Hydrological and Environmental River Basin Modelling

Technische Universität München Ingenieurfakultät Bau Geo Umwelt Lehrstuhl für Hydrologie und Flussgebietsmanagement Univ. Prof. Dr.-Ing. Markus Disse

Hydrological and Environmental River Basin Modelling Lecture Notes

Winter Term 2015/16

Content

Content Content

III

List of Figures 1 Introduction

VIII 1

1.1

Organisation ................................................................................................................... 1

1.2

Course Contents ............................................................................................................ 1

1.3

About these Lecture Notes ............................................................................................. 1

1.4

Examination.................................................................................................................... 2

2 Hydro-Environmental Models and their Use 2.1

3

Model concepts .............................................................................................................. 3

2.1.1

Model distinction based on process description ........................................................ 3

2.1.2

Model distinction based on spatial representation..................................................... 5

2.1.3

Model distinction based on aspect of randomness.................................................... 5

2.2

SWAT Model .................................................................................................................. 5

2.2.1

Hydrologic Similarity and the HRU Concept.............................................................. 6

2.2.2

Water Balance in SWAT ........................................................................................... 7

2.3

Case Study: Ethiopia .................................................................................................... 11

2.3.1

NeXus: Water, Food & Energy................................................................................ 11

2.3.2

Description of the Research Area ........................................................................... 11

2.3.3

Data Collection and Management........................................................................... 12

2.3.4

Model Set Up: DEM ................................................................................................ 13

2.3.5

Research Outline .................................................................................................... 14

2.4

Further Reading ........................................................................................................... 15

3 Climate Input 3.1

16

Precipitation.................................................................................................................. 16

3.1.1

Measurement equipment ........................................................................................ 16

3.1.2

Snow ...................................................................................................................... 17

3.2

Temperature ................................................................................................................. 21

3.2.1

Air temperature ....................................................................................................... 21

3.2.2

Soil temperature ..................................................................................................... 22

3.2.3

Water temperature.................................................................................................. 26 III

Content

3.3

Elevation bands............................................................................................................ 26

3.4

Water Vapour and Relative Humidity ............................................................................ 28

3.5

Wind Speed ................................................................................................................. 30

3.6

Nomenclature ............................................................................................................... 31

4 Evapotranspiration

34

4.1

Penman Monteith ......................................................................................................... 36

4.2

Priestley Taylor ............................................................................................................ 45

4.3

Hargreaves .................................................................................................................. 45

4.4

Actual Evapotranspiration in SWAT.............................................................................. 46

4.4.1

Evaporation of intercepted rainfall .......................................................................... 46

4.4.2

Transpiration .......................................................................................................... 46

4.4.3

Sublimation and evaporation from the soil .............................................................. 47

4.5

Further Reading ........................................................................................................... 52

4.6

Nomenclature ............................................................................................................... 52

5 Infiltration 5.1

55

Matrix Potential and Soil Hydraulic Characteristics ....................................................... 55

5.1.1

Matrix potential ....................................................................................................... 55

5.1.2

Soil Hydraulic Characteristics ................................................................................. 55

5.2

Darcy Richards Approach ............................................................................................ 56

5.3

SCS-CN Method .......................................................................................................... 57

5.3.1

Background ............................................................................................................ 57

5.3.2

Theoretical Model ................................................................................................... 58

5.3.3

SCS Curve Number (CN) in SWAT ........................................................................ 60

5.4

Green and Ampt ........................................................................................................... 65

5.4.1

Approach................................................................................................................ 65

5.4.2

Parameter Estimation ............................................................................................. 68

5.5

Nomenclature ............................................................................................................... 69

6 Vertical Soil Water Movement 6.1

72

Soil Characteristics and Soil Water Content in SWAT .................................................. 72

6.1.1

Soil Charcteristics .................................................................................................. 72

6.1.2

Soil Water Content ................................................................................................. 73

6.2

Water Uptake by Plants................................................................................................ 74

6.3

Percolation ................................................................................................................... 76

IV

Content

6.4

Bypass Flow ................................................................................................................. 77

6.5

Perched Water Table.................................................................................................... 80

