CHAPTER 1 - INTRODUCTION

The quantity of water available to the Earth is constant, water is only flowing through different phases of the hydrologic cycle and its physical conditions are changing. The following 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 following trends:

• 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 grown more than 10 times. The water consumption for industry has almost doubled. Beside the withdrawal due to consumption, the water losses have been increased due to the irrigation and higher evaporation from water storage (1.800 km³/a)

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

• 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 water movement and its recycling on the earth.

Water is constantly being cycled between the atmosphere, the ocean and land. The power of the sun together with the gravity forces are driving this cycle and are responsible for continuous changes of the physical conditions of water. The water cycle is composed of the following processes:

• Evaporation from oceans and surface • Advection of the water vapour • Condensation and precipitation • Precipitation that reaches ground surface, or is intercepted by vegetation cover and

evaporated back to atmosphere. • Infiltration through the ground surface, recharge and discharge of groundwater flow to

the water bodies.

The hydrological cycle allows the relationship between precipitation and streamflow expressed in a very general way. As in natural conditions the streams receive water only from their own catchment area, thus the continuity equation in the form of the water balance equation may be applied:

Inflow = Outflow +/- ∆Storage

Water balance equation is expressed as:

N = A + V +/- ∆S

Where:

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

Solving this equation, a quantitative assessment of the movement of the water through the catchment area is possible. Hydrologic cycle is illustrated in Figure 1.1.

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 concerned with the class of natural phenomena governed by general laws. Hydrology is the science that deals with the waters on the Earth, their occurrence above, on and beneath the surface, circulation and distribution, their chemical and physical properties and their reaction with the environment including their relation to living things.

It is of essential importance to understand water circulation processes and interrelations between them, as well as the human impact on water use in order to achieve a sustainable balance between protecting ecosystems and meeting human needs.

As hydrology deals with those processes, it is of great importance for Water Resources Engineering and sustainable management.

Definition: Water Resources Engineering (Water Management deals with the water use by humans as well as with the water protection. Further, it can be divided into Qualitative and Quantitative WRE. This course predominantly deals with the Quantitative WRE.

The following tasks are considered within the Water Management approach:

• Monitoring of quantity and quality of the water: o Yearbooks o Water quality report

• Setting up statistics: o Derivation of calculation parameters o Estimation of frequency

• Forecasting of discharge water level and quality with the purpose of flood protection and risk estimation:

o By regression analyses o By process modelling

• Management of Reservoirs for water supply, flood protection and power production: o Assessment of the available water o Design of the reservoirs o Presentation of case scenarios o Operation and monitoring 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:

o Determination of the discharges parameters o Setting up drainage concepts o Monitoring and operation of drainage mechanism

An integral approach in river basin management should be considered, therefore, water saving concepts must be set up in addition, 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

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

Legal Background on WRE

A sustainable water resource management should be aligned with legal regulations set under a global, regional and national level. Guidelines have been set up in Europe and also in Germany, below an example of the main legal requirements:

EU Water Framework Directive (WFD): tends to harmonize water management. Its aim is protect water bodies and improve their quality and quantity under a sustainable concept of water use. The principal objectives of the Directive are to achieve a good ecological and chemical status of surface water and a good chemical and quantitative status of groundwater within each European river basin until 2015.

German Water Framerwork Directive (WHG): related to water resources management measures (management of water quantity and quality). It requires that waters are to be safeguarded as a component of the natural balance and as a habitat for fauna and flora. The use of water for human activities must avoid any impairment of their ecological functions.

1.4 Importance of modelling in Water Management

1.4.1 The problem

Nowadays, the world is facing several problems related to an inappropriate use of the natural resources, which can be reflected in the following examples:

• Increase of flood hazards worldwide causing lots of physical, economical and human losses.(Figure 1.2)

Figure 1.2 Examples of Flood Hazards

• Erosion and reduced Groundwater Recharge by intensive farming causing drought problems and therefore economical crop losses.(Figure 1.3)

Figure 1.3 Intensive Farming consequences

• Exploitation of Groundwater Resources and surface water which interferes with the potable water supply resulting in reduction of water quality and quantity. (Figure 1.4)

Figure 1.4 Exploitation of surface waters

• Exploitation of Nature Resources due to industrialization and river regulation.(Figure 1.5)

Figure 1.5 Exploitation of Nature Resources

In order to solve the problems mentioned above and reduce the possibilites of damages by any flood event, a Sustainable Development of WRE must be implemented considering the following guidelines below:

• Catchment Based Water Management: o Restore Natural Surface Retention: by structural improvement of farmland,

converting cultivated land back into forest and grasslands. o Restore Natural Groundwater Recharge: by sealing up driveways, streets,

courtyards, and constructing retention ponds, furrows, pits and pools in the city area.

• Floof-plain Management: o Integration of flood aspects in spatial planning. o Restore natural flood plains: by pulling back of dikes. o Maximize flood plain storage by polders: by implementing flow polders and a

good land use management system. • River Management:

o Restore Natural structures o Avoid Obstructions in the River

• Flood Management: o Increase Awareness of flood risk o Efficient flood forecast and warning system o Flood prevention by structural measures

Besides an Integrative Flood Management system should be implemented, based on the following measures:

• Holding Back of Water: o Upstream detention o Extensifying of agriculture o Unsealing urban areas o Decentralizing stormwater management

• Giving river more space: o Restoration of natural rivers o Reclaim of natural inundation areas o Controlled retention by polders

The measures outlined above have the aim of protecting urban areas of any flood event by means of avoiding the increase of peak flood flow, preserve the flow capacity, determine inundation areas and obtain a better resistance to flooding.

1.4.2 Modelling

In order to accomplish the tasks of Water Management described in the subsection 1.4.1, it is of great importance to obtain hydrological information and quantify the processes in the catchment areas.

The type of hydrological information needed to achieve high level of flood protection, are mainly: determination of the peak flow and discharge hydrograph along the river, determination of flood stages, determination of flood probability, determination of inundation areas, and evaluation of groundwater recharge and the effect of flood control measures on the hydrological system.

For quantification, mathematical models are widely used for the simulation of hydrological processes in catchments and flow processes in rivers. One differentiates between stochastic models (inductive), that only take into account the random aspect of natural phenomena and deterministic models (deductive) that assume that all physical, chemical, and biological parameters of a watershed are known. (Figure 1.6)

Figure 1.6 Hydrological Modelling

They can be further subdivided according to their characteristics:

Deterministic Model

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

Distributed Model: estimates the spatial distribution of moisture, energy fluxes, and runoff generation by subdividing the model domain into small computational grid elements using the spatial resolution of an underlying digital elevation model.

Detailed model: Based on the physical interrelations between initial conditions and output values. (e.g. rainfall runoff models). This model is subdivided in:

• Short term model: The simulation of rainfall runoff processes is limited to flood simulation.

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

1.4.2.1 Application of the Models

The above mentioned models can be applied in Water Management to solve the following problems:

• Short term simulation models (Rainfall Runoff Model) can be used for: o Design of dams and flood control storage (retention basins) o High water analysis o Analysis of the water regime by agriculture (sealing of the surfaces)

• Long term simulation model (Water Balance Model) can be applied for: o Intake for potable water from retention basins o Intake from rivers o Optimisation of operation of a dam o Low-water analysis o Change of the water regime by groundwater use

This course predominantly deals with the rainfall runoff models, their scope of application, and especially their contribution to efficient flood management as well as their applicability for the low water calculations.

CHAPTER 2 - HYDROLOGICAL CYCLE 2.1 General considerations The hydrological cycle is the fundamental principle of hydrology. It is the idealized form of water circulation and its recycling on earth. Water cycle is continuous process by which water is transported from the oceans to the atmosphere, to land and back to the sea. Because the total quantity of water available in the earth is finite, hydrologic system can be looked upon as closed. In that system, the processes, such as, evaporation, advection of the water vapor, formation of precipitation and back flow as surface and groundwater discharge, are in equilibrium. This is expressed in the water 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] Figure 2.1 is a schematic drawing of the hydrological cycle showing the water pathways.

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 real processes in nature. The hydrological cycle is made up of many different factors; therefore, it can become quite complicated when trying to analyze the relationships between those factors. A complete mathematical description of the hydrological processes is one of the most difficult tasks in engineering and natural sciences and has not been completely solved yet. It comprises the setting up of an integral model system, in which the components such as global climate

model, surface runoff model and groundwater model of saturated and unsaturated soil layers are coupled together. Considering the fact, that in each of those model components, there are still unexplained physical processes and small-scaled natural processes, this simplification of the hydrological cycle becomes reasonable. Further, a closed mathematical model that describes the water cycle is set and it is composed of relevant physical processes in nature. Processes over the see surface are not considered, and processes like vapor advection or condensation are also excluded. Those phenomena are meteorological and their simulation in Hydrology is not of great importance. But, the important process for this simulation is temporal and areal distribution of the precipitation over the surface. This information is provided by the weather service or any other meteorological institution. Finally, considering those modifications and simplification of the real system, one can distinguish the following processes relevant for the Hydrological Simulation: They are listed in Table 1.

Layer Vertical Process / Runoff Formation

Horizontal Process / Runoff concentration

Rainfall formation

Snow storage ablation Surface Layer

Interception

Surface Runoff

Infiltration

Transpiration & Evaporation Soil/Groundwater Storage

Unsaturated soil layer (Zone of aeration

Percolation (Groundwater Recharge)

Interflow

Groundwater Discharge Saturated soil layer Baseflow

Water course Flood wave

Table 1. Processes of the Hydrological Cycle

As it is shown in the Table above, one can distinguish two types of processes: horizontal and vertical. And as part of the runoff formation the following figure shows the flow processes involved.

Figure 2.2 Vertical and horizontal flow processes 2.2 Main Processes of the Hydrological Cycle 2.2.1 Precipitation (rainfall) Precipitation is the main factor which controls the hydrology of a region. Therefore, it is of great importance to know its spatial and temporal distribution to get a better understanding of the soil moisture, groundwater recharge and river flows of a region. During the precipitation event, 2 types of processes are predominantly occurring over the surface. Concerning the state of water one can distinguish the following forms of precipitation.

They are given in Table 2.

Form Type of formation Liquid Solid

Snow Snow grains

Drizzle (drops cca

0,5mm) Snow pellets Graupel Hail (1-6 mm) Hail (ice balls size 5-50mm)

Direct condensation or sublimation in the atmosphere

(precipitation from clouds, rainfall) Rainfall

Ice needles(1.5mm) Frost Hoar

Indirect condensation or sublimation of water vapour close to the surface

( precipitation as condensation) Dew

Glazed frost

Table 2. Different forms of Precipitation

In comparison to the other meteorological parameters such as temperature or solar radiation that are relatively constant, precipitation often varies in space and time. 2.2.1.1 Precipitation mechanisms Precipitation occurs when a mass of moist air is sufficiently cooled to become saturated, and then condensate. Since, the most important cooling mechanism is due to uplift of air, there are three ways of precipitation causing the vertical air motion:

• Convective precipitation is the most heterogeneous and is typical of the tropics. It is formed when the air is heated near ground which then expands and rises. It cools as rises and becomes saturated. This lead to locally intense precipitation of limited duration. An example is a summer thunderstorm which is very intensive and appears locally.

