar flow. In order to obtain the numericalsolution to those equations, the whole catchment is sub divided into raster elementsand for example, applying the finite element method, it is solved for all raster cells.


- 2 -

The wide application of this hydrodynamic model for description of the rainfallrunoff processes has its shortcomings, predominantly related to:- thorough data as basis for the model is usually not available- technology is not completely developed- the results that are obtained through this method are not significantly better than

the ones obtained applying other methods

The last point underlines the fact that this method as well as the other ones fail to acertain extent in describing accurately the basic mechanisms of the runoff formation.It is obvious that it is necessary to apply more detailed numerical decomposition andmore accurate spatial data.Both of those requirements is not possible to achieve in the foreseeable future.Therefore, for the calculation of the runoff formation , the so called „hydrologicmodel approaches“ will dominate. They assume that the discharge concentration canbe decomposed into translation and retention process, where both of them can belinearly superposed.The translation refers to the process of the temporal delay of the water. In atranslation model outflow is delayed for

)()( Lttptq ��

Applying this translation model, it is possible to describe time delay of the runoff(discharge) streamlines in the water network and subcatchments.

Lines of the same translation time, that correspond to the flow duration of a waterparticle from one point of the catchment area to the outlet node is referred to asisochrone.Beside that, the discharge depends on the retention characteristics. Thus, thedischarge concentration is also influenced by the storage capacity of the catchmentand not only by the translation process. The retention capacity depends on the soiltype and cover as well as on the slope of the terrain between 0,7 and 8,0 mm andrepresents considerable amount of water that should not be neglected especially inflat catchment areas. It particularly influences the shape of the rising limb ofhydrograph.

In the following text, the concept of translation and retention will be introduced.

2. Translation model

In translation model, the concentration time to the outlet node for discrete segmentsof the catchment area is determined. Relating those segments to the flow time, thetime-are function is obtained. Multiplied by the discharge height of 1mm this diagramcorresponds to the rising limb of the hydrograph for an effective precipitation of amillimetre exclusive of translation. It is also referred to as Unit Hydrograph. One


- 3 -

can say the unit hydrograph represents direct runoff at the outlet of a basin resultingfrom one unit of precipitation (of 1mm) excess over the basin.If the time steps are discretised, the unit hydrograph can be mathematically expressedin the following way:

� �

e

i

1h

1i

1kt At

t)1i(A)ti(A

)ti(U��

��

��

� ��

�

�

where:

A(i� t) = Segments that are discharging at the time i� tAe = Catchment area� t = Time intervalu(i� t) = Ordinate of the unit hydrograph at time i� t

For runoff from plain surfaces, the time-area function is rectangle.If duration of the rainfall is equal to the concentration time of the catchment tc , thehydrograf is in form of isosceles triangle.

Nowadays, the concepts for flood protection that are used in water management inurban areas are based on this principle. Natural catchments are considerablyinhomogeneous in terms of translation, as the relief in such areas substantially varies.For calculation of the concentration time, lot of more or less empirical formulas are inuse. For the scope of this course, the approach based on the kinematic wave isapplied.

33,0so

4,0eff

6,0

Stc i

kL2184,9t ��

�

��

�

where:

tc = Concentration timeL = Flow pathkSt = Strickler-constantieff = Intensity of the effective rainfallIso = Average gradient (terrain)

This formula also considers the average slope of the terrain and surface roughness. Ifthe terrain is considerably heterogeneous , this formula gives only the approximativevalue. In this case it would be more accurate to divide the catchment into thesegments with homogenous gradient and roughness and the concentration time iscalculated as the sum of those individual segments, as follows:


- 4 -

��

�

n

1i i

ic v

st

wheresi = Length of the segment in = Number of segments to the outletvi = Mean flow velocity in segment i

Mean flow velocity can be calculated assuming the quasi-stationary flow fordischarge and applying universal flow laws. The Gauckler-Manning-Strickler law iscommonly in use.

2/1i,so

3/2ii,Sti hkv ��

where kSt = Strickler-coefficienthi = Flow depth

For the empirical parameter and adopted flow depth h, some data is taken from theliterature.

According to Pasche/Schröder and the research and analysis of the hydrologic datausing GIS 1994, the following values are obtained.

Agricultural crop land(arable):

kSt = 4,5 m 1/3s –1

Grassland(meadows):

kSt = 4,5 m 1/3s –1

Forest kSt = 5,5 m 1/3s –1

Water depth h = 0,03m

If the catchment area is heterogenous, neither flow paths nor concentration time ispossible to calculate manually due to the complexity of such a system. This fact opens room for application of Geographic Information System (GIS).

It the basic data for the catchment is available, it is possible to automatise thegeneration of the time-area function for each catchment type and structure. Further,based on the unit hydrograph derived from the time-area function, it is possible tocalculate the discharge for any effective rainfall with the intensity ieff(t) for eachcatchment, as follows:

� ��

t

0eff d)(i)t(Ac)t(Q


- 5 -

Again, if the constant discharge is discretised over time with the time step t� , thanthe equation above can be transformed as following.

��

��

k

1iieff )ti(i)t)1i(ti(Ac)ti(Q

tc

��

6,31

It is important to mention that for this approach, the retention effects on dischargeflow are not considered. Therefore, it is applicable only in small catchments withhigh sealing rate.