6.6

Nomenclature ............................................................................................................... 81

7 Lateral Flows 7.1

84

Surface Runoff ............................................................................................................. 84

7.1.1

Peak Runoff Rate ................................................................................................... 84

7.1.2

Time of concentration ............................................................................................. 84

7.1.3

Runoff Coefficent .................................................................................................... 87

7.1.4

Rainfall intesity ....................................................................................................... 87

7.1.5

Modified Rational Formula ...................................................................................... 88

7.1.6

Surface Runoff Lag................................................................................................. 88

7.1.7

Transmission Losses .............................................................................................. 89

7.2

Subsurface Flow and Lag ............................................................................................. 90

7.2.1

Subsurface Flow ..................................................................................................... 90

7.2.2

Flow Lag ................................................................................................................. 92

7.3

Base Flow .................................................................................................................... 94

7.3.1

Shallow Aquifer ...................................................................................................... 94

7.3.2

Deep Aquifer .......................................................................................................... 97

7.4

River Routing................................................................................................................ 98

7.4.1

Open Channel Flow and Characterisitcs ................................................................. 98

7.4.2

Flow Rate and Velocity ......................................................................................... 100

7.4.3

Variable Storage Routing Method ......................................................................... 101

7.4.4

Muskingum Flood Routing Method ....................................................................... 102

7.4.5

Transmission Losses, Evaporation and Bank Storage .......................................... 105

7.5

Further Reading ......................................................................................................... 107

7.6

Nomenclature ............................................................................................................. 107

8 Soil Erosion

111

8.1

Background ................................................................................................................ 111

8.2

USLE and MUSLE ...................................................................................................... 112

8.2.1

USLE .................................................................................................................... 112

8.2.2

MUSLE ................................................................................................................. 112

8.3

Excursion: SWAT Erosion Modelling in Ethiopia ......................................................... 118

8.3.1

Background .......................................................................................................... 118 V

Content

8.3.2

SWAT Model ........................................................................................................ 119

8.4

Further Reading ......................................................................................................... 120

8.5

Nomenclature ............................................................................................................. 120

9 Crop Growth and Crop Yield

121

9.1

Potential Heat Units ................................................................................................... 121

9.2

Dormancy................................................................................................................... 122

9.3

Biomass production and Crop Yield ........................................................................... 123

9.3.1

Biomass Production ............................................................................................. 123

9.3.2

Nitrogen Uptake ................................................................................................... 124

9.3.3

Crop Yield ............................................................................................................ 125

9.4

Stress factors ............................................................................................................. 127

9.4.1

Water Stress ........................................................................................................ 127

9.4.2

Temperature Stress.............................................................................................. 127

9.4.3

Nitrogen Stress .................................................................................................... 128

9.4.4

Phosphorus Stress ............................................................................................... 129

9.5

Actual Growth ............................................................................................................ 129

9.6

Further Reading ......................................................................................................... 129

9.7

Nomenclature ............................................................................................................. 130

10 Water Quality Modelling

132

10.1 Problems with Nitrification and Eutorphication ............................................................ 132 10.2 Nitrogen and Phosphorous Cycles ............................................................................. 132 10.2.1 Nitrogen Cycle...................................................................................................... 132 10.2.2 Phosphorous Cycle .............................................................................................. 133 10.3 Processes in Soils ...................................................................................................... 134 10.3.1 Nitrogen Fixation .................................................................................................. 134 10.3.2 Degradation of Nitrogen ....................................................................................... 134 10.3.3 Degradation of Phosphorus .................................................................................. 137 10.4 Processes on the Land Surface ................................................................................. 139 10.4.1 Nitrate Pathways .................................................................................................. 139 10.4.2 Soluble Phosphorus Movement ............................................................................ 141 10.4.3 Nutrient Lag in Surface Runoff and Lateral Flow .................................................. 143 10.5 Instream Processes ................................................................................................... 144 10.5.1 Algae .................................................................................................................... 144 VI