• Cyclonic precipitation is considered as long-lasting precipitation. It is formed when

cold air mass meets warm air mass. Warm air is less dense and is forced upward resulting in cooling and precipitation. Cyclonic precipitation can be classified as frontal and non frontal. Cold advancing fronts move fast bringing intense localized storms. Warm advancing fronts move more slowly creating disperse and less intense precipitation. Those kinds 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 650 km.

• Orographic precipitation is formed by mechanical lifting of moist air over natural

barriers such as mountain ranges or islands in oceans. It has different intensity and prolongation, depending on the changes in topography. One can distinguish the following intensity grades and respectively the type of the precipitation:

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 the following condition is fulfilled:

( )224/5 ttN −⋅≥ Where is: N = rainfall height in mm t = duration of the rainfall event in min There is a very important dependence between the intensity and duration of a precipitation event. The shorter duration of the precipitation event, the more intensive it is. Contrary, long lasting rainfall has very low intensity. 2.2.1.2 Analysis of Precipitation Since in hydrology interpretation of precipitation is very important for water balance and rainfall-runoff analysis, it is necessary to obtain data from separated gauges. In order to determine the spatial and temporal distribution of rainfall, the network of gauging station is established. However, they provide only information about the height of precipitation at fixed points. There are large number of techniques that can be used to calculated the spatial distribution of rainfall such as: polygonal weighting, isohyetal (lines of equal rainfall), trend surface analysis, analysis of variance and kriging. The selection of the most appropriate technique depends on several factors, such as: the time available, expertise, density of gauge network, and the known spatial variability of the rainfall field. Thiessen polygon Method In order to achieve accurate estimation of the spatial distribution of rainfall, it is necessary to use interpolation methods, for this, the Thiessen* method is considered as the most important in engineering praxis. This method assigns weight at each gauge station in proportion to the catchment area that is closest to that gauge. The method of constructing the polygons implies the following steps: 1. Gauge network is plotted on map of the catchment area of interest. 2. Adjacent stations are connected with lines. 3. Perpendicular bisectors of each line are constructed (perpendicular line at the midpoint of each line connecting two stations) 4. The bisectors are extended and used to form the polygon around each gauge station. 5. Rainfall value for each gauge station is multiplied by the area of each polygon. 6. All values from step 5 are summed and divided by total basin area.

An example of spatial precipitation distribution according to Thiessen method can be appreciated in Figure 2.2.

Figure 2.3 Construction of Thiessen polygon

Although it is widely used in engineering praxis, this method has its shortcomings. For example, in mountainous areas, an irregular spatial distribution of precipitation can be formed over small distances, and for such circumstances the Thiessen method can yield erroneous results. In order to overcome these problems, more accurate methods are used, in which the whole area is “rastered” and each of those rasters are calculated according to the quadrant method. For each of the four quadrants is the nearest gauging station determined and the precipitation at the raster point is weighted according to the following relation:

Nii

jiNj hWh ⋅= ∑=

4

1

where is: wij = weight of the gauge station regarding the raster point j hNj = precipitation height at the raster point j hNi = measured precipitation heights at the gauging station i

In order to achieve a homogenous field, it is necessary to calculate the weight ij w as spatial distance according to the following formula:

∑=

−= 4

1

1

ii

iij

d

dW

If weather conditions differ considerably between gauging station and raster point (e.g. river, mountains) or there are significant topographic changes between them, it is necessary to consider those differences using correction parameters and applying one of the convenient methods (e.g. Kriging-method). Other methods that measure areal and temporal precipitation continuum are still in testing phase, as for example: precipitation radar. Finally, DWD weather forecast service of Germany and Europe provide information about the area with rainfall 48 hours in advance. The quantitative distribution of this forecast quality value is, however, interesting only for water management purposes. 2.2.2 Interception Precipitation that is falling on the surface is partly retained by the canopy and only a part may actually reach the ground beneath. This storage of water above the ground surface, mostly in vegetation is called interception. Depending on the density of the vegetation cover a proportion of the rain is intercepted by the leaves and stems and temporarily stored on its surface. First of all, the leaves are becoming wet as fine drops are collecting on their surface and at the end; the intercepted water is drained from the leaves. The intercepted water can be drained in different ways. It can drip down from the leaves or can be drained along the stem of the plant. Parallel to this drainage process, part of the intercepted water is evaporated back into the atmosphere, and so take no part in the hydrological cycle, this is called interception loss. Before the precipitation really reaches the ground, it is further partly retained by near ground plants and leaves. This process is similar to the interception of bigger vegetation species. And the water which reaches the ground constitutes the net rainfall. Interception plays an important role in the calculation of water balances in hydrology, because the net rainfall is generally less than the total rainfall falling in top of the vegetation cover, due to almost one third of the total precipitation can be retained as interception and than evaporate producing considerable interception loss. The storage capacity and consequently the interception capacity, heavily depends on the vegetation cover. The most important vegetation parameters that shape the interception process are: specific area under vegetation cover, lifestage of the plants and the characteristics of the land. Interception is highly seasonal dependent (seasonal landuse). For the maximal interception capacity, the following values are considered:

Deciduous without leaves with leaves

till 1mm 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 Throughfall and stemflow do not begin until the storage capacity of the canopy is completely reached. Once storage capacity is exceeded, then additional 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 some additional storage capacity available, so the drip rate decreases temporarily. The main processes related to interception are given in Figure 2.4.

Figure 2.4 Main interception processes Where Pa Actual precipitation TF Throughfall SF - Stemflow I - Interception E – Evaporation Since the necessary data for this differential approach is usually not available, Interception loss is usually calculated according to the threshold value method. In this method, the whole vegetation cover is considered as storage and the following continuity equation is obtained:

)()()(´)( titETtidt

tdInan

c −−=

Where:

Ic(t) Actual content of the interception storage mm i‘n(t) Intensity of the precipitation on the soil mm/h ETa(t) Actual evapotranspiration rate from the interception storage mm/hin(t) Intensity of the rainfall that reaches soil mm/h

2.2.3 Snow-hydrological processes The snow is a form of precipitation and it is formed if the air in a cloud is below freezing point. The longer temperature stays below zero (expressed as mean daily temperature), the thicker the snow cover is (accumulation phase). If the temperature rises over freezing point, warm air compresses the snow cover and increases its compactness. It is about the same effect that is caused by rain. Dry snow usually has a water content of 10% at temperatures below zero. However, when snow starts being melted at temperatures above zero, it will not release water before it achieves the “Maximal compactness” (~ 45% of water content). This maximum compactness is achieved if it is not possible to increase density of the snow pack any more. The excessive amount of water, in form of thawed snow, penetrates the ground (ablation). The snow melt processes can be mathematically described by the Snow-Compaction-Method. This method is based on the physical processes during the snow melting, while the water loss through evaporation is neglected as the order of those values is too small (0.1-0.8 mm/d). In this method, the influence of wind speed, humidity and temperature corrections, depending on the elevation are considered as averaged parameters.

Figure 2.5 Snow Processes

2.2.3.1 Accumulation The increase in the snow depth ∆l occurs when the precipitation occurs at temperatures below zero. Mean daily values are considered for temperature. The snow depth resulting from the precipitation is calculated by the absolute amount of precipitation and the water content of snow according to the following equation:

100)()( WAtPtl ⋅

=∆

where:

P(t) Precipitation [mm] WA Initial water content in snow [%]

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

WD PP ⋅−= 4774.04.147 where:

PD Snow height in % of the initial height [%]

PW Accumulated water of the water equivalent of the dry snow (initial water content) [%]

Considering the above mentioned relation, the compression ∆h is calculated as following:

100)4774.04.147(

100WADA PhPhh ⋅−⋅

=⋅

=∆

where:

hA Initial snow height [mm] ∆h Compression of the snow pack [mm]

The indication of accumulated water PW (compactness) requires the estimation of the potential snow melt rate. It can be quantified according to the Temperature-Factor-Method. In this method, the snow melt rate is divided into the temperature dependent and the temperature independent part. The temperature dependent part is proportional to air temperature and the temperature independent part is represented by a constant. This constant is estimated based on the average rates of the absorption of the snow pack, wind speed, relative humidity and the cloudiness. Further, it is necessary to consider the heat flux that reaches the snow cover by precipitation. These dependencies are shown in the following equation (MEUSER 1989):

)()(0125.0)()( tTtitTaati PTus ⋅⋅+⋅+= where:

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]

ip Precipitation intensity [mm/h] Accumulated snow water content PW is dependent on the snow melt rate, according to the following equation:

)()()()(

tWtPtWtP

rW

+=

where:

W(t) Absolute water content in the snow pack [mm] Wr(t) Water content reduced by portion of melted snow Wr(t)=W(t)-is(t).∆t ∆t Time between the simulations [h]

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

( tPhtWtiA ) ∆⋅−= /100/)()( max , falls W(t) > h . Pmax where:

iA Water removal in [mm/h] Pmax Maximal snow compactness [%] (i.d.R. 45%)h Snow depth [mm]

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

2.2.4 Evapotranspiration 2.2.4.1 General Concepts The physics of the evaporation process is based on the process of providing sufficient energy for breaking the bonds between water molecules. The fact of supplying the system with heat, causes water molecules to become increasingly moveable which results in increase in distance between them. The higher temperature is, the more water molecules escape the surface into the lower layers of the air (liquid water transformed into gaseous state) . The physical cause for this phenomenon is called Brownian movement. For changing the aggregate state from liquid to gaseous, it is necessary to provide higher amount of energy (2450 J/g) and then for the melting process (340 J/g). In other words, Evaporation is loss of water from a wet surface through its conversation into 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). Another important mechanism which is influenced the mass transfer of water is the transpiration process, which is based on the concept of a continuous pumping (by growing plants) of water from the soil which is then moved up to leaves and lost, through biological processes in vegetation, into the atmosphere (DIN 4049,1994). However under field conditions it is not possible to separate evaporation from transpiration totally, thus we are generally concerned with the water loss or evapotranspiration from a basin. The sum of soil, interception and open water evaporation and transpiration is called evapotranspiration. The term Potential evapotranspiration (Maximal) “Etp” is by definition the rate of evapotranspiration from an extended surface of an 8 - 15cm tall green grass cover, actually growing, completely shading the ground and not short of water (DIN 4049, 1994). This maximal evapotranspiration is also called reference evapotranspiration. The term, Actual evapotranspiration “Eta” is by definition “the rate of the evapotranspiration from a surface under field conditions and with limited water supply.” Under those conditions, the actual evapotranspiration is much less then the potential, especially if the 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 of precipitation is 64%. At the global scale, evapotranspiration and precipitation are the principal elements of the hydrological cycle. The evapotranspiration rate depends on different parameters (areal and temporal variable). Those parameters are defined considering atmosphere, soil and type of vegetation cover. While it is possible to assess accurately those parameters for open water and atmosphere, it is very difficult to quantify forces in soil and vegetation cover. In soil, for example, the forces depend on radius in capillaries, and in vegetation this process is regulated through small pores on the leaves (stomata). Soil, plant and atmosphere form parts of a continuous flow in which water moves at varying rates. In soil, water moves under the influence of moisture gradient toward the roots of the plant. It is then absorbed and travels up the plant stem to the plant

leaves from where it is finally vaporized. Plants can control transpiration by varying the opening of stomata. In that way they can prevent dehydration. 2.2.4.2 Methods to determine evapotranspiration Despite the crucial importance of evapotranspiration in hydrological cycle, it is very difficult to measure and quantify it. In order to assess evapotranspiration, the following methods are applied:

• Direct methods • Indirect methods • Computational methods

Direct measuring of evapotranspiration is not possible; as such a method has to manage turbulent mass transport of water vapour, which technically has not been solved yet. Measuring techniques, such as evaporimeter, are although considered as direct methods, but can give only reference values. In this method, evapotranspiration is calculated from water balance equation, where the other elements (variables) of the equation are measured. Indirect methods are based on the interrelation between directly measured meteorological parameters and water vapour transport or heat flow of the evaporation process in the air layer close to the surface. As the measuring methods that are mentioned above are very complex, computational methods are developed. Computational methods calculate evapotranspiration using energy budget equation and aerodynamic principles. 2.2.4.2.1 Direct Methods The results obtained by the instruments for direct measuring of evapotranspiraton (Evaporimeter) do not coincide with the real potential or actual values since those results depend on the type of the instrument and local conditions (i.e place where the equipment is located). It is however possible to calculate reference values using correction factors and formulas. a) Atmometer(called evaporimeter) Atmometers (Evaporimeters) are of two types, those that measure the evaporation rate from a free water surface and those that measure it from a continuously wet porous surface. a.1) Piché evaporimeter As part of the instruments using a porous surface to measure evaporation, the “Piché evaporimeter” is used. It consits of a small tube reservoir filled with water from which water evaporates through a porous material (filter paper disc) simulating an evaporating surface. The amount of water evaporated through the paper is read at the graduated tube reservoir (in ml) and this value is to be corrected as a function of the filter paper size. Due to the different physical conditions the rate of evaporation of the Piché evaporimeter is higher than actual evaporation of free water or bare soil surface. Therefore this correction is necessary.

Because evaporation rates are so sensitive to the water supply, and the nature of the evaporating surface, data collected by evaporimeters often do not reflect true evaporation processes; hence, evaporimeters have limited use. a.2) Evaporation pans A more useful quantitative measure may be obtained from a tank with a free water surface. Evaporation pans are open vessels, which are filled with water. Because of its simplicity of manufacture and operation, it is probably the instrument used most widely to estimate potential evaporation. The values for water loss obtained for an evaporation pan are used as reference values for water bodies or potential evapotranspiration (ETp). Although this method is very simple and widely used it has its shortcomings which have to be considered. The evaporation from a pan can differ significantly from that from an adjacent water body or land covered with vegetation. Therefore it is necessary to accommodate these differences using empirical pan coefficients. Those coefficients vary significantly with siting and pan design as well as with climatic factors. The most widely used instrument is the “Class A” pan introduced by U.S. Weather Service, which has the characteristic of being installed above ground. 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

Figure 2.6 “Class A” Evaporation Pan a.3) Raft evaporation tanks

The methods that explained above, give us only reference or approximate values of evaporation over land. Another method is measuring the evaporation over the free water 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. Different construction materials can be used, such as: brass tank with copper bottom, white-enamelled or with silver, bronze painted sheet iron. The results obtained during those measurements should also been corrected. This correction is between 3-5%. b) Lysimeter Lysimeter is an instrument for determination of the water balance of an isolated block of soil and vegetation growing on it, with known dimensions and characteristics. Then evaporation is calculated from a mass balance of the difference between precipitation and drainage from the base of the block, together with measurements of changes in soil water storage. It is designed in a way that reproduces the soil type and profile, moisture content, and type and size of the vegetation of the surrounding area. Soil water changes can be determined by differente types of lysimeters: weighing and non weighing lysimeters. Weighing lysimeters, measure the weight of the entire mass of soil in regular time intervals (e.g. daily), the change in water storage is determined by weight difference:

∆W = (WE - WA) Further, the equation for the actual evaporation (ETa) in function of drainage from the soil volume (SW) and precipitation (P in 1m height) is derived from water balance equation:

ETa = (P - SW - ∆W) / ∆t Where: P: precipitation (mm) SW: soil drainage volume (mm) ∆W: change in water content (mm) Non weighing lysimeters (drainage gauges-measurers) are used to control the lateral water movement through the soil, and are suitable for determination of the average values of evaporation over a longer period of years. It is calculated as a difference between precipitation (total water input) and drainage water (collected at the bottom of the lysimeter), as in this case the water content of the soil 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 to provide the natural conditions in the vessel. It is achieved by filling the lysimeter with soil in layers as they are stratified in the natural compression.

Figure 2.7 Lysimeter c) Determination of water content in the soil Evaporation of bare soil, if the groundwater is far enough from the sample area, can be determined by termin measurements of water content in soil (W). There is a variety of practical methods for those measurements of soil moisture, from which the following ones are most widely in use: c.1) Gravimetrical methods: Soil sample is extracted from the field and the water content is determined in the laboratory by measuring the mass of water content to the mass of dry soil (after being dried in an oven at 105°C). When converting the water content into soil moisture value, the compaction of the soil layers is to be considered. Shortcoming of this method is contained in the fact that the natural soil profile is disturbed (disrupted). The following methods are overcoming this deficiency.

c.2) Tensiometer This device measures the pressure potential or matrix potential (capillary tension) in soil at various depths. This is the force with which water is held in the soil. Pressure potential or tension (which is the positive value of pressure potential) is a measure of this force. The main disadvantage is that it works only from zero to about ~0.8 bar, which represents a small part of the entire range of available water.

Figure 2.8 Examples of tensiometers c.3) Neutron scattering (Nuclear Technique) This technique is widely used for estimating volumetric water content. The operation principle is based on the following: fast neutrons emitted from a radioactive source are slowed down (thermalized) by hydrogen atoms in the soil. Since most hydrogen atoms in the soil are components of water molecules, the proportion of thermalized neutrons is related to soil water content. Thus, the gradient of deceleration of neutrons measured is proportional to the soil moisture. This technique has its shortcomings such as: radiation hazard, insensitivity near the soil surface and insensitivity to small variations in moisture content.

Figure 2.9 Measuring of water content with a neutron scattering device

c.4) Time-Domain-Reflectometry (TDR)- Method (Electromagnetic Technique) TDR defines the dielectric relationships between soil, rock and other materials, and their moisture content in situ. It converts the travel time of a high frequency, electromagnetic pulse into volumetric water content. The operation principle is based on measuring the velocity of propagation of an electromagnetic pulse along the transmission lines (wave guides that are inserted or buried in the soil). The velocity of propagation (travel time) of the high frequency in soil is influenced primarily by the water content and electrical conductivity. This instrument helps to estimate the apparent dielectric constant of the soil and allows the determination of its water content. This method is limited only to loose rocks. In order to determinate the evaporation of the soil, it is necessary to consider following two cases:

• 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 according to the following equation:

tWPEta ∆

∆−=

Where P is the precipitation (mm) ∆W is the change in water content (mm)

• If the water shed is in the root layer, the plants take up a certain amount of water as

well as a part of ground water discharge. Therefore, total evaporation can be calculated according to the following formula:

( )

tSWWWPEt uo

a ∆−∆+∆−

=

where: ∆Wo Change in water content above the watershed ∆Wu Change in water content below the watershed SW Soil drainage volume

In order to determine SW it is necessary to determine permeability of an unsaturated soil layer and afterwards the drainage (infiltration) according to Darcy law. c.5) Turbulence-Correlation method (Eddy Flux) This is the most direct measure of water flux with minimum theoretical assumptions and therefore is the preferred micrometeorological technique. The principle is that near the surface, turbulent eddies, within the body of the moving air, cause movements of the evaporated water. The measurement of the flux of water vapor in turbulent air requires instantaneous and simultaneous measurement of vertical velocity and vapor density (or air temperature). Vertical movement of water vapor can be calculated from the temporal averaging of the products of the fluctuations. To determine the fluctuations, different measuring instruments are used (e.g. ultrasonic anemometer for wind velocity, LYMAN alpha or infrared hygrometer for humidity.

2.2.4.2.2 Indirect Methods a) Gradient measurements in the near the surface air layer This method assumes that evaporation causes horizontal and vertical turbulent exchange. It causes the flow of water vapour, heat and impuls and can be calculated applying diffusion equation. b) Energy Balance (Budget) Method This method assumes that the water balance between earth-atmosphere can be treated as the mass of water conserved at all times, where evaporation is the connecting link between the system's water and energy balances. According to this method, it is possible to calculate evaporation as:

HGLERn ++= where:

LE Latent heat flux out (Wm-2) Rn Net radiation (Wm-2) G Soil heat flux (Wm-2) H Sensible heat flux (Wm-2)

G

H LE

Rn

Figure 2.10 Transfer of energy scheme (Energy Balance) The amount used in Evaporation (E) can be calculated as :

LGHR

LLEE n −−

==

where:

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

The individual components of the energy balance equation are determined by the following measuring methods: b.1) Solar radiation It is measured directly by special radiation balance measures with range of 1,5-2,0m. Those devices measure the difference between upwards and downwards oriented radiation from short and long wave radiation. Alternatively, radiation can be determined as a difference between other wave lengths. b.2) Measurements with pyranometer (Global radiation) Shortwave solar radiation and sky (diffuse) radiation is detected on the horizontal sensor surface with a thermo-sensitive element (wavelength range ~ 300 to 3000 nm). Alternatively, it is possible to measure relative sunshine duration S/S0, where the sunshine duration is registered by the instrument called sushine autograph. It consists of a glass bowl with special effect, so that the sunshine is registered on the adhesive tape.

Figure 2.11 Pyranometer b.3) Heat Flux It is determined with so called ” Heat Flux Plates ” under closed conditions, or by the gradients of the soil temperature applying the heat conduction equation. Perceptible Heat Flow: is the transfer of heat that we can feel. By conduction, air close to the surface is warmed and by convection the warmed air is transport away from the surface to the surroundings. With application of the fluctuation method, wind and temperature components under dissolution of their turbulent fluctuation sizes are measured and set in correlation to the perceptible heat flow. Latent Hat Flow: Through evaporation of liquid water at the surface of the earth, the surface is cooled (i.e. loses energy). Through condensation of water vapor in the atmosphere, air is warmed (i.e. gains energy). Together this transport of latent heat acts to take energy away from the surface and transfer it to the atmosphere.