3. Retention model

In hydrology, the retention model is usually simplified. A very simple and widelyused model is linear-reservoir. According to this approach, the outflow from theretention is directly proportional to the reservoir content.

)()( tQktV out��

where

V (t) = Reservoir content at time tQout(t) = Outflow from reservoir at time tk = Retention constant

Further, considering the continuity equation i.e.Inflow = Outflow + Change of the storage,the equation above gives:

dttdQktQtQ out

outin)()()( �

��

whereQin(t) = reservoir inflow in time t

General solution to the equation gives:

��

��

��

t

t

ktz

kttoutout dte

kQetQtQ

0

0 /)()(0

1)()()(�

�

�

for Qout(to) = 0 and to = 0 this equation is as follows:

��

��

tkt

inout dtek

QtQ0

/)(1)()( �

�


- 6 -

If Qin( � ) is the inflow and Qout(t) outflow, considering the equation above it is toconclude that, the outflow is derived directly from the inflow and multiplied by a unitflow. The unit flow reflects the transmission characteristics and thus the retentionfeatures of the catchment.

k/)t(R e

k1)t(U ��

This unit hydrograph corresponds to the discharge hydrograph , with evenlydistributed precipitation of N=1mm. If the duration of the precipitation 0�� , theInstantaneous Unit-Hydrograph (IUH) is obtained.Again, if the continuos discharge flow is discretised over time and the intensity of theeffective rainfall in time t� is multiplied by the total catchment area:

eeffin AiiQ �� )()( ��

the following summation formula for the discharge flow due to the retention gives:

� ��

��

n

1iReffea tt)1i(tnu)ti(iA)tn(Q

where:

� � � � k/t)1i(tnek1t)1i(tnu ��

According to this equation, the discharge at time tn �� is obtained through thesuperposition of all catchment responses that are developed from one inflowgenerated during the effective rainfall within the time period 0t� to tnt �� .Individual responses are product of the effective intensity at time � t and the axis �=t(folding) of the Unit Hydrograph.

The retention constant, with unit ��

��

�

h1 can be directly determined from the recession

limb of the hydrograph.

)2(Qln)1(Qlnttk

AA

12�

�

�

4. Combination of translation and reservoir models

As it was already induced, discharge concentration in a catchment area is subject totranslation as well as to the retention. Combination of these two different physicalapproaches can be mathematically described.The time-area function is through a linear reservoir led to the outlet of the catchment i.e. the time-are function is folded by the instantaneous unit hydrograph.


- 7 -

� ��

t

0RTTR d)t(u)(u)t(u

where is:

eT A

)(A)(u �

��

Another possibility to describe accurately the discharge concentration is to introducethe system of the reservoir chain. The translation of the catchment is considered bythe number of the reservoirs n. The outflow from the n reservoir of the linearreservoir chain as reaction to the inflow of 1mm rainfall gives:

k/t1n

sk ekt

!)1n(k1)t(u �

�

��

��

�

�


- 1 -


SURFACE HYDROLOGY


Chapter 5- Subsurface runoff (Interflow and groundwater runoff)

In addition to the runoff processes on the surface, the subsurface runoff processes arealso to be considered. The ones relevant for the water balance assessment areinterflow that occurs in the upper soil layer, and groundwater outflow and base flow.The subsurface runoff is delayed comparing to the surface runoff and in thehydrology is modelled independently.

5.1 Interflow

In soil water regime modelling, an unsaturated layer is divided into horizontallamellas depending on the hydro-geologic characteristics. For setting up the conceptof the interflow the following assumptions are to be considered:- in lamella i interflow occurs only if the lamella below the lamella i is already

saturated - boundary between lamellas is inclined.

In the soil water regime modelling, the soil water is balanced for each hydrotop.(seesemi-distributed model).The discharge, however, regards the entire subcatchment area, which consists of nHydrotops.The interflow rate Intfij from each Hydrotop and each lamella is aggregated to a totaldischarge QIntf within the subcatchment area:

)t(IntfA6,3/1)t(QIntf ijn

1j

m

1iji ��

� �

Surface Hydrology-Subsurface runoff

- 2 -

where is:Aj area [km2] of the Hydrotop j

Interflow behaves similar to the surface runoff in terms of Translation and Retentioneffects. Thus, it is possible to draw the stream lines of the interflow applying theisochrone method as following:

� ��

��

n

1iTRa tt)1i(tnu)ti(QIntf)tn(Q

� � � ��

��

��

n

1i

k/t)1i(tn

eTR e

k1

A)ti(Atnu

However, it is to be considered that subsurface translation and retention occur tomuch larger extent than in case of surface runoff.

So, the time-area function A(t) and retention constant k that are assessed for thesurface runoff are not the same ones that are to be used for the subsurface runoff.They should be derived considering the physico geographical and geologicalconditions of the soil.

In principle, the derivation of the time-area function (t) for subsurface runoff is muchmore difficult than from the surface due to the complexity of the media.

It is often the case that the time-area function for the surface runoff is also taken forthe Interflow and simply multiplied by a constant and in that way linearlytransformed.