Content

10.5.2 Organic Nitrogen .................................................................................................. 144 10.5.3 Ammonium ........................................................................................................... 145 10.5.4 Nitrite .................................................................................................................... 147 10.5.5 Nitrate................................................................................................................... 147 10.5.6 Organic Phosphorus ............................................................................................. 148 10.5.7 Soluble Phosphorus ............................................................................................. 148 10.6 Streeter – Phelps Equation ......................................................................................... 149 10.7 Further Reading ......................................................................................................... 152 10.8 Nomenclature ............................................................................................................. 152 11 Literature

159

VII

Content

List of Figures Figure 2-1 Model concepts .......................................................................................................................... 3 Figure 2-2 Different types of water quality models ...................................................................................... 4 Figure 2-3 Classification of the SWAT model .............................................................................................. 6 Figure 2-4 Spatial discretization structure in SWAT .................................................................................... 7 Figure 2-5 The Hydrological Cycle after Neitsch et al. (2011), (slightly modified) ...................................... 8 Figure 2-6 The land phase of the hydrological cycle, (Neitsch et al. 2011) ................................................ 9 Figure 2-7 Water Routing Phase, Van Griensven (2010),modified .......................................................... 11 Figure 2-8: Location of the Blue Nile Basin in Ethiopia ............................................................................. 12 Figure 2-9: DEM of Abbay basin at 90 m resolution, from CGIAR-CSI. .................................................... 13 Figure 2-10: Subbasins distribution of Abbay catchment, based on an automatic delineation ................. 14 Figure 3-1 Measurement equipment for precipitation ................................................................................ 17 Figure 3-2 Phase transition of water .......................................................................................................... 18 Figure 3-3 Snow distribution ...................................................................................................................... 18 Figure 3-4 Areal depletion curve................................................................................................................ 20 Figure 3-5 Diurnal variation of air temperature for August 2013 over two days (Osterseeon) .................. 22 Figure 3-6 Four-year average air and soil temperature at College Station, Texas [1] .............................. 22 Figure 3-7 Temperature fluctuations in soil [2] .......................................................................................... 25 Figure 3-8 Water vapor pressure vs. air temperature vs. relative humidity ............................................... 29 Figure 3-9 Psychrometer ........................................................................................................................... 30 Figure 3-10 Wind sensor ........................................................................................................................... 31 Figure 4-1 Energy for evaporation ............................................................................................................. 34 Figure 4-2 Graphical representation of processes of evaporation ............................................................ 35 Figure 4-3 Measurement device “Class A Pan” ......................................................................................... 35 Figure 4-4 Energy budget of the earth ....................................................................................................... 37 Figure 4-5 Distance between sun and earth .............................................................................................. 38 Figure 4-6 Declination of the earth ............................................................................................................ 39 Figure 4-7 Albedo values ........................................................................................................................... 40 Figure 4-8 Kipp & Zonen Pyranometer; Rolf Gegenbach Messtechnik..................................................... 41 Figure 4-9 Wind profile .............................................................................................................................. 44 Figure 4-10 LAI values for different vegetation types ................................................................................ 47 Figure 4-11 Soil evaporative demand distribution with depth .................................................................... 50 Figure 4-12 Soil evaporative demand distribution with depth (esco factor) .............................................. 51 Figure 5-1: Typical soil water retention curves for relatively coarse- (solid line), medium- (dashed line), and fine-textured (dotted line) soils. .......................................................................................................... 56 Figure 5-2: Typical curves of the hydraulic conductivity K, as a function of the pressure head (left) and water content (right) for coarse- (solid line), medium- (dashed line), and fine-textured (dotted line) soils. ................................................................................................................................................................... 56 Figure 5-3: Relationship between rainfall, Infiltration and Runoff .............................................................. 58 Figure 5-4: CN curve sheet ........................................................................................................................ 60 Figure 5-5: CN values for different moisture conditions ............................................................................ 63 Figure 5-6: Different curve numbers for different slopes ........................................................................... 65 Figure 5-7: Green and Ampt based on Darcy ............................................................................................ 66 Figure 5-8: Comparison of moisture content distribution modeled by Green and Ampt and natural observed distribution. ................................................................................................................................. 67 Figure 6-1: Depth distribution of water uptake ........................................................................................... 75 Figure 6-2: Dry vertisol with huge cracks (source: www.geography.hunter.cuny.edu) ............................. 78 Figure 6-3: Perched water table ................................................................................................................ 81 Figure 7-1: Influence of surlag and tconc on fraction of surface runoff released ....................................... 89 Figure 7-2: Behavior of the water table as assumed in the kinematic storage model. .............................. 91 Figure 7-3: Influence of TTlag on the fraction of lateral flow released. ...................................................... 93 Figure 7-4: Difference between subsurface and ground water / base flow (source: criticalzone.org) ...... 94 Figure 7-5: Evaporation from Shalow Aquifer ............................................................................................ 96 Figure 7-6: Trapezoidal channel dimensions ............................................................................................. 98 Figure 7-7: Flood plain dimension ........................................................................................................... 100 Figure 7-8: Prism and wedge storages in a reach segment .................................................................... 103 Figure 7-9: Bank Storage ......................................................................................................................... 106 Figure 8-1: The Nile Basin (left) and the Blue Nile Basin (right) (MSc Theses H. Huber)....................... 118 VIII