2.2.4.3 Calculation of evaporation To calculate the rate of evapotranspiration the following methods are considered:

• Aerodynamic method (Dalton-process): deals with the upwards diffusion of water vapor from the surface, and is concerned with the “drying power” of the air, including its humidity and the rate at which water vapor can diffuse away from the evaporating surface into the atmosphere.

• Energy balance methods: or thermodynamic approach. Estimates the latent energy

available for water to change from liquid to gas. • Combination of the above mentioned methods

2.2.4.3.1 Determination of evapotranspiration of an area with vegetation cover In most cases, only total evaporation from an area plus transpiration (from vegetation) is of real interest. Penman- Monteith Model is a method that combines energy balance and mass transport approaches for determining the evapotranspiration. It assumes that the heat flux into and out of the soil (G) is small enough (~2% of total incoming energy) to be neglected and that the plants are taking the whole available energy for evapotranspiration. Thus, the net solar radiation is distributed between heating air and evaporation. Evapotranspiration is limited by three conditions: small pores on leaves-stomata (physiological influence of vegetation cover), aerodynamic and meteorological conditions. 2.2.4.3.1.1 Reference or potential evapotranspiration If those conditions are known, it is possible to determine potential evapotranspiration ETo according to DVWK, booklet 238 the grass reference evapotranspiration according to FAO-Standard:

( ))34.01(

)(27375.3

2

2

Vs

eTeVTL

GRsEt

sN

o ⋅+⋅+

−⋅⋅+

+−

=γ

γ

where (according to DVWK-bulletin 238/1996):

ETo mm Grass-/Reference evapotranspiration (FAO-Standard) s hPa/K Slope of the saturation vapour pressure curve for water RN W/m² Net solar radiation G W/m² Heat flux between soil and atmosphere L Heat required for evaporation of 1mm water T °C Air temperature V2 m/s Wind speed, at 2m above the ground γ hPa/K Psychrometric constant (= 0,65 hPa/K)

To solve the equation above it is necessary to consider the following equations: G can be neglected γ is a constant (γ = 0,655 [hPa/K]). Heat required for evaporation of 1 mm water

L = (249,8 - 0,242 * T) [J / (cm² ( mm)] Saturation vapor pressure:

+⋅

⋅= TT

s Te 27348.7

1011.6)( Slope of the saturation vapor pressure curve for water:

( )212.2434284)(

TTes s +

⋅=

Net solar radiation RN:

( ) ( ) ( )5.04 044.034.09.01.015.2731 eSSTRR

oGN ⋅−⋅

+⋅+⋅−⋅−= ςα

where:

ς = 0,49 ( 10 -6 [(J : cm2 × k4)] γ Joule/m2k4 Stefan-Bolzmann-Constant α diml. Albedo for grass and agricultural used land S h Sunshine duration So h Astronomic possible sunshine duration RG W/m² Global solar radiation R0 W/m² Extraterrestrial radiation ς diml. Function considering the selected day in year φ Grad Latitude

Global radiation:

( )5.00 044.034.055.019.0 e

SSRR

oG ⋅−⋅

⋅+⋅=

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

⋅+⋅=

oG S

SRR 55.019.00

Extraterrestrial radiation:

[ ])1(sin)51(18.0sin08.79.92450 −⋅−⋅+⋅+⋅= ξϕξR Where ς is function that considers the day in year (JT):

39.10172.0 −⋅= JTξ If there is no available measuring data for sunshine duration, it is possible to calculate it according to the following formula:

−⋅= 35.082.1

0RRSS G

o with S = 0 for S < 0

Astronomic possible duration of sunshine:

−

+⋅+=6

513.4sin3.12 ϕςoS

This theoretical analysis clearly shows that, just by measuring air temperature, sunshine duration, wind speed and relative moisture as daily values it is possible to determine evapotranspiration.

Grass reference crop (well watered grass)

=

Climate: • Temperature • Radiation • Wind speed • Relative Moisture

+

ET0

Figure 2.12 Main data required for ET0 calculation 2.2.4.3.1.2 Correction of ET0 with vegetation The calculated value for reference evapotranspiration ET0 is corrected with the correction factor Kc for crop evapotranspiration under standard conditions (ETpfl). The Kc coefficient represents the difference in evapotranspiration between the crop and the reference grass surface.

opfl ETKcET ⋅= Factor Kc can be found in Table 6.2 in DVWK-bulletin 238/1996. In order to consider forest conditions additional correction factor for higher values of interception 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 on the age and type of forest.

One important difference between the evaporation from open water and from vegetation cover is that in open water the evaporation is not determine by the water regime of the plants. However, the heat transport occurs also in water body and has to be considered for the balance equation. 2.2.4.3.1.3 Determination of the actual evapotranspiration The crop evapotranspiration under non-standard conditions (EVA) is the evapotranspiration from crops grown under management and environmental conditions that differ from the standard conditions (ETpfl). The EVA considers, besides the vegetation cover (ETpfl), the soil water tension (reduction of soil moisture in root layer). Therefore, the EVA is a function of the actual soil moisture and the vegetation specific transpiration as daily values. WENDLING proposes the following balance equation:

( ) ipfl RETtPtPEVA ⋅−−= )()( Where:

P(t) mm Precipitation Ri diml. Reduction factor

Reduction factor Ri, considers the deviation ratio between the soil water content and the wilting point, is calculated as follows:

1( ) 0.1

1

root

ipfl

WP hETPSW tR WP ET

FC

⋅−

= + ⋅−

, where 0 < Ri < 1

Where: WP % Mean wilting point in effective root layer hroot dm Effective root depth SW(t) mm Soil moisture in the effective root layer in period t FC % Mean field capacity in the effective root layer

mm/d Mean long-term value of ET pfl in the vegetation period Further, according to WENDLING (DVWK-bulletin283/196) it gives: Ri = 1, P(t) > ETpfl Ri = 1, Ri > 1,0 Ri = 0, Ri < 0,0 From the relations showed above, it can be deduced that: if the water content reaches the Wilting point (WP), the reduction coefficient will approximate nearly to zero and therefore the EVA is only dependant on precipitation. On the other hand, if the water content in the soil reaches the field capacity (FC), the reduction factor will be ~1 and the EVA will not be influenced by precipitation and the maximum EVA will be reached (ETpfl).

2.2.4.3.2 Determination of evaporation from open water surface The rate of open water evaporation is determined by different factors: meteorological (energy, humidity, wind) and physical (size of water body, water depth). The evaporation from open water can be determined according to Dalton’s law.

( )eTevfEW WOs −⋅= )()( in [mm/d] Where: TWO [°C]: temperature of water surface e: vapor pressure es: saturation vapor pressure f(v) : wind function, expressed as follows:

cvbavf ⋅+=)( Where constants “a” and “b” (according to Richter, see DVWK-bulletin 238/1996) are calculated depending on the wind speed. Temperature of water surface Temperature of the water surface is rarely measured. It is usually calculated as a function of the measured mean air temperatures T [°C] in the following way:

• For the period without ice April - December

qTpTWO +⋅= * Empirical constants “p” and “q” can be found in the Table 5.2 of DVWK-bulletin 238/1996 and:

∑+

+⋅+

=mn

mnTmn

T1

)(1*

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

0.404.1 +⋅= zm Where “m” depends on the water depth z [m]

• For winter months January till March:

12.0*98.0 −⋅= TTWO

where m is set to constant = 10, thus T* is given as follows: ∑+

⋅=111

1)11(

111* TT

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 and that 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 infiltrate into the soil. To which extent the infiltration in the soil occurs, depends on the physical characteristics 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 soil moisture modelling. It implies calculation of the actual soil moisture SW (t) using continuity equation for the unsaturated soil layer. Elements of the vertical water flow are shown in Figure 3.1.

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

As it can be observed there are many different processes that operate either from the ground surface (infiltration, evaporation) or from the bottom layers of the aeration zone (vadose zone) such as groundwater recharge or capillary uprise. This development of soil water potential gradients in the vertical direction creates a predominant vertical movement of soil water in the aeration zone. 3.1.1 Soil layer distribution Below the surface the soil pores contain both air and water. This region is know as vadose, unsaturated or aeration zone. At the top of the unsaturated zone is the zone where the roots of plants can reach the soil water. Under some conditions (slope, soil characteristics) water can flow laterally in the unsaturated zone, which is called interflow. Excess water in the unsaturated zone can migrate downward by gravity recharging the grounwater zone. There is an intermediate zone where the pores are filled with capillary water which can be uptaken by the roots depending on some conditions (soil texture, distance between roots and water table). At some depth, the pores are saturated with water, the top of the saturation zone is called water table and the water stored in this zone is known as groundwater. In this zone water moves as groundwater flow through the soil layers in a horizontal direction where the

pressure gradient is the driving force of the flow. Figure 3.2, shows the different types of soil layers explained above.

Figure 3.2 Distribution of the soil layers 3.1.2 Modelling soil moisture in unsaturated zone For the purposes of mathematical modelling the soil is divided into layers with homogeneous structure- lamellas (as shown in Figure 3.3).

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

to soil moisture (Source: Ostrowski)

Thus, the 1D water balance equation gives:

[ ]( ) ( ) ( ) ( ) ( ) ( )ii i i i

dSW t Inf t Perc t Intf t Eva t CU tdt

= − + − +

where:

SW Available water (for plants) [mm] Inf Infiltration [mm/h] Perc 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 determined from inflow components: Infiltration rate (Inf) and capillary uprise (CU) as well as Percolation (Perc) and Evapotranspiration (Eva). Before each of those components is thoroughly analysed it is necessary to be introduced to soil characteristics and soil water. 3.1.3 Soil water constants Soil moisture is the water content that can be removed via drying the soil sample at 105° C in an oven. Seasonal differences in water inflows and losses in soil are summarised under the term “soil moisture regime”. The amount of rainfall that infiltrates into the soil, percolates only partly to groundwater. The rest remains in the unsaturated soil layer i.e remains above the groundwater level. In the case of saturated soil layer, water fills all available pore space in that layer. Important parameters (constants) considering the extent to which the water is attached to the soil particles and size of pores are :Wilting point (WP), Field capacity (FC), Available Field Capacity (aFC), Air Capacity (AC), Total Pore Volume (TPV) and Maximal Water Capacity (SWMAX). They are defined according to the Bodenkundlicher Kartieranleitung, Arbeitsgruppe Boden, 1982. They are used to facilitate comparisons between different hydrological status of different soils. Field Capacity (FC): It is defined as the amount of water that remains 2-3 days after the saturation of a soil with water (after gravity movement of water has largely ceased). As a rule, soil moisture tension of 101.8 and 102.5 mbar occurs in soil. It varies considerably depending on the clay and silt content, soil texture, content of organic matter, humus form, coherency of soil and ion characteristics in soil colloids. Field capacity rate depending on soil texture is as follows:

Sand<Loam<Silt<Clay<Peat

Permanent Wilting Point (PWP) Permanent wilting point (PWP) is the lowest amount of water that is hold in the soil and which plants are unable to use. Soil water that is incorporated into soil structure with higher soil moisture tension than 104.2 mbar is not useful for majority of the plants through root system. The remained soil moisture is considered as “dead water”. Available Field Capacity (aFC): Available Field Capacity is the difference between Field Capacity (FC) and wilting point of a soil. It is considered as water available for plants to extract from the soil moisture zone. According to the definition of WP and FC, the values for the soil moisture tension (SMT) are between 101.8 and 104.2 mbar. Air Capacity (AC): It specifies the size of the drained macro-pore space. The water percolating in the soil has flown downwards and is not available for the plants any more. It coincides with the size of the pore space that is filled with air at the field capacity conditions. Total Pore Volume (TPV): The amount of all cavities in soil expressed in Volume-% Maximal water capacity (SWMAX): It is the maximal water content of soil (by total water saturation). If it is expressed in Volume-% it coincides with the total pore volume (TPV). 3.1.4 Forms of soil water According to BEAR, the following soil water types are considered in the unsaturated soil layer. (See Figure 3.4, BEAR). Hygroscopic water Hygroscopic water is held in the soil between air dry and oven dry and can only be removed from the soil through heating. Unavailable water remains when soil is drier than wilting point. Unavailable water is soil water held so firmly to soil particles by adsorptive soil forces that it cannot be extracted by plants. Neither evaporation nor percolation into deeper soil layer is possible. 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 occurs in this zone. The capillary forces are held by formation of menisci at the contact points with the mineral particles, so that neither percolation into deeper soil layers nor discharge between different soil layers is possible.