Better and at the same time physically based approach assumes that the flowvelocities of the Interflow are behaving proportional to the Darcy’s law:

GfIntf Ikv ��

where is:kf = permeability coefficient for the saturated soil [m/s] andIG = mean gradient of the terrain along the longest flow path [-].

Physical processes are also simplified in this method, as the parameter is averaged forthe whole catchment and kf value of the saturated soil layer is taken also for theunsaturated layer. As it is assumed that the flow velocity is constant in the wholecatchment area, the time-area function becomes a direct function of the areas with thesame flow path, as following:


- 3 -

��

��

m

1i

m

1i]t)1n[(A]tn[A)tn(A

where is:�� ]tn[A total area of the subcatcment with the same flow path

Intfvtn]tn[s ��

In order to estimate the retention constants k there is no approved method yet.Therefore, it should be done by calibration from the rising limb of a floodhydrograph, whereby the considered times t1 and t2 should lie on the recession limbof the flood hydrograph.

)2(Qln)1(Qlnttk

AA

12�

�

�

5.2 Discharge concentration in aquifer (groundwater flow)

The water that percolates from the last soil lamella into deeper layers represents theground-water formation, which flows to the ground-water reservoir. The calculationsof the ground-water reservoir is performed integrally for the total catchment area.All hydrotps with recharge to the groundwater reservoir are aggregated.One can distinguish 3 different types of the groundwater reservoir (GWR):- upper GWR- lower GWR- carst or deep GWR

Upper and Lower GWR are illustrated in Figure 5.1.

Figure 5.1 Cross section GWR -river


- 4 -

Upper and lower GWR can be considered together as they differ from each other bythe height of the groundwater level in the reservoir. But, the deep GWR should beconsidered as an independent element.Upper GWR is in direct contact with the channel. Outflow of this reservoir flows tothe channel. Infiltration from the channel to the upper GWR is not considered.On the other side, the discharge from the lower GWR flows to the GWR of theadjacent catchment. It is possible that max 3 catchments receive the discharge fromthe one catchment, only the weighting of those inflows can be different. It is tomention that the capillary uprise from the GWR is neglected in the model. The deepGWR is, as already adduced, an independent element that drains into a recepient. Thelocation where it occurs can be chosen arbitrary.GWR receives the inflow from the soil water as well as from the GWR of theadjacent catchments. Inflow is proportionally divided into GWR and deep GWR, asfollows:

)(6.3

1)(inf; tPercAtQ HHH ��

)()1()()( inf;inf tQITRtQtQ HHGWNGGW zu � ��

)()( ,inf tQITRtQ HinHHydrotop

TGW ��

where:QGWNGin : Inflow to the groundwater reservoir (GWR) from the adjacent

(GWR) [m3/s],AH : Hydrotop area [km2],PercH(t) : Percolating water from the soil water storage [mm/h],QGWinf(t) : Inflow to the GWR (from above and below) [m3/s], QGWTinf(t) : Inflow to the deep GWS (characteristical for carst areas) [m3/s]ITRH : weighting factor [-] depending on the Hydrotop.

Theoretical background for the concept is the continuity equation of the inflow,outflow and changes in the storage. Inflow is composed of the discharge of theadjacent element (GWR) and does not depend on the actual water content in thereservoir. On the other side, the outflow from the observed GWR depends on thewater quantity in it.The outflows from the GWR are, for example, in the program package BCENAiterratively calculated, according to the following equation:

)()()( tQouttQindt

tdV�� (Balance equation)

dttQk

dttdV out )()(

�� (Linear reservoir)

where is:


- 5 -

V(t) : actual water content in the reservoir [m3],Qin(t) : Inflows to the GWR [m3/s],Qout(t): Outflows from the GWR [m3/s]k : retention constant of the linear reservoir [s].

In its developed form, the balance equation gives:

it is illustrated in Figure 5.2.

Figure 5.2 Water balance for the catchment area 2

5.2.1 Discharge from the Upper and Lower GWR

In the model BCENA, upper and lower GWR are joined and considered as one unit.According to the model concept, it is considered that the channel runs over the watertable of the lower GWR.Thus only an exchange of water of the ground-water reservoir and the channel ispossible starting from a certain water level height in the GWR.The range of the GWR without contacts to the channel is referred to as lower GWRand the one overlying the channel bad is called upper GWR. The program BCENAenables indication of the lower (hGu) and upper height (hGo), for the definition of thesize of the two reservoirs.If the isotropic i.e. homogeneous aquifer is presupposed, that is characterised by verypoor spatial variability, that the reservoir volume can be determined depending on the


- 6 -

groundwater level. An additional assumption is that the groundwater level is uniformfor the subcatchment.

610)()( �� APorstGWRtV

where isV(t) :actual water content of the reservoir[m³],GWR(t) :water height in the reservoir element (m),Pors :porosity of the aquifer[-] A :area of the catchment (km²).

Figure 5.4 Upper and lower groundwater reservoir

Considering the concept of the model, the following cases are possible concerning thegroundwater level and its relation to the hGu and hGo. (refer to Fig 5.4)

1. .case: GWR(t) < hGu Groundwater height is below the lower groundwater level. Thus, the total watervolume is assigned to the lower groundwater reservoir. It can be described as follows:

0)(10)()( 6

�

��

tVPORSAtGWRtV

GWup

scGWlo

2. case: GoGu htGWRh �� )(The groundwater level is between hGu and hGo

� �

� � 6

6

105,0)(

105,0)()(

��

��

�

�

��

��

� �

��

��

�

�

��

PORSAHGUGWRHGUHGOHGUGWRtV

PORSAHGUHGOHGUGWRtGWRtV

scGWup

scGWlo

3. case: GWR(t) > hGo

Groundwater level is above the upper height. The volume of the lower groundwaterreservoir is not increasing.