Content

Figure 9-1: Daily temperature profile and base temperature for a specific crop; growth will occur above the base temperature ...............................................................................................................................121 Figure 9-2: Variation in optimal harvest index (HIi/HIopt) with fraction of growing season (frPHU) .......126 Figure 9-3: Impact of mean air temperature on plant growth for a plant with T base=0°C and Topt=15 °C .128 Figure 10-1: SWAT nitrogen pools and processes that move nitrogen between the pools .....................133 Figure 10-2: SWAT soil phosphorus pools and processes that move P between the pools ...................134 Figure 10-3: Decrease of nitrate concentration with depth ......................................................................135 Figure 10-4: Temperature dependent rate coefficients σ and β ...............................................................145 Figure 10-5: Nitrification inhibition factor at low oxygen concentrations ..................................................146 Figure 10-6: Streeter – Phelps Oxygen sag curve ...................................................................................149 Figure 10-7: Temporal evolution of oxygen deficit (D), oxygen demand (C), and dissolved oxygen (DO) ..................................................................................................................................................................151

IX

1 Introduction 1.1 Organisation

1 Introduction 1.1 Organisation Module Appropriation M.Sc. Environmental Engineering

Module Number: BGU54008

Lecturer Prof. Dr.-Ing. Markus Disse, [email protected]

1.2 Course Contents The main goal in this module is to give a comprehensive overview in the aspects of ecohydrological modeling. Structure and interaction of different components of an ecohydrological model will be explained as well as associated calculation methods. Water quality aspects and influencing factors from land use and land management practices will be discussed. Additionally, mathematical descriptions for crop growth and the related water and nutrient demand of different plants will be introduced. In addition to lectures, students will apply their theoretical knowledge in guided exercises. Using an ecohydrological software model (SWAT) they will setup, calibrate and validate a model for a real catchment. At the end of the module, students are able to understand the transfer of process flows from natural hydrologic and nutrient cycles to an ecohydrological software model. In addition to that they will be able to understand different methods for the calculation of single components of ecohydrological processes and their interplay. Moreover students will be able to use an ecohydrological model and to analyse model outputs. The students will be capable to identify different influencing factors and to evaluate the meaningfulness of model results.

1.3 About these Lecture Notes These lecture notes are intended as a more detailed supplementary material, which go along with the lecture slides. They contain all necessary theory from the lecture, the relevant equations and also some parts we consider as basic knowledge to master this module and understand the SWAT model. The notes are based on the original SWAT Theoretical Documentation (Neitsch et al., 2009 http://swat.tamu.edu/media/99192/swat2009-theory.pdf), which can be consulted for additional information. At the end of some chapters there are recommended some scientific papers for further reading. It´s not meant that you study the papers in detail. The intention is rather that you extract the relevant information, which is good scientific practice, e.g. 1

1 Introduction 1.4 Examination

with respect to your final thesis. Furthermore, it helps you to set the chapter´s content into the context of current research questions and the possible applications of SWAT. However, this part is completely voluntary and the content of those papers will NOT be part of the exam.