Gravitational water (excess soil water) This zone is above the Field Capacity. Gravitational water drains or percolates readily by gravitational force. Since drainage takes time, part of the excess water may be used by plants before it moves out of the root zone. Alternatively, the water flows horizontally forming interflow. Distribution of those zones depending on the soil moisture can be appreciated in Figure 3.4.

Figure 3.4 Porosity according to Bear Within those soil moisture zones, it is possible to simplify the dependencies of the Infiltration, Percolation, Transpiration and Interflow rates and show them as linear functions of soil moisture which can be appreciated in Figure 3.5.

Figure 3.5 Infiltration (Inf), Evapotranspiration (Eva), Percolation (Perc) and Interflow (Intf) express as 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 (FC) as well as between Field Capacity (FC) and maximal soil saturation (SWMAX). 3.1.2 Infiltration Water that falls on the ground in form of precipitation can either infiltrate or runs off as overland flow. Infiltration rates mostly depend on the soil type, surface cover, actual water content and surface structure. In case of sealed areas, almost 100% of the rainfall is converted into overland flow and no infiltration occurs. In case of natural areas, this rate depends predominantly on the surface soil structure, which is further shaped by the pore volume in the soil. It is 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 are illustrated in Figure 3.6.

Figure 3.6 Pore distribution in the soil Macro pores are important for the infiltration process since water percolates 100-400 times faster through macro pores than through the micro ones. The macro pores are created by plant roots, channels and paths of the soil fauna (e.g. mice, moles) soil aggregation and the soil cultivation. The most important factor that shapes soil porosity created by fauna is earth worms, which make a uniform vertical pore system. Beside that, physical soil parameters, such as granulation, particle shape, soil structure and soil compaction are also of relevance. As it can be deduced from the figure 3.5, “Cinf” determines the rate of infiltration. The higher values for Cinf are, the smaller portion of precipitation is converted into overland flow. Due to the large number of relevant parameters are required for infiltration, an accurate determination of the infiltration capacity (Cinf) is possible only by performing field tests. Approximately, it is possible to estimate the infiltration capacity from soil type, effective compaction according to Bodenkundlicher Kartieranleitung, Arbeitsgruppe Boden, 1982 and

hydraulic conductivity obtained for soil saturation conditions kF and presuming loose soil (Ld 1-2):

Cinf = kF.1/2 Real infiltration is determined as function of the soil structure, slope inclination and actual water content. In the zone of Wilting point, this value coincides with the infiltration capacity. It is linear till it reaches the point of maximal soil capacity. Concerning the values for soil moisture portions corrected for Wilting point, the potential Infiltration capacity curve of the soil “Infp”, considers the following conditional equation:

WPSWtSWSW

CtInfpi

iiii −

−⋅=

max

maxinf

)()(

Where the actual infiltration “Inf” depends on the intensity of the rainfall and percolation rate of the soil layers. For soil layer i = 1:

Inf1(t) = Infp1(t) if iN,eff(t) > Infp1(t)Inf1(t) = iN,eff(t) if iN,eff(t) ≤ Infp1(t)

For soil layer i > 1:

Infi(t) = Infpi(t) if Perki-1(t) > Infpi(t) Infi(t) = Perki-1(t) if Perki-1(t) ≤ Infpi(t)

3.1.3 Percolation Percolation is water trickling downward through the cracks and pores in the soil and subsurface material, similar to the infiltration process. However, it reaches maximum at the field capacity conditions. Percolation capacity (Cperc) heavily depends on the soil structure and texture, air capacity and soil compactness. It can be estimated by hydraulic conductivity and for higher compactness of the soil, this equation gives:

Cperc = kF,4/5 Considering those assumptions, potential percolation “Percp” from soil lamella i can be calculated as follows: , if SWi < FCi 0

=)(tPercpi

ii

iiiperc FCSW

FCtSWC−−

⋅max

)( , if SWi ≥ FCi

3.1.4 Capillary Uprise Capillary uprise from ground water in the effective root layer close to the groundwater horizon (level) is of great importance for water balance assessment. It can be calculated based on the water conductivity and suction power (force) and is expressed as Capillary Uprise (CU). It varies as a function of the depth of the groundwater level (GW) below the lower bound of the effective root layer and the soil characteristic. For sand the capillary uprise reaches 3mm/d if the groundwater level is 5dm below the lower boundary of the effective root zone. If this distance is 8dm, capillary uprise is only 0.2mm/d. In order to estimate the capillary uprise rates, die Bodenkundliche Kartieranleitung and DVWK bulletin 238 are used. 3.1.5 Interflow It is the part of the precipitation which infiltrates the surface soil and moves laterally through the upper soil horizons above the water table toward surface waters. It is also called subsurface runoff (see baseflow). The interflow “Intf” from the soil lamella i can be calculated as the difference between potential percolation “Percp” and actual Percolation Perc.

)()()( tPerctPercptIntf iii −= In order to consider this lateral movement of water in the soil, this value is multiplied to the ordinate of the time-area function (see chapter 4), that discharges in the time interval ∆t.

)(6.3

1)( tIntfAtQ iintfiI ⋅⋅=

where:

Intfi Interflow through the lamella i [mm/h] QIntfi Interflow through the lamella i [m³/s]

Ai Contact surface area [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 moisture of the lamellas above the observed lamella i. At the lower boundary it is necessary to correct the value for the soil moisture as there are no chargeable lamellas bellow it:

, if iSWmax iiii SWtIntfAtSW max)()1()( >⋅−+ =)(tSWi

, if )()1()( tIntfAtSW iii ⋅−+ iiii SWtIntfAtSW max)()1()( ≤⋅−+ If the first condition is fulfilled, then is:

iiii SWtIntfAtSW max)()1()( −⋅−+ returned to lamella i-1.

3.2 Mathematical solution to soil water equation The continuity equation for inhomogeneous, linear differential equation in its general form is expressed as:

12 )()( CtSWCdt

tdSW=⋅+

Constants C1 and C2 are calculated in dependence of the Infiltration, Percolation, Evapotranspiration and Interflow. CHAPTER 4 - PROCESSES OF DISCHARGE CONCENTRATION 4.1. General Information

The discharge formation can be mathematically described as a predominantly vertical process and therefore, the catchment area is divided into individual homogeneous vertical layers (columns).

The next step of the rainfall runoff modelling would be the mathematical definition of the runoff formation. For this to be modelled, the physical process considers the transformation of the effective precipitation of a catchment area into a discharge hydrograph at the outlet node of this catchment area. In one detailed rainfall runoff model, each discharge component is separately considered and afterwards linearly superpositioned. It means that each rain input generates a response component (hydrograph) which after, all individual response components are superimposed to form one hydrograph for the rainfall event (see Unit Hydrograph).

In principle, it is possible to describe accurately the process of the discharge concentration applying the partial differential equations. But the closed solution to this equation is only rudimental due to the high numerical complexity of the system. For example, in the SHE-Model (Système hydrologique europienne) for the individual components such as groundwater, interflow and surface runoff a system of differential equations is solved and the interaction between those components is implemented in the form of boundary conditions. For example, for the surface runoff the 2 dimensional depth-averaged shallow water equations for quasi laminar flow is applied. In order to obtain a numerical solution for those equations, the whole catchment is sub divided into raster elements and for example, applying the finite element method, it is solved for all raster cells. The wide application of this hydrodynamic model for description of the rainfall runoff 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 others fail to a certain extent in describing accurately the basic mechanisms of the runoff formation. It is obvious that it is

necessary to apply more detailed numerical decomposition and more accurate spatial data. Both of those requirements are not possible to achieve in the foreseeable future.

Therefore, for the calculation of the runoff formation, the so called “hydrologic model approaches” dominates. It assumes that the discharge concentration can be decomposed into translation and retention process, where both of them can be linearly superposed. As it can be modelled by the equation:

( )0

( ) ( )t

e ffQ t i u t A dτ τ τ= ⋅ − ⋅ ⋅∫

The translation refers to the process of the temporal delay of the water. In a translation model outflow is delayed for:

q(t) = p(t-tL)

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 water particle from one point of the catchment area to the outlet node is referred to as isochrones.

On the other hand, the discharge also depends on the retention characteristics. Thus, the discharge concentration is also influenced by the storage capacity of the catchment and not only by the translation process. The retention capacity depends on the soil type and cover as well as on the slope of the terrain between 0,7 and 8,0 mm and represents considerable amount of water that should not be neglected especially in flat catchment areas. It particularly influences the shape of the rising limb of hydrograph.

4.1.2. General Concepts 4.1.2.1 Hydrogaph The term hydrograph refers to the pattern of streamflow that occurs over a certain period (runoff event). The section of the hydrograph where an increase of flow occurs is called rising limb and the section where the flow decreases is called the falling limb. A hydrograph can be separated into two main components: direct runoff (volume of water produced from the rainfall event) and baseflow (volume of water representing the groundwater contribution). This separation can be observed in figure 4.1.

Figure 4.1 Hydrograph separation

There are several factors which influence the shape of the hydrograph, some of them are:

• Drainage efficiency of the catchment: influenced by the slope and length of the upland surface, drainage patterns and depressions, permeability and mositure content of the soil, vegetation cover, etc.

• Catchment shape • Drainage density

As examples: catchments with a good drainage system will have a shorter time of concentration than other with lakes, reservoirs or other surface depressions. Rivers with a steeper longitudinal profile show a more rapid response (higher peak discharge) than one that is not as steep. Shorter and wider catchments produce a faster rise and fall limb than longer and narrower catchments because of the shorter travel time. Grassland and agricultural land show faster rise limb than woodlands and urban areas lead to increase total runoff, show a higher peak discharge and shorter times of concentration.