- 7 -

� ��

� � 6

6

1021)(

105,0)()(

��

��

PORSAHGUHGOtV

APORSHGUHGOtGWRtV

TEGWlo

GWup

The outflows from the upper and lower GWR depend on the actual water content inthe reservoir according to the linear reservoir concept and considering thecorresponding retention constants RetBas and RetGW.

5.2.2 Deep groundwater outflow

Deep groundwater outflow can be calculated from the system of the differentialequations that has already been introduced. Additionally, the retention constant forthe deep groundwater reservoir RetTGW, is introduced.This outflow can be assigned to any node, not necessarily to the outlet node. Further,it is possible to construct the net independent from the one for the surface runoff andcan for example, simulate the flow paths in carst areas.


- 1 -


SURFACE HYDROLOGY


Chapter 6- Flood wave formation in the channel

In the hydrologic cycle the following components are of relevance for the modelling:surface runoff (overland flow), interflow and basic flow. Additionally, the flow inchannel is considered as a part of the inflow:

)()()()()( tQtQtQtQtQ ChannelBasisInterflowSurfacein �� (1)

For flood wave propagation in natural channels, two important phenomena should beconsidered: attenuation of the peak flow and time lag that leads to the modification ofthe discharge hydrograph. Figure 6.1 shows the transformation of the flood wave.This discharge hydrograph is modified if the water section flows through retention.

Figure 6.1 Inflow/Outflow hydrograph of a flood wave

Surface Hydrology-Flood wave formation in channel

- 2 -

The computation methods that are applied to describe the processes in the watercourse, are summarised under the name Food Routing. Flow routing may beclassified as either hydrologic (lumped) or hydraulic (distributed).The hydraulic computation methods employ both, the equation of continuity and theequation of motion (momentum conservation).They model the integral movement process of the water in the channel, but inengineering practice one and two-dimensional models are commonly in use.In the rainfall runoff models are currently in use only hydrologic methods. They donot deal with the overall motion process, but they concentrate of the water strand andtreat it as a linear reservoir of the volume V(t) that is changing with the outflowQout(t) as follows:

)()( tdQoutktdV �� (2)

Thus, those methods determine only the resulting process that is generated as areaction to the inflow.

6.1 Flood Routing applying the linear reservoir model

Generally, flood routing is a mathematical method used to predict the temporal andspatial variation of a flood wave, at one ore more points along a water course (river orchannel). The watercourse may be a river, stream, reservoir, estuary, canal, drainageditch or storm sewer. In case of a stationary discharge flow, there is a clear connection between the waterdepth h and discharge Q for each water section.

)(hfQ � (3)

To each water strand corresponds only one volume, and therefore it is possible todefine a unique relation between volume and discharge:

)(VfQ � (4)

Further, for the stationary flow of the linear reservoir model, it becomes:

)()()( hdQouthKhdV �� (5)

In the equation above, the parameter k(h) represents the time necessary for water toflow through the channel subsection.


- 3 -

Different analysis of the river courses showed that the parameter k changesinsignificantly over the discharge range, so that it can be considered as a constant.

K(h) = k = const. (6)

With this assumption, the conditions for applying the liner reservoir concept arefulfilled, as the outflow is linearly related to the storage. The integration of thisequation over the water depth gives:

)(10VV

kQout �� (7)

where, 0V is the volume that remains in the scours of the river bed in dry season. Thisvolume is not of importance and can be neglected.

Again, the continuity equation is applied:

dVdtQdtQ ain �� , (8)

where Qin is the inflow, Qout is the outflow and dV represents the changes in thestorage.Inserting the Eq (7) in the Eq (8), the following is obtained:

zQVkdt

dV��

1 (9)

For the special case, that is 0�inQ , the equation can be modified as follows:

dtkV

dV 1�� (10)

Integrating this equation (0,t) it gives: C

ktV ��ln (11)

And in its exponential formkt

eCV�

�� (12)

For t=0 is V= V0 so that the equation becomes:

kt

eVV�

�� 0 (13)


- 4 -

If one replaces V with kQout � and 0V with ktQout �� )0( 0 , the following is obtained:

kt

outout etQQ�

�� )( 0 (14)

In the second step, the special case 0�inQ is introduced to the general solution0�inQ , in the Eq 12. The first derivation of this equation is as follows:

kt

kt

ek

CedtdC

dtdV ��

��1 (15)

Introduced in the Eq 9, this equation gives:

zkt

kt

kt

Qek

CeCk

ekdt

dC��

�� 111 (16)

orkt

in eQdtdC

�� (17)

Integrating this equation over time and replacing the constant C with kt

eV � (Eq 12),leads to:

))((0

0 dtetQVeVt

kt

inkt

� ��

�

(18)

Again, if V is replaced with kQa � , and 0V with ktQa �� )0( 0 , the following is obtained.