1.4 Examination The Assessment will be divided in two parts. In the first part the student has to write a small document in which the resulting model is described at the end of the term. The second part is a written exam. The theoretical part contains roughly one third and the calculation part two third of the total points. Exam duration: 90 min.

2

2 Hydro-Environmental Models and their Use 2.1 Model concepts

2 Hydro-Environmental Models and their Use Hydrological models are simplified representations with approximations of the hydrological cycle in reality. They are primary used for predictions and the understanding of hydrological processes. In order to set up a model as well as to parameterize it, it is essential to have forcing data, the system characteristics, the observations of states and the associated system responses. One can speak from a good performance model when it provides reliable predictions in a realistic way and the model system is completely clear. A model is always as good as the data input it gets.

2.1 Model concepts Hydrological models can be variously classified. There are many computer codes available for catchment hydrology simulation and analysis, dependent on the specific application. These differ significantly on their way of process schematization and conceptualization. Figure 2-1 represents the different hydrological model concepts.

Figure 2-1 Model concepts

2.1.1 Model distinction based on process description A physically based model takes the most important physical laws governing the phenomena into account. Based on the balance equations, it describes explicitly the potential gradients and resistances that determine water flows along multiple paths. Nonetheless, it needs a high amount of information and is computationally demanding.

3

2 Hydro-Environmental Models and their Use 2.1 Model concepts

An empirical or Black Box model does not aid in understanding physical, hydrological problems hence it contains parameters that have little direct physical significance. It is built upon observations and derived relations (functions) of input and output but does not show the process of conversion. Nevertheless it can serve as a helpful tool in decision making due to accurate answers. A conceptual model, also called grey box models, describes the most important processes. Physical laws and assumptions about main flow processes at catchment scale are included but in a highly simplified form. It needs reasonable amounts of input data and the model parameters are usually determined through calibration against observed runoff. Nevertheless it should be taken care that the runoff process is presented in a realistic manner. Conceptual models are easier to construct than physically based models, quick to implement with a moderate amount of information and are computationally efficient. Conceptual models lie in the middle between physically based and empirical model structures.

All types of process models represent options to model reality in different levels of approximation but for different ranges of application. Different circumstances afford different usages, depending on effectiveness, objective of the work, data availability, the degree of complexity of the modelling question, the available resources and the desired degree of accuracy. The more physically based a model is, the more the parameters consider the measurable landscape characteristics. The more conceptual a model is, the more the parameters consider the functional aspect of the whole watershed and the less they are influenced by the measurable landscape characteristics. Figure 2-2 shows that physically based models are usually applied in smaller catchments than conceptual models. Moreover, the performance of conceptual models rises with catchment size whereas the physically based model performs best on very small catchments.

Figure 2-2 Different types of water quality models

4

2 Hydro-Environmental Models and their Use 2.2 SWAT Model

2.1.2 Model distinction based on spatial representation Regarding the spatial discretization and resolution, an ascending scale of accuracy and sophistication can be identified. Lumped models treat the complete basin as one homogenous area where the spatial variations of characteristics and processes are averaged or ignored. The next higher group of models are the semi- distributed models that subdivide the catchment in further separated areas or subbasins to compute flow contributions. The highest resolution offers distributed models, where spatial variations are considered explicitly. Since the whole area is separated in elementary unit areas, like a grid net, flows can be passed from one grid point to another.

2.1.3 Model distinction based on aspect of randomness Deterministic models include parameters and processes which are free of random variation. In contrast, stochastic models describe random variation and incorporate variables having probability distributions.

2.2 SWAT Model The hydrological model SWAT, Soil and Water Assessment Tool, is a semi- distributed model and works within a user friendly ArcGIS environment with a graphical interface. Nowadays the public domain model has a large user community. It is a continuous time model that is able to model on an hourly time step. It was developed for the USDA (United States Department of Agriculture) to quantify the impact of land management practices in large and complex watersheds over a long time. Therefore SWAT is useful to make predictions about the hydrological cycle, the nutrient, bacteria and sediment transport as well as the yield. So it’s particularly helpful in watersheds where agriculture is the main land use. SWAT originates from the CREAMS model, where the daily hydrology, erosion and nutrients at the field scale come from, combined with the EPIC crop growth model and the GLEAMS pesticide fate component. In addition SWAT takes up a multiple subbasins approach such as subsurface and groundwater flow, flood and sediment routing, a weather generator and reservoir storage. Figure 2-3 shows the classification of the SWAT model concept.