Figure 4.2 Examples of shapes of hydrographs

4.1.2.2 Unit hydrograph (UH)

Unit hydrograph represents the direct runoff at the outlet of a basin, resulting from one unit of precipitation (of 1mm) excess over the basin, in other words, it is the pulse response of the catchment to the rainfall input. It represents the combination of surface and subsurface runoff, which results from a relatively short and intense rain. The UH predicts the relationship between the rainfall and surface runoff for any storm event and is very useful to for design flood prediction.

Figure 4.3 Unit Hydrograph construction

As it will be explained in detail in the following subsection the UH is described by the formula:

( ) /1( ) t ku t ek

ττ − −− = ⋅

As it can be observed in the equation above, the UH is a function of the retention coefficient (k) that is characteristic for each subcatchment which determines the shape of the curve (i.e. flat or steep). Since “k” is considered as a constant value for the catchment , the shape of the curve will always remains the same, and only the scale of the curve will be affected by the effective rain which generates overland flow (Qin), thus for any sequence of effective rainfalls in a period t, an estimation of the surface runoff can be obtained. It can be observed by the following equation.

∫ −−⋅⋅=t

ktinout dte

kQtQ

0

/)(1)()( ττ

4.1.2.3 Time of concentration (tc)

It is the time required for a particle of water to travel from the the farthest point in the catchment to the point of collection (measuring point).

4.1.2.4 Time area function

The Time-area function method was developed due to the great importance of time distribution of rainfall on runoff. The subject of the time-area method is the time-area histogram, which indicates the distribution of subareas of the watershed contributing to runoff at the outlet as a function of travel time. The subareas are bounded by isochrones: contour lines joining those points in the watershed that are located from the outlet by the same travel time.

4.2. Translation model

Translation refers to the delay of water in the water network. In the translation model, the concentration time to the outlet node for discrete segments of the catchment area is determined. Relating those segments to the flow time, the time-area function is obtained. Multiplied by the discharge height of 1mm this diagram corresponds to the rising limb of the hydrograph for an effective precipitation of a millimetre exclusive of translation (Unit Hydrograph). If the time steps are discretised, the unit hydrograph can be mathematically expressed in the following way:

( )[ ]

e

i

h

i

kt At

tiAtiAtiU

⋅∆

∆−−∆=∆

∑ ∑=

−

=1

1

11)(

)(

where:

A(i∆t) Segments that are discharging at the time i∆t Ae Catchment area ∆t Time interval U(i∆t) Ordinate of the unit hydrograph at time i∆t

For runoff on plain surfaces, the time-area function is rectangle. If duration of the rainfall is equal to the concentration time of the catchment tc, the hydrograph is in form of isosceles triangle. Nowadays, the concepts for flood protection that are used in water management in urban areas are based on this principle.

Natural catchments are considerably inhomogeneous in terms of translation, as the relief in such areas substantially varies. For calculation of the concentration time, lots of more or less empirical formulas are in use. For the scope of this course, the approach based on the kinematical wave is applied:

33,04,06,0

2184,9 −− ⋅⋅

= soff

stc Iie

kLt

where:

tc Concentration time L Flow path kSt Strickler-constant ieff Intensity of the effective rainfall Iso Average gradient (terrain)

This formula also considers the average slope of the terrain and surface roughness.

Considering the following conditions:

• If the terrain is considerably heterogeneous , this formula gives only the approximate value. In this case it would be more accurate to divide the catchment into the segments with homogenous gradient and roughness and the concentration time is calculated as the sum of those individual segments, as follows:

∑=

=n

i i

ic v

st1

where:

si Length of the segment i n Number of segments to the outlet vi Mean flow velocity in segment i

Mean flow velocity can be calculated assuming the quasi-stationary flow for discharge and applying universal flow laws. The Gauckler-Manning-Strickler law is commonly in use.

2/1,

3/2, isoiisti Ihkv ⋅⋅=

where: kSt,i Strickler-coefficient hi Flow depth

For the empirical parameter and adopted flow depth h, some data is taken from the literature. According to Pasche/Schröder and the research and analysis of the hydrologic data using GIS 1994, the following values are obtained.

Agricultural crop land (arable) kSt 4,5 m1/3s-1

Grassland (meadows) kSt 4,5 m1/3s-1

Forest kSt 5,5 m1/3s-1

Water depth h 0,03m • If the catchment area is heterogeneous, neither flow paths nor concentration time is

possible to calculate manually due to the complexity of such a system. This fact opens room for application of Geographic Information System (GIS).

• If the basic data for the catchment is available, it is possible to automatise the

generation 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 to calculate the discharge for any effective rainfall with the intensity ieff(t) for each catchment, as follows:

( )0

( ) ( )t

e ffQ t c A t i dτ τ τ= − ⋅ ⋅∫

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

( )( )∑=

∆⋅∆−−∆⋅=∆k

iieff tiititiActiQ

1)(1)(

tc

∆⋅=

6,31

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

4.3. Retention model

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

)()( tQktV out⋅=

where:

V(t) Reservoir content at time t Qout(t) Outflow from reservoir at time t k Retention constant

Further, considering the continuity equation i.e.

Inflow = Outflow + Change of the storage,

the equation above gives:

dttdQktQtQ out

outin)()()( ⋅

+=

where:

Qin(t) Reservoir inflow in time t

General solution to the equation gives:

∫=

−−−− ⋅⋅+⋅=t

t

ktz

kttooutout

o

o dtek

QetQtQτ

ττ /)(/)( 1)()()(

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

∫ −−⋅⋅=t

ktinout dte

kQtQ

0

/)(1)()( ττ

If Qin(τ) is the inflow and Qout (t) outflow, considering the equation above it can be concluded that, the outflow is derived directly from the inflow and multiplied by a unit flow. The unit flow reflects the transmission characteristics and thus the retention features of the catchment.

ktR e

ktU /)(1)( ττ −−⋅=−

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

effein AtiiQ ⋅∆= )()(τ

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

[ ] ttitnUtiiAtnQn

iReffea ∆∆⋅−−∆⋅∆=∆⋅ ∑

=1)1()()(

where:

[ ] [ ] ktitnek

titnu /)1(1)1( ∆−−∆−⋅=∆−−∆

According to this equation, the discharge at time n.∆t is obtained through the superposition of all catchment responses that are developed from one inflow generated during the effective rainfall within the time period t = 0 to t = n∆t. Individual responses are product of the effective intensity at time ∆t and the axis τ = t (folding) of the Unit Hydrograph.

The retention constant, with unit 1/h can be directly determined from the recession limb of the hydrograph.

)2()1(12

AA InQInQttk

−−

=

4.4. Combination of translation and reservoir models As it was already induced, discharge concentration in a catchment area is subject to translation as well as to the retention. Combination of these two different physical approaches 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.

∫ −⋅=−t

RTTR dtUUtU0

)()()( ττττ

where:

eT A

AU )()( ττ =

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

ktn

sk ekt

nktu /

1

)!1(1)( −

−

⋅

−=

CHAPTER 5 - SUBSURFACE RUNOFF (Interflow and Groundwater runoff) In addition to the runoff processes on the surface, the subsurface runoff processes are also to be considered. The relevant elements for the water balance assessment are interflow that occurs in the upper soil layer, and groundwater outflow and base flow. The subsurface runoff is delayed compared to the surface runoff and therefore in hydrology it is modelled independently. 5.1 Interflow In soil water regime modelling, an unsaturated layer is divided into horizontal lamellas depending on the hydro-geologic characteristics. For setting up the concept of 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.(see semi-distributed model). However, the discharge regards the entire subcatchment area, which consists of n Hydrotops. The interflow rate Intfij from each Hydrotop and each lamella is aggregated to a total discharge QIntf within the subcatchment area:

,1 1

1 ( )3,6

n m

Intf i j ijj i

Q A= =

= ⋅ ⋅∑∑ Intf t

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

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

[ ]1

( ) ( ) ( 1)n

a Intf TRi

Q n t Q i t u n t i t t=

∆ = ∆ ⋅ ∆ − − ∆ ∆∑

[ ] [ ]( 1) /

1

( ) 1nn t i t k

TRi e

A i tu n t eA k

− ∆ − − ∆

=

∆∆ = ⋅ ⋅∑

However, it is to be considered that subsurface translation and retention occur to much larger extent than in case of surface runoff. Thus, the time-area function A(t) and retention constant “k” that are assessed for the surface runoff are not the same ones that are to be used for the subsurface runoff. They should be derived considering the physical, geographical and geological conditions of the soil. In principle, the derivation of the time-area function A(t) for subsurface runoff is much more difficult than from the surface due to the complexity of the media. But in practice, it is often the case that the time-area function for the surface runoff is also taken for the Interflow and simply multiplied by a constant and in that way it is linearly transformed. At the same time physically based approach assumes that the flow velocities of the Interflow are behaving proportional to the Darcy’s law:

Intf f GV k I= ⋅ where:

kf Permeability coefficient for the saturated soil [m/s] IG Mean gradient of the terrain along the longest flow path [-]

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

[ ] [1 1

( ) ( 1)m m

i iA n t A n t A n t

= =

∆ = ∆ ∆ − ∆ − ∆∑ ∑ ]

where:

∆A[n∆t] Total area of the subcatcment with the same flow path s[n∆t] = n∆t . VIntf

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 flood hydrograph, whereby the considered times t1 and t2 should lie on the recession limb of the flood hydrograph.

2 1

(1) (2)A A

t tkInQ InQ

−=

−

5.2 Discharge concentration in aquifer (groundwater flow) The water that percolates from the last soil lamella into deeper layers represents the ground-water formation, which flows to the ground-water reservoir. The calculations of the ground-water reservoir are performed integrally for the total catchment area. All hydrotops with recharge to the groundwater reservoir are aggregated. One can distinguish 3 different types of the groundwater reservoir (GWR):

• upper GWR (Hgo) • lower GWR (Hgu) • carst or deep GWR

Upper and Lower GWR are illustrated in Figure 5.1.

Figure 5.1 Cross section GWR –river-Model approach for base flow Upper and lower GWR are considered together for modelling and they differ from each other by the height of the groundwater level in t e reservoir. But, the deep GWR should be considered as an independent element. Upper GWR

h

is in direct contact with the channel. Outflow of this reservoir flows to the channel. Infiltration from the channel to the upper GWR is not considered . On the other hand, the discharge from the lower GWR flows to the GWR of the adjacent catchment. It is possible that maximum three catchments receive the discharge from one catchment, only the weighting of those inflows can be different. The capillary uprise from the GWR is neglected in the model. The deep GWR is, as already adduced, an independent element that drains into a recipient. The location

where it occurs can be chosen arbitrary.