))(1)(0

0 dtetQek

eQtQt

kt

inkt

kt

outout � ��

��

(19)

According to the result of the derivation, the temporal discharge variation consists of:

- discharge due to taking the water form the reservoir that is available at time t=0- discharge as consequence of the inflow

Since closed-form solutions to the complete hydraulic routing differential equationsdo not exist, it is solved by discretising inflow hydrograph over time, with intervals�t and with the assumption Qin=const in those time intervals. Now, it is possible towrite the equation above as a summation in the following way:

tetmQek

etQtnQn

m

ktm

ink

tnk

tn

outout ��

��

��

�

��

��

1)(1)0()( (20)

or:


- 5 -

tek

tmQetQtnQn

m

ktmnin

ktnoutout ��

�

��

��

�

��

1

/)(/ 1)()0()( (21)

where, )( tnQin � is the inflow and )( tnQout � is the result and this transmissioncharacteristics can be described in terms of impulse response function as follows:

ktmnek

tmnu /)(1])[( ��

�� (22)

It formally coincides with the Unit Hydrograph, that is already introduced in theChapter 5, when explaining the processes of the discharge concentration. So, the wave transformation in channel is the process analog to the dischargeconcentration on the surface.

6.2 The method of Kalinin-Miljukov

In the previous text the stationary state flow for discharge was assumed. However,this assumption is hardly applicable to the flood wave situation, as the storm state in aflood wave is always instationary, that is indicated as temporal change of the meanvelocity v and water depth h along the channel as follows:

0

0

�

�

dtdhdtdv

The results derived for the stationary state, applied for the instationary (unsteady)storm is not fully correct, since in the case of instationary flow, there is no uniquevolume/discharge relation.

Differently from the stationary state, discharge depends not only on the water levelbut also on the slope. It can be denoted as following:

),(),(IVfQIhfQ

�

�

Consequently, for each section of the channel, exists a so called Hysteresis curve, thatmeans: storage versus outflow is not a single valued function. (Fig 6.2)


- 6 -

Figure 6.2a Hysteresis curve

Figure 6.2 b: Hysteresis curve

As it depends on the flood wave propagation (transformation), can not be a prioridetermined. Considering the Hysteresis curve (6.2), to each discharge )( 1tQout it ispossible to assign 3 different water depths; on the rising and falling limb of theinstationary flow and the one for the case of the stationary (steady) flow.Kalinin Miljukov, however, showed that it is possible to define an unique functionbetween discharge and volume in spite of the dependency on the slope if the channelis represented by a cascade of n linear reservoirs with the length L (characteristicsection). The unsteady flow in those sections can be approximated by a quasi-stationary flow.For one single reservoir is assumed that the time required for the increase of thedischarge volume �Q from the inflow side l to the outflow side r corresponds to theflow time in the stationary storm state.Further, it is coupled with the assumption of a weak instationary flow, so that thedischarge increase �Q becomes so small and its flattening can be neglected.

The replenishment of this additional water volume �V occurs with the increase �Q.The time required for this process can be estimated according to the linear reservoirmethod as following:


- 7 -

QVkinst

�

�� (23)

Assuming little changes in the inflow, then the time instk corresponds to the flow timeof the flood wave peak.

Further, an unique discharge curve for the characteristic channel reach is obtained, ifwe consider the characteristic of the Hysteresis curve that the time lag �t of dischargeon the rising as well as on the recession limb of the flood, can be observed on thegauge.If the discharges of the cross section r are assigned to the water levels in the crosssection m, it becomes:

tvlm �� , (24)

so that the discharges of the stationary discharge curve at the crossection m can beassigned to the corresponding instationary ones at the cross section r.The length of the characteristic section is:

mlL 2� (25)

is calculated by the discharge volume and the characteristics of the water course.Further, the instationary discharge Q in the cross section r is divided into a stationaryand the additional discharge:

dstatinst QQQ �� (26)

In the same way, discharge volumes and water surface slope can be divided:

dstatinst VVV �� (27)dstatinst III �� (28)

Kalinin-Miljukov Method is based on the considerations of the validity of the linearreservoir assumption also for the unsteady flow:

dd

dstatdstat

kdQdVbzw

QQdkVVd

�

��

.)()(

(29)

Further, assuming that the flow resistance coefficients of the instationary stormcorrespond to the same ones for the stationary flow for the same water depth, and that


- 8 -

energy line is parallel to the water surface line, it is possible to compute the dischargeQ applying the Darcy Weisbach law.