5


Figure 2-3 Classification of the SWAT model

Typically the usage of SWAT is optimized for North-American conditions, so the crop and soil database are adapted to these requirements and climate data acquisition is optimized for data pools in the U.S.A. The SWIM model (Soil and Water Integrated Model), developed at the Potsdam Institute for Climate Impact Research (PIK), aims to adapt to European conditions. Versions and modifications of SWAT are also available. In order to overcome the weaknesses in the process description, they assure an improving performance in certain scenarios. •

SWAT-G, Eckhardt et al. (2002): •

•

SWAT-N, Pohlert (2006): •

•

SWAT overestimates base flow and underestimates interflow especially for low mountain range catchments. SWAT- G puts more weight on interflow and improves the groundwater modelling.

SWAT overestimates nitrate losses due to denitrification. Therefore SWAT- N was developed to lower the influence of denitrification.

SWAT-E, Vandenberghe, et al. (2002): •

SWAT only uses input data on a daily time step. With the SWAT- E model it is possible to use hourly data which improves the accuracy of the modelling process.

2.2.1 Hydrologic Similarity and the HRU Concept As a spatial discretization the watershed is partitioned into several subwatersheds or subbasins for the modeling process. The benefit of this is that there is a distinction between subbasins with different land uses or soils that are influencing the hydrology in a different way. Subbasins are spatially distributed, which means that there is a reference between different areas of the watershed. The main input information for the subbasins are the climate, ponds or wetlands, groundwater and the reach, draining the subbasin. Furthermore SWAT characterizes the subbasin by generating aggregated, conceptual units of uniform characteristics, so called hydrolog6


ical response units (HRU). A HRU is defined by the unique combination of land use, soil type and slope class. It is meant to be that HRUs with same characteristics act in a similar way during a rainfall- runoff process. HRUs are non- spatial subdivisions, meaning that they are independent of position in the catchment. So there is no explicit routing of subsurface or surface flow between the HRUs because they are not connected to each other. Besides it is impossible to model nuances within a subbasin because of the missing interpretation at the pixel level. Therefore SWAT is a semi-distributed model approach where the calculations are based on the aggregated HRU area and not on a cell-by-cell basis. Fine DEM resolution results in a high number of HRUs and thus leading to a time-consuming parameterization. The advantages of the HRU approach are a better depiction of the processes governing surface runoff („SCS CNmethod“) and an improvement of management specification depending on plant and slope (e.g. contour tillage on steep slopes).

Figure 2-4 Spatial discretization structure in SWAT

2.2.2 Water Balance in SWAT In order to get a faithful prediction for the movement of pesticides, nutrients or sediments, the hydrological cycle simulated by the model has to match to the processes in the watershed in reality. The driving force behind every model concept in SWAT is the water balance, which therefore has to be taken into account carefully. A proper calibration of the hydrologic cycle is precondition for further calibration on sediment erosion and nutrient transport.

Simulation of the hydrological processes is partitioned in a land phase and a water phase. The land phase controls the amount of water, sediment, nutrient and pesticide loadings into the reach of each subbasin. On the other hand, the water phase describes the movement of water and sediments through the channel network, the so called routing phase. 7


Figure 2-5 The Hydrological Cycle after Neitsch et al. (2011), (slightly modified)

Land Phase The land phase of the hydrological cycle follows the water balance equation: 𝑖

𝑆𝑊𝑡 = 𝑆𝑊0 + ∑(𝑅𝐷𝑎𝑦 − 𝑄𝑠𝑢𝑟𝑓 − 𝐸𝑎 − 𝑤𝑠𝑒𝑒𝑝 − 𝑄𝐺𝑊 )

(2-1)