GWR receives the inflow from the soil water as well as from the GWR of the adjacent catchments. Inflow is proportionally divided into GWR and deep GWR, as follows:

inf, ( ) ( )3,6H H HQ t A Perc t= ⋅ ⋅

1

ITRH

,inf inf,( ) ( ) (1 ) ( )GW GWNGin H HQ t Q t ITR Q t= + + ⋅∑ inf ,( ) ( )TGW H in HQ t ITR Q t= ⋅∑

Hydrotop

On the other hand, the o

where:

Q Inflow to the groundwater GWNGin

reservoir (GWR) from the adjacent catchment (GWR) [m³/s]

AH Hydrotop area [km²]

QGWTinf(t) Inflow to the deep GWS (characteristical for carst areas) [m³/s] Weighting factor [-] depending on the Hydrotop

As part of the theoretical background for the concept of the GWR the continuity equation of the inflow, outflow and changes in the storage is used. Inflow is composed of the discharge of

e adjacent element (GWR) and does not depend on the actual water content in the reservoir. utflow from the observed GWR depends on the water quantity in it.

Also, the Linear Reservoir approach is used to obtain the water content on the GWR. It ssumes that the aquifer acts as a huge reservoir and the Qout is only a function of the actual

of the reservoir. The outflows from the G the program package Kalypso iteratively calculated, according to the following equations:

PercH(t) Percolating water from the soil water storage [mm/h] QGWinf(t) Inflow to the GWR (from above and below) [m³/s]

th

avolume

WR are, for example, in

( ) ( ) ( )in outV t Q t Q t= −∑ Balance Equation dt

d

( )( ) outQ tdV t k= ⋅dt dt

where:

V(t) Actual water content in the reservoir [m³]

Linear Reservoir

ntion constant of the linear reservoir [s]

Qin(t) Inflows to the GWR [m³/s] Qout(t) Outflows from the GWR [m³/s] k rete

In its developed form, the balance equation gives:

( ) ( ) ( ) ( ) ( ) ( )totalsoil GWin Base GWout deep

dV t Q t Q t Q t Q t Q tdt

= + − − −

It is illustrated in Figure 5.2.

.2.1 Dischar

the mccording to t e lower GW

hannel is pos R. The range of the W R and the one overlying the

hannel botto

he program Go), for the efinition of ogeneous aquifer is resupposed, e an be determn additional ent.

Figure 5.2 Water balance for the catchment area 2

ge from the Upper and Lower GWR

odel Kalypso, upper and lower GWR are joined and considered as one unit. he model concept, it is considered that the channel runs over the water table ofR. Thus only an exchange of water of the ground-water reservoir and the sible starting from a certain water level height in the GW

R without contacts to the channel is referred to as lower GWm is called upper GWR (see figure 5.1).

BCENA enables indication of the lower (hGu) and upper height (hthe size of the two reservoirs. If the isotropic i.e. homthat is characterised by very poor spatial variability, that the reservoir volumined depending on the groundwater level. assumption is that the groundwater level is uniform for the subcatchm

5 InAthcGc TdpcA

6( ) ( ) 10scV t GWR t Pors A= ⋅ ⋅ ⋅

V(t) Actual water content of the reservoir[m³]

where:

GWR(t) Water height in the reservoir element (m)Pors Porosity of the aquifer[-] A Area of the catchment (km²)

Additionally, the area of the subcatchment (Asc) is assumed to be of a geometrical shape, where:

scA lm ba= ⋅

Figure 5.3 Dimensions of the subcatchment

onsidering the concept of t e following cases are possible concerning the roundwater level and its relati

. Case: GWR(t) < hGu

roundwater height i e is

Figure 5.4 Upper and lower groundwater reservoir

he model, thon to the hGu and hGo. (Refer to Fig 5.4)

Cg

1 G s below the lower groundwater level. Thus, the total water volumassigned to the lower groundwater reservoir. It can be described as follows:

6( ) ( ) 10GWloV t GWR t Pors A= ⋅ ⋅ ⋅

( ) 0GWupV t =

2. Case: hGu ≤ GWR(t) ≤ hGo

etween hGu and hGo

The groundwater level is b

6GWR HGU − ( ) ( ) 0,5 10GWlo scV t GWR t A PorsHGO HGU

= − ⋅ ⋅ −

⋅

( ) 0,5GWupV t ( ) 610scGWR HGU GWR HGU A PorsHGO HGU

− ⋅ − ⋅ ⋅ − = ⋅

. Case: GWR(t) > hGo

roundwater level is above the upper height. The volume of the lower groundwater reservoir

3 Gis not increasing.

1( ) ( )GWloV t HGO HGU= + 6102 scA Pors⋅ ⋅

⋅

⋅

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

.2.2 Deep groundwater outflow

eep groundwater outflow can be calculated from the system of the differential equations that

e outlet node. Further, it is possible to construct the net independent from the one for the surface runoff and can for example, simulate the flow paths in carst areas.

6( ) ( ( ) 0,5( )) 10GWup scV t GWR t HGO HGU A Pors= − − ⋅ ⋅

5 Dhas already been introduced. Additionally, the retention constant for the deep groundwater reservoir RetTGW, is introduced. This outflow can be assigned to any node, not necessarily to th

CHAPTER 6 - FLOOD WAVE FORMATION IN THE CHANNEL In the hydrologic cycle the following components are of relevance for the modelling: surface

noff (overland flow), interflow and basic flow. Additionally, the flow in channel is onsidered as a part of the inflow:

ruc

Qin(t) = QSurface(t) + QInterflow(t) + QBasis(t) + QChannel(t) (1) For flood wave propagation in natural channels, two important phenomena should be onsidered: attenuation of the peak flow and time lag that leads to the modification of the ischarge hydrograph. Figure 6.1 shows the transformation of the flood wave. This discharge ydrograph is modified if the water section flows through retention.

cdh

ave

he computation me scribe the processes in the water course are ummarised under the nam y be classified as either ydrologic (lumped) or putation methods employ oth, the equation of entum conservation). It odels the integral m ter in the channel, but in engineering practice

ne and two-dime the rainfall runoff hods” are currently in use. They do not deal ith the overall mo centrate on the water strand and treat it as a near reservoir of the out(t) as follows:

Figure 6.1 Inflow/Outflow hydrograph of a flood w T thods that are applied to de

e Food Routing. Flow routing mahydraulic (distributed). The hydraulic com

continuity and the equation of motion (momovement process of the wa

nsional models are commonly in use. models only “hydrologic mettion process, but they con volume V(t) that is changing with the outflow Q

out(t)

ines only the resulting pr

shbmoInwli

dV(t) = k . dQ (2) Thus, this method determ ocess that is generated as a reaction to the

flow. The Flood Routing plied to asses the

pacts brought by land development or management. For this reason Stormwater anagement is the common practice for mitigation of these impacts by managing the size and

in concept is of great importance because it can be ap

imMmovement of the flood wave.

6.1 Flood Routing applying the linear reservoir model Generally, flood routing is a mathematical method used to predict the temporal and spatial variation of a flood wave, at one ore more points along a water course (river or channel). The watercourse may be a river, stream, reservoir, estuary, canal, drainage ditch or storm sewer. In case of a stationary discharge flow, there is a clear connection between the water depth h and

ischarge Q for each water section: d

Q = f(h) (3) To each water strand corresponds only one volume, and therefore it is possible to define a unique relation between volume and discharge:

Q = f(V) (4) Further, for the stationary flow of the linear reservoir model, it becomes:

dV(h) = K(h) . dQout(h) (5) In the equation above, the parameter K(h) represents the time necessary for water to flow through the channel subsection. Different analysis of the river courses showed that the parameter k changes insignificantly

ver the discharge range, so that it cao

n be considered as a constant.

K(h) = k = const. (6) With this assumption, the conditions for applying the linear reservoir concept are fulfilled, as he outflow is linearly related to the storage. Tht

de integration of this equation over the water

epth gives:

Q = 1/k (V-V ) out 0⋅

dV/V = -(1/k)dt (10

(7) where, V is the volume that remains in the scours of the river bed in dry season. This volume is not o neglected.

0f importance and can be

Again, the continuity equation is applied:

Q dt - Q dt = dV in out Q

here Q is the inflow, Q is the outflow and dV represents the cha

(8)

wA

in out nges in the storage. dding Eq (7) in Eq (8), the following is obtained:

dV/dt + (1/k) V = Q ⋅ in

For the special case, that is Q = 0, the equation can be modified as follows:

(9)

in

)

Integrating this equation (0,t) it gives:

ln V = -(t/k) + C (11) And in its exponential form:

V = C ⋅ e-t/k (12) For t = 0 is V = V so that the equation becomes: o

V= V . e-t/k o (13)

one replaces V with Q k and V with Q (t =0) k, the following is obtained: If out o out 0

⋅ ⋅

t/k

Q = Q (t ) . e t/k out out 0 (14)

the second step, the special case Q = 0 is introduced to the general solution Qin ≠ 0, in the on is as follows:

In inEq 12. The first derivation of this equati

ktek

Cedtdt

/1 −⋅⋅−⋅= (15)ktdCdV /−

Introduced in the 9, this equation gives:

Eq

zkkkdt(16)ktktkt QeCeCedC

=⋅⋅+⋅⋅−⋅⋅ −−− /// 111

r o

dC/dt = Qin . e (17)

tegratin his ion over time and replacing the constant C with V et/k (Eq 12), leads to: In g t equat ⋅

))(( // dtetQVeV ktt

kt ⋅+= ∫− 0

ino (18)

Again, if V is replaced with Qa k, and Vo with Qa(to = 0) ⋅ ⋅ k, the following is obtained.

dtetQek

eQtQ ktin

ktktoutout

/

0

//0, )()( ⋅⋅⋅+⋅= ∫ (19)

t1 −−

ccording 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 • quence of the inflow

A

discharge as conse

Since closed-form solutions to the complete hydraulic routing differential equations do 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 to write the equation above as a summation in the following way:

tetmQek

etQtnQm

inoutout ∆⋅

⋅∆⋅⋅+⋅==∆ ∑=1

)()0()( (20)n

ktmktnktn ∆∆−∆− /// 1

or:

tetmQetQtnQn

ktmnin

ktnoutout ∆⋅

⋅⋅∆+⋅==∆ ∑ ∆−−∆− /)(/ 1)()0()(

s wav

(21)km =1

wh , stics can nction as follows:

ere Qin(n∆t) is the inflow and Qout(n∆t) is the result and this transmission characteri be described in terms of impulse response fu

( )[ ] ktmnek

tmnu /)(1 ∆−−⋅=∆− (22)

It formally coincides with the Unit Hydrograph that is already introduced in the Chapter 5,

hen explaining the processes of the discharge concentration. So, the wave transformation in channel is the process analogue to the discharge concentration on the surface. 6.2 The method of Kalinin-Miljukov

In the previous section the stationary state flow for discharge was assumed. However, this assumption is hardly applicable to the flood wave situation, as the storm tate in a flood e is always instationary, that is indicated as temporal change of the mean velocity v and water

epth h along the channel as follows:

w

d

dt0dv

≠

0dhdt

≠

e discharge depends not only on the water level but

Q = f(V,I) Consequently, for each section of the channel, it exists a so called Hysteresis curve, that means: storage versus outflow is not a single valued function. (Fig 6.2)

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 unique volume/discharge elation. r

ifferently from the stationary state, thD

also on the slope. It can be denoted as following:

Q = f(h,I)

Figure 6.2a Hysteresis curve

determ

Figure 6.2 b: Hysteresis curve

As it depends on the flood wave propagation (transformation), it can not be a priori ined. Considering the Hysteresis curve (6.2), to each discharge Qout(t1) it is possible to

assign 3 different water depths: one on the rising and falling limb of the instationary flow and one for the case of the stationary (steady) flow.