2/12/1, )()8( instrhy

rrinst IrgAQ ��

� (30)

In the cross section r, the following relation of the instationary storm state to thestationary flow with the same hydraulic radius resp. water depth, can be concluded:

stat

inst

stat

inst

II

QQ

� (31)

Applying the Eq 28, the instationary water table slope can be transformed asfollowing:

)211(1

stat

dstat

stat

dstatdstatinst I

IIIIIIII �� (32)

Placing this expression in the Eq 31, the following is obtained:

)211(,

stat

drstatinst I

IQQ �� (33)

Further considering Eq 26 and 33, it becomes:

stat

drstatd I

IQQ �� ,21 (34)

Qd corresponds to the additional volume Vd, that can be represented only by smallincrease/decrease of the water level by multiplying the water level with the stationarydischarge Qstat and the mean depth of the additional water level hd.

dstatd hQV �� (35)

Further, if it is assumed that the slope is constant, along the characteristic section, itbecomes:

2/LIQV dstatd �� (36)

If the Eq 34 and 36 are differentiated over Id, it gives.

dstatd dILQdV �� 2/ (37)


- 9 -

and

dstat

rstatd dI

IQ

dQ ��,

21 (38)

Inserting the Eq 37 and 38 into Eq. 29, it becomes:

dstatdstat

rstat dILQkdII

Q�� 2/

21 , (39)

Transforming and inserting the d

d

dQdV

k�

1 , finally gives conditional equation for the

characteristic channel section L:

statstat

d

drstat

QIdQdVQ

lL�

��

,

2 (40)

This equation can be applied to channels with any cross section. It can be simplifiedif the changes of the volume predominantly depend on the water level.

mdstatd dhQdV ,�� (41)

In this case, the Eq 40 becomes:

statstat

mdstat

dQIdhQ

lL�

�

��

,2 (42)

By very small changes of the profile along the channel reach, the changes of thewater level in the balance point m can be approximated by the one of the crossectionr.

statstat

rdstat

dQIdhQ

lL�

�

��

,2 (43)

6.3 Application of the Kalinin-Miljukov Method to a large channel reach

Until now, only the channel reaches with the limited length L were considered, wherethe length L was derived from the hysteresis curve and from the linear reservoirmethod.It means, that if we consider a channel strand with the length LG, applying theKalinin-Miljukov method, it has to be divided into n equal sections with the length L.


- 10 -

Each section represents one independent reservoir, and the Eq 22 describes the waveformation for it. The outflow from the reservoir n is the inflow to the reservoir n-1.

)()1( nQnQ outin �� (44)

where the numbering of this reservoir chain starts on the lower boundary of the totalchannel length LG and is proceeded against the flow direction.Inserting the Eq 44 into Eq 22, it becomes:

tek

tmQtnQn

m

ktmnnana ��

��

��

�

��

�

1

/)(,1,

1)()( (45)

The total channel reach can be understood as a unit, if the reservoir chain representedas a summation of those individual reservoirs. It is expressed as follows:

ktnnza e

ktnQ

nkttnQ /1)1()(

)!1()( ��

��

�� , (46)

where the number of reservoirs is calculated in the following way:

LLn G

�


- 1 -


SURFACE HYDROLOGY


Chapter 7- Application aspects of the Kalinin-Miljukov Method

7.1 Basic Equations for the discharge calculation in channel

According to the Kalinin-Miljukov method, the integral channel reach is subdividedinto individual sections of the length L. Each of those sections is represented by alinear reservoir cascade with n single reservoirs.In principle, this reservoir chain can be successively calculated from the beginning ofthe section to the lower boundary (run out) for the individual reservoir using thefollowing equation:

ttmnutmQtnQn

mjjjout ��

�

��

��

�

�

11, )1[)()(

where, ktmn

jj e

ktmnu /)1(1])1[( ��

��

j =1,2,....nS index of the single reservoir, starting from the lowest one

nS = total number of the reservoirs in one subsection

As the intermediate results are usually not of relevance, the superpositioned reservoirequation is used.


- 2 -

])1[()()( tmnutnQtnQ inout �� where is

k/t)1mn(1n

Se)

k1(

)!1n(kt]t)1mn[(u S ��

��

��

In these equations, the retention constant k corresponds to the flow time t in a singlereservoir j. It is assumed, that each single reservoir of the cascade has the samehydrologic characteristics.The total number of the reservoirs nS can be calculated according to Kalinin/Miljukovmethod, considering the characteristic length L, as follows:

LLn G

S �

where:

statstat

m,dstat

dQIdhQ

L�

�

�

To bear in mind: The above shown conditional equation for L is derived consideringthe simplification, that the changes of the reservoir volume V is predominantlyfunction of the water depth h. If it is not the case, it is necessary to apply the extendedform of the equation:

statstat

d

dstat

QIdQdVQ

L�

�

Further, applying the Kalinin/Miljukov method, the retention constant k is to bedetermined. It is done assuming that the flow of a flood wave is a weak instationaryprocess. In that case, the flow time tF corresponds to the following expression:

QhhQ

dQdVkt j

d

djF

�

��

)(

where )()( hbLhQj �� (with the water depth h corresponding water levelsurface)

7.2 Extending the basic equation for channel flow for the characteristic profile

Most of the water courses, overflow its banks by flooding. As a consequence, anextended channel profile is formed, that consists of the left and right flood planes andthe main (dominant) channel.


- 3 -

The extended profile is shown in Figure 7.1.

Figure 7.1 Flood cross section

Where is:Flood Plain: The flat area of land adjacent to a stream that is formed by current floodprocesses.Bankfull Stage: The stage at which water starts to flow over the flood plain; theelevation of the water surface at bankfull discharge.

At the very beginning of the overflow in the river, the water surface changes, more orless discontinuely.The h-Q-Function shows no linearity in case of those discharges. In this case, it is notnecessary to apply the extended form of the equation for L, instead of the linearchain, the parallel reservoir cascade is used.