𝑡=1

Where 𝑆𝑊𝑡 is the final soil water content [𝑚𝑚 𝐻2 𝑂], 𝑆𝑊0 is the initial soil water content [𝑚𝑚 𝐻2 𝑂], 𝑅𝐷𝑎𝑦 is the amount of precipitation [𝑚𝑚 𝐻2 𝑂], 𝑄𝑠𝑢𝑟𝑓 is the surface runoff [𝑚𝑚 𝐻2 𝑂], 𝐸𝑎 is the amount of evapotranspiration[𝑚𝑚 𝐻2 𝑂], 𝑤𝑠𝑒𝑒𝑝 is the amount of water entering the vadose zone [𝑚𝑚 𝐻2 𝑂] and 𝑄𝐺𝑊 is the amount of water returning to the rivers as base flow[𝑚𝑚 𝐻2 𝑂], all indications on day i. Evapotranspiration is calculated on subbasin basis, which enables the model to reflect differences due to different crops or soil types. A schematic representation of the land phase is shown in Figure 2-6.

8


Figure 2-6 The land phase of the hydrological cycle, (Neitsch et al. 2011)

The climate is responsible for the moisture and energy that is needed to control the water balance and to set the processes with the main impact on the hydrological cycle. The climate data that is required as an input for SWAT is the daily maximum/minimum temperature, the wind speed, the humidity, the precipitation and the solar radiation. There are some functions for customization, like elevation bands that take into account the orographic effects, or climate change variations which allows the user to set some adjustment factors. Hydrology is especially complex and particularly difficult to model due to several different pathways of water movement in a HRU. Precipitation either directly contributes to surface runoff, mostly if the rate of water application exceeds the rate of infiltration. For the calculation of the surface runoff, the (NRCS) Curve Number Method is used to simulate flood events. It is a widely used method for estimating effective rainfalls, developed by the U.S. Natural Resources Conservation Service (formerly the Soil Conservation Service (SCS)). The discharge can be estimated via rainfall records and area specific runoff parameters for vegetation and soils with respect to soil moisture, especially if there are only a few or no rainfall-runoff relationships available. SWAT makes the assumption that hydrological soil groups have the same runoff potential under similar storm events and land cover. Another approach to calculate the runoff volume is the Green- Ampt method of infiltration, which assumes that water that does not infiltrate becomes surface runoff. The portion of rainfall that does not turn into surface runoff is divided by infiltration and evaporation. Precipitation that infiltrates into the ground can go back to the main reach by lateral flow through the soil profile or as base flow coming from the aquifer. The groundwater system is partitioned into two aquifers: one unconfined, which contributes to the surface stream, and a deep confined aquifer, where infiltrated water goes out of the system. The Penman- Monteith method is used to calculate potential evapotranspiration which includes evaporation from plant canopy and the soil, transpiration and sublimation. SWAT first evaporates water from the plant canopy and then 9


calculates the maximum amount of transpiration and sublimation/soil evaporation to determine the actual amount of sublimation/soil evaporation. Evapotranspiration is very important in the water balance due to the fact that it is the primary removal mechanism of water from a catchment and exceeds runoff in quantity in most river basins. A single crop growth model is used by SWAT to determine the removal of water and nutrients from the root zone, transpiration and biomass/yield production. Erosion and sediment yield are calculated with the MUSLE equation, a modified form of the Universal Soil Loss Equation (USLE). The runoff serves as an indicator of erosive energy for each HRU. Nutrients and pollutants are entering the channel by runoff or lateral flow. SWAT models the movement and transformation of nitrogen and phosphorus and keeps also track of the pesticide movement in the stream as well as in the soil profile. It is possible to implement different management operations in SWAT, such as planting, irrigation, fertilization, tillage and harvest applications. By defining the timing, amount or type, the user can specify the management practices for each HRU. This chapter just outlines the basic processes implemented in SWAT and how the model basically subdivides its single components. All these concepts mentioned above are further explained in the following chapters.

Water Phase In the land phase SWAT models the amount of water, sediments, nutrients and pesticides and in the water or routing phase these loadings are routed through the channel network of the catchment, as shown in Figure 2-7. It is not just keeping track of the loadings, there are also different in-stream processes involved; biodegradation, diffusion, sorption and deposition take place in the SWAT modeling process. For the flood routing itself, the variable storage coefficient method or the Muskingum routing method is used. A more detailed description is given in the following chapters.