Kalinin Miljukov, however, showed that it is possible to define a unique function between discharge and volume in spite of the dependency on the slope if the channel is represented by a cascade of n linear reservoirs with the length L (characteristic section). The unsteady flow in those sections can be approximated by a quasi-stationary flow.

6.3 Characteristic length for Kalinin-Miljukov Method

or one single reservoir it is assumed that the time required for the increase of the discharge olume ∆Q from the inflow side l to the outflow side r corresponds to the flow time in the

stationary storm state. Further, it is coupled with the assumption of a weak instationary flow, so that the discharge increase The replenishm ∆Q. The time

ethod as following: Qout(t

Fv

∆Q becomes so small and its flattening can be neglected.

ent of this additional water volume ∆V occurs with the increase required for this process can be estimated according to the linear reservoir m

1)

QV inst ∆

∆=K (23)

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

m, it ecomes:

Further, an unique discharge curve for the characteristic channel reach is obtained, if we consider the characteristic of the Hysteresis curve that the time lag ∆t of discharge on the rising as well as on the recession limb of the flood, can be observed on the gauge. If the discharges of the cross section r are assigned to the water levels in the cross sectionb

lm= V * ∆t (24) so that the discharges of the stationary discharge curve at the cross section m can be assigned to the corresponding instationary ones at the cross section r. The length of the characteristic section is:

L = 2 lm (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 stationary and the additional discharge:

Qin = Q + Q st stat d (26)

In the s scharge volumes and water surface slope can be divided: ame way, di

Vinst = Vstat + Vd (27)Iinst = Istat + Id (28)

Kalinin-Miljukov Method is based on the considerations of the validity of the linear reservoir assump for the unsteady flow:

d d

urther e flow resistance coefficients of the instationary storm correspond to e same ones for the stationary flow for the same water depth, and that energy line is parallel

ying the Darcy

tion also

d(Vstat + Vd) = k . d(Qstat + Qd) bzw. dV = k . dQ

(29)

F , assuming that ththto the water surface line, it is possible to compute the discharge Q appl

eisbach law. W

2/12/1, )() instrhy

r

IrA ⋅⋅ 8g(rinstQ ⋅=λ (30)

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

statIinstinst IQ

= statQ

(31)

pplying the Eq 28, the instationary water table slope can be transformed as following: A

)11(1 dstat

dstatdtatnst

IIIIIIsIi ⋅+⋅=+⋅=+= (32)2 statstat II

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

)11( dIQQ ⋅+⋅= 2,

statrstatinst I

(33)

urther considering Eq 26 and 33, it becomes: F

stat

drstatd I

IQQ ⋅⋅= ,21 (34)

d corresponds to the additional volume Vd, that can be represented only by small increase/decrease of the water level by multiplying the water level with the stationary discharge Qstat and the mean depth of the additional water level hd.

Q

Vd = Qstat . hd (35) Further, if it is assumed that the slope is constant, along the characteristic section, it becomes:

Vd = Qstat . Id . L/2 (36) If the Eq 34 and 36 are differentiated over Id, it gives.

dVd = Qstat . L/2 . dId (37)and

drstat

d dlQ

dQ ⋅⋅= ,1 (38)statI2

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

dstatd dlLQkdlI

⋅⋅⋅=⋅⋅22

(39)rstatQ1 ,

stat

Transfo an g the 1/k = dVd/dQd , finally gives conditional equation for the characteristic channel section L:

rming d insertin

statstat

drstat

QI

Q

⋅

,dV

ddQlL == 2 (40)

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

dVd = Qstat . dhd,m (41) In this case, the Eq 40 becomes:

mdstat dhQ ⋅ ,

statstat dQIlL

⋅== 2 (42)

By very small changes of the profile along the channel reach, the changes of the water level in the balance point m can be approximated by the one of the cross section r.

rdstat

dQIdhQ

lL⋅

⋅== ,2 (43)

statstat

haracteristic length can be calculated depending on the staC

dtionary outflow. The term

hstat/dQ represents the increase of water level-discharge-relation, which is, in general, not constant. The retention constant can be calculated in the following way:

stat

ma min

max min

steady

steady

dh h hk B L B LdQ Q Q

−= ⋅ ⋅ ≈ ⋅

− x

Once more, the continuity equation will be introduced: dV Q Q= − in outdtwith linear relation between discharge and volume:

where:

A V= ⋅ Q kIn discrete form, it can be transformed as follows:

1 2( ) ( ( ) ( )) ( )out in in AQ t t c Q t t Q t c Q t+ ∆ = ⋅ + ∆ + + ⋅

1tc ∆

=+ ∆ 2

2k tc − ∆=

+ ∆

2k t 2k tinally, this method has advantages comparing to, for

Ft

example, Muskingum Method, since he application of this method does not require measurements, as it is possible to obtain

parameters L and K from the channel geometry. Measurements also showed that the results obtained with Kalinin-Miljukov method are sufficiently accurate.

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

ited length L were considered, where the ngth L was derived from the hysteresis curve and from the linear reservoir method. It means

that if we consider a channel strand with the length LG, applying the Kalinin-Miljukov ethod, it has to be divided into n equal sections with the length L. Each section represents

ependent reservoir, and the Eq 22 describes the wave formation for it. The outflow from the reservoir n is the inflow to the reservoir n-1.

6

Until now, only the channel reaches with the limle

mone ind

Qin(n - 1) = Qout(n) (44) where the numbering of this reservoir chain starts on the lower boundary of the total channel length LG and is proceeded against the flow direction. Inserting the Eq 44 into Eq 22, it becomes:

nktmn ∆−− /)(1 te

ktmQtnQ

mnana ∆⋅

⋅⋅∆=∆ ∑−1

,1, )()( (45)

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

n 1−

=

ktnza e

ktnQttnQ /1)()( ∆−⋅

⋅∆

∆=∆ (46)

nk )!1( −

where the number of reservoirs is calculated in the follo GL nL

= wing way:

INEAR RESERVOIR

inear reservoir model represents the catchment area as a reservoir, in which the inflow in form of precipitation is stored and with time lag discharged. It is also considered as a black box method, as the processes in the system are not completely described, but merely the transmission characteristics of the system, especially the correlation between the input and

utput, are taken into account.

One can conclude that the linear reservoir is of great importance in hydrological modelling hich purpose is to define the universal relation between rainfall and the resulting discharge ydrograph for one catchment area.

L L

o

wh

[ ]( ) ( )Q t f i t= In the LRM, the overall system is divided into subsystems and the individual components that are forming the total discharge, such as overland flow, interflow, seepage and groundwater recharge are considered as linear reservoir and are linearly superposed. For the purposes of mathematical formulation of the linear reservoir, the behaviour of the atchment area is simplified, ac ssuming that the catchment area behaves like a container

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

( ) ( )outV t k Q t= ⋅ (8) where:

V(t) Volume in Reservoir at time t QA(t) Outflow from the reservoir at time t k Retention constant

his is illustrated in Figure 1.

T

Figure 1. Linear reservoir and corresponding outflow and volume curve

Further, applying the continuity equation for the reservoir

( )( ) ( )in outV tQ t Q t

dt= + (9)

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

( )( ) ( ) outin out

k dQ tQ t Q tdt

⋅= + (10)

where:

Qin(t) Reservoir inflow at time t

his equation can be solved by multiplying both sides by the factor et/k. It becomes:

T

/ / /( ) 1 1( ) ( )t k t k t kAout in

dQ te e Q t edt k k

+ = Q t (11)

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

/ /1( ( ) ) (t k t kout in

d Q t e e Q tdt k

⋅ = ) (12)

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

(0)

( )

0

1out

out

Q t t

Q

(13)/ /( ( ) ) ( )t k t k

out ind Q t e e Q dk

τ τ ∫ ∫ ⋅ =

The general solution to this equation is :

0out out ink

(14)/ /1( ) (0) ( )t

t k t kQ t e Q e Q dτ τ ⋅ − = ∫

and after transformation :

0

1(t

t

d=

(15)/ ( ) /( ) (0) )t k t kout out inQ t Q e Q e

kττ τ− − −= ⋅ + ⋅ ⋅∫

For Qout(0) = 0 this equation becomes :

( ) /

0

1tt kτ− −( ) ( )out inQ t Q e d

kτ τ= ⋅∫ (16)

If the Qin(τ) is „inflow impulse“ and Qout(t) „outflow impulse“, then according to the equation above, the „outflow response“ is directly derived from the inflow impulse, multiplying it by a

unit impulse. Transferred to the catchment area, Qout(t) corresponds to the discharge ydrograph at outlet node and the Qin(τ) to the precipitation N(τ)=Ae*i(τ) and the intensity i(t)

off the effective rainfall, as following:

h

( ) /1( ) ( )t

t koutQ t N e dττ τ− −= ⋅∫

(19)

(17)0 k

The system function

( ) /1( ) t ku t e ττ − −− = k

represents the transmission characteristics of the catchment. It corresponds to the discharge hydrograph generated for one catchment for the effective rainfall of 1mm that is spatially and during the time t uniformly distributed over the catchment and shifted for the (t- t ). It is also alled the Unit Hydrograph (UH).

(18)

c

0

( ) ( ) ( )t

outQ t N u t dτ τ τ= ⋅ −∫

his equation is basiT

Ins for the Linear reservoir model. If the duration of the rainfall τ→0,

stantaneous Unit-Hydrograph (IUH) is obtained. Until now, the continual temporal field was presumed. If time is discretised with the time steps ∆t, the effective rainfall of the m. interval (corr. to the time point t=(m-1)∆t) becomes:

( 1)

( ) ( ) ( )m t

e em t

N m t A i d A i m t tτ τ∆

− ∆

∆ = ⋅ = ⋅ ∆ ⋅ ∆∫ (20)

The time shift between inflow and outflow impulse for the time intervalt = n∆t is:

( 1int t n m ) tτ= − = − + ∆ The Unit Hydrograph is discretised over time:

n m t− + ∆

[ ]( 1)

( )

1( 1) ( )inn m t

u n m t u t dt

τ− ∆

− + ∆ =∆ ∫ (21)

here: w

int t Vτ= − = is the time lag If the Eq. 20 and 21 are inserted the following summation formula is obtained:

( )1

( ) ( ) 1out em

Q n t t A N m t u n m t−

∆ = ∆ ⋅ ∆ ⋅ − + ∆ ∑ (22)n

where:

( ) ( )1 /11 n m t ku n m t ek

− − + ∆ − + ∆ =