� � tt)1mn(u)tm(Q)tn(Qn

1mza ��

�

��

��

�

where is� � ]t)1mn[(u)1(]t)1mn[(ut)1mn(u 21 ��

and

� � 11,S k/t)1mn(1n

11,S11 e)

k1(

)!1n(ktt)1mn(u ��

��

��

� � 2k/t)1mn(1n

22,S22 e)

k1(

)!1n(ktt)1mn(u 2,S ��

��

��

The first reservoir cascade represents the discharge into the main channel and thesecond cascade, the discharge to the foreshore (flood plain). The parameter �corresponds to the discharge distribution between flood plains and the main channel

tot

fp

QQ

��1�

where: fpchanneltot QQQ �� (Total flow in the channel)Qfp : total discharge from the flood plain [m³/s],Qchannel : discharge from the channel [m³/s]


- 4 -

7.3 Determination of the parameters for the Kalinin/Miljukov method

The empirical parameters of the Kalinin/Miljukov method require lots of data as wellas the execution of the complex hydraulic calculations, since neither h-Q-relation ofthe subsection (from measurements) nor the discharge distribution between floodplain and the main channel is available.As basic data, the channel and profile data are considered. They have to be located onthe places that can relevantly describe the processes in retention.In general, calculation of the hydrologic Flood Routing does not require so denseprofile distribution that is necessary for computation of the water table, since for theweak non uniform flow conditions, the h-Q-B relations is sufficiently accuratedescribed by stationary-uniform flow calculations for the Kalinin-Miljukov method.Therefore, every 500 m of the channel and valley profiles are required.

It is often the case that the Flood Routing analysis is followed by the water tablecalculation in order to assess the influence of the flood volume, determined by theFlood Routing method, to the water course. In order to accomplish that, the channeland valley profiles are required, only in this case the density of the profiles has to beconsiderably higher, since the irregularity of the flow can not be neglected.Therefore, the distance between 2 profiles should not exceed 50-100m.When defining the profiles for the water table calculation, it is possible to use theones already used for the Flood Routing Method.In principle, two different approaches are possible:

1. Determination of the parameters using the mean profile

Within the subsection of the channel, for which the Flood Routing calculation isperformed, all profiles that are in this subsection are pondered(integrated) and onemean profile is defined.For this profile, the stationary-uniform flow is assumed (IE = IWsp = ISo) and the h-Qrelation as well as V-Q relation is calculated. Finally, the parameters nS and k arederived, both for the flood plains and for the main channel, as well as distributionfactor �, between the sub cross sections.The following example illustrates this procedure:

Discharge Channelreservoir

Flood plain reservoir width height

n k n k[cbm/s] [-] [h] [-] [h] [m] [m+NN]0,171 30,0 0,002 - - 1,54 173,261,729 23,3 0,001 - - 3,08 173,615,466 14,7 0,001 - - 4,62 173,969,447 12,7 0,001 30,0 0,005 16,11 174,1631,316 10,1 0,001 27,6 0,001 30,08 174,43


- 5 -

Sum 90,80 57,60Mean value ~18 0,001 ~29 0,002

Distribution factorα

0,70

2. Determination of the parameters for each profile and afterwards aggregation

For each profile within the subsection of the channel, the relation h-Q as well as V-Qusing the water table calculation method and the corresponding parameters nS and kas well as � are calculated as mean values of the profile according to the aboveshown procedure.Further, those values for the single profiles, are aggregated to one value that appliesfor the whole subsection, as follows:

Retention constant:

�

�

�

�

�

�

p

p

n

1jj

j

n

1jj

l

lkk

Number of the reservoirs: ��

�

pn

1jj,SS nn

Distribution factor:

�

�

�

�

��

��

p

p

n

1jj

j

n

1jj

l

l

Bankfull discharge:�

�

�

�

�

�

p

p

n

jj

j

n

jjbankfull

bankfull

l

lQQ

1

1,

This way of aggregating is not mathematically correct, especially if the geometricdeviations of profiles are considerably high. If this is not acceptable, it is then necessary to divide the section into subsectionsconsidering the profiles j, j-1and j+1.


- 1 -


SURFACE HYDROLOGY


Linear Reservoir model

-Appendix-

Linear reservoir model represents the catchment area as a reservoir, in which theinflow in form of precipitation is stored and with time lag discharged. It is alsoconsidered as a black box method, as the processes in the system are not completelydescribed, but merely the transmission characteristics of the system, especially thecorrelation between the input and output, are taken into account.

One can conclude that the linear reservoir is of great importance in hydrologicalmodelling which purpose is to define the universal relation between rainfall and theresulting discharge hydrograph for one catchment area.

)]t(i[f)t(Q �

In the LRM, the overall system is divided into subsystems and the individualcomponents that are forming the total discharge, such as overland flow, interflow,seepage and groundwater recharge are considered as linear reservoir and are linearlysuperposed.

For the purposes of mathematical formulation of the linear reservoir, the behaviour ofthe catchment area is simplified, assuming that the catchment area behaves like a

Surface Hydrology-Linear Reservoir

- 2 -

container (reservoir) with the linear relation between actual volume in the reservoirand discharge:

)()( tQktV A�� (8)

where

V (t) = Volume in Reservoir at time t

QA(t) = Outflow from the reservoir attime t

k = Retention constant

This is illustrated in Figure 1.

Figure 1. Linear reservoir and corresponding outflow and volume curve

Further, applying the continuity equation for the reservoir

dttVtQtQ outin)()()( �� (9)

the first order differential equation for the linear reservoir is obtained:

dttdQktQtQ out

outin)(

)()(�

�� (10)

where Qin(t) = reservoir inflow at time t.

This equation can be solved by multiplying both sides by the factor k/te . Itbecomes:


- 3 -

)t(Qek1)t(Qe

k1

dt)t(dQe z

k/tA

k/tAk/t�� (11)

Applying the product rule this equation can be rewritten as follows:

)t(Qek1)e)t(Q(

dtd

zk/tk/t

A �� (12)

Integrating this equation in range QA(to=0) and QA(t) it gives:

��

��

��

� d)(Qek1)e)t(Q(d t

0 zk/k/t

A(t)Q(0)QA

A(13)

The general solution to this equation is :

��

��

��

� d)(Qek1)0(Qe)t(Q t

0 zk/

Ak/t

A (14)

and after transformation :

��

��

t

0

k/)t(z

ktAA de

k1)(Qe)0(Q)t(Q (15)

For QA(0) = 0 this equation becomes:

� ��

t

0

k/)t(zA de

k1)(Q)t(Q (16)

If the Qz( � ) is „inflow impulse“ and QA(t) „outflow impulse“, then according tothe equation above, the „outflow impulse“ is directly derived from the inflowimpulse, multiplying it by a unit impulse.

Transferred to the catchment area, )(tQout corresponds to the dischargehydrograph at outlet node and the )(�Qin to the precipitation )(iA)(N e �� andthe intensity i(�) off the effective rainfall, as following:

��

t

0

k/)t(A de

k1)(N)t(Q (17)

The system function


- 4 -

k/)t(ek1)t(u �� (18)

represents the transmission characteristics of the catchment. It corresponds to thedischarge hydrograph generated for one catchment for the effective rainfall of1mm that is spatially and during the time � uniformly distributed over thecatchment and shifted for the (t-�). It is also called the Unit Hydrograph (UH)

� ��

t

0A d)t(u)(N)t(Q (19)

This equation is basis for the Linear reservoir model.

If the duration of the rainfall 0�� , Instantaneous Unit-Hydrograph (IUH) isobtained.

Until now, the continual temporal field was presumed. If time is discretised withthe time steps �t, the effective rainfall of the m. interval (corr. to the time point�=(m-1)�t) becomes:

� � ttmiAd)(iA)tm(N etm

t)1m( e ��

��(20)

The time shift between inflow and outflow impulse for the time interval t=n�t is:

�� n(ttv m+1)�t.

The Unit Hydrograph is discretised over time:

��

��

��

t)1mn(t)mn( v d)t(ut

1]t)1mn[(u (21)

where Vttv �� is the time lag

If the Eq 20 and 21 are inserted the following summation formula is obtained:

� ��

��

n

1meA t1mnu)tm(NAt)tn(Q (22)

where is

� � � � k/t)1mn(ek1t)1mn(u ��


- 5 -

In order to have a better overview of the main properties of a hydrograph, theyare illustrated in following Figure:

(source: Florida International University, Department of Civil and Environmental Engineering)

Table of Contents:

Chapter 1: Introduction

1.1 Importance of Freshwater1.2 Water cycle and Water balance1.3 Tasks of the Hydrology and Water Management1.4 Importance of modelling in Water Management

Chapter 2: Hydrologic Cycle

2.1 General considerations2.2 Precipitation (rainfall) 2.3 Interception2.4 Snow-hydrological processes

2.4.1 Accumulation2.4.2 Compression2.4.3 Ablation

2.5 Evapotranspiration2.5.1 Methods to determine evapotranspiration2.5.2 Determination of potential evapotranspiration of an area with vegetation cover2.5.3 Determination of the actual evapotranspiration

Chapter 3: Modelling of the soil moisture regime

3.1 Hydrologic processes in the unsaturated soil layer3.1.1 Soil moisture3.1.2 Infiltration3.1.3 Percolation3.1.4 Determination of the actual evapotranspiration3.1.5 Capillary Uprise3.1.6 Interflow

3.2 Mathematical solution to soil water equation

Chapter 4: Processes of discharge concentration

4.1. General Information4.2. Translation model4.3. Retention model4.4. Combination of translation and reservoir model

Chapter 5: Subsurface runoff (Interflow and groundwater runoff)

5.1 Interflow5.2 Discharge concentration in aquifer (groundwater flow)

5.2.1 Discharge from the Upper and Lower GWR5.2.2 Deep groundwater outflow

Chapter 6: Flood wave formation in the channel

6.1 Flood Routing applying the linear reservoir model6.2 The method of Kalinin-Miljukov6.3 Application of the Kalinin-Miljukov Method to a large channel reach

Chapter 7: Application aspects of the Kalinin-Miljukov Method

7.1 Basic Equations for the discharge calculation in channel7.2 Extending the basic equation for channel flow for the characteristic profile7.3 Determination of the parameters for the Kalinin/Miljukov method

Appendix: Linear Reservoir Model

Documents

49Surface Hydrology