10

2 Hydro-Environmental Models and their Use 2.3 Case Study: Ethiopia

Figure 2-7 Water Routing Phase, Van Griensven (2010),modified

2.3 Case Study: Ethiopia 2.3.1 NeXus: Water, Food & Energy Africa is a relative dry and water scare continent, where about 300 million people suffer from water shortages. A continent with arid and semi-arid climate and where a huge fraction of the crops are in dry region where irrigation is not sustainable. The Nile River Basin specifically suffers from increasing water stress due to population increase and competing demands on limited resources (water, food, energy). Currently the water use management is a need and not a subject to any rational and sophisticated decision support system. A research project at TUM is focused on several aspects of hydro climatology, water-food security, land use management and distribution, water reservoirs availability, remote sensing, and the interaction of the people and the surrounding environment in the Blue Nile Basin in Ethiopia. It is intended to develop an integrated management tool based on the present and projected future hydrological conditions of Ethiopia, the availability of water in reservoirs, groundwater and soil moisture to optimize the overall water use of the Nile River Basin, and to maximize its ecological, economic and social stability as well. For further information about the TUM NeXus project, please visit: www.nexus.wasser.tum.de .

2.3.2 Description of the Research Area The Nile River is considered to be of international importance due to its condition of water supplier for several countries in Africa: Congo, Ethiopia, Eritrea, South Sudan, Sudan, Tanzania, 11


Uganda, Egypt and others. The sources of the Nile River are the White and the Blue Nile rivers. The Blue Nile Basin also called Abbay basin was after several discussions with my respective advisors selected for my research due to its local and international importance. Abbay basin is located in the north-western part of Ethiopia, approximately between Latitude 7 40’N and 12 51’N, and Longitude 34 25’E and 39 49’ E (see Figure 2-8). The total area of Abbay basin is approximately 199,812 km2, however two of the subbasins in the northern region of Abbay basin are shared with Sudan, and therefore this area is not counted in this study, so the total study area in this research is 172,767 km².

Figure 2-8: Location of the Blue Nile Basin in Ethiopia

2.3.3 Data Collection and Management Since the scope of this project is very broad and as in any other research, data collection is one of the most important tasks. This data is necessary with the objective of gaining familiarization with the research area, obtaining enough information about the current condition of the Blue Nile Basin, understanding what kind of researches were already made in the area and to determine the main problems that the Blue Nile Basin is currently facing. After a better knowledge of the area is acquired and the required data from the Ministry of Water and Energy of Ethiopia and from the National Meteorological Agency of Ethiopia is obtained, it is possible to plan the next steps of the research and determine more detailed goals and how these goals can be achieved. As one of the objectives of this research is to assess the social, economic and environmental vulnerability of the Blue Nile Basin, it is to mention that there is not an universal method to per12


form this assessment. Therefore this research will intend to evaluate different hydrological models and propose the best model that can meet an equilibrium point between the studied factors.

2.3.4 Model Set Up: DEM

Figure 2-9: DEM of Abbay basin at 90 m resolution, from CGIAR-CSI.

A SRTM DEM of the Blue Nile at 90 meters resolution was downloaded from the CGIAR-CSI (Consultative Group on International Agricultural Research-Consortium for Spatial Information) website. Based on this DEM, the altitude of the Blue Nile Basin in Ethiopia, ranges from 483 meters above sea level downstream in the border with Sudan, to 4,248 meters above sea level in the eastern part of the Ethiopian highlands (see Figure 2-9). In this DEM, it can be seen that Abbay basin has two very distinctive topographic regions, the eastern region is very mountainous, covering altitudes approximately from 1,956 up to 4,248 meters above sea level. Whereas the western region, has mostly lowlands, with altitudes between 483 and 1,955 meters above sea level approximately. Officially the Blue Nile Basin in Ethiopia is divided into 14 subbasins. However, for this research an automatic watershed delineation was done, where the flow direction, flow accumulation and streams network were automatically determined based on the 90 meters resolution DEM. The outlets for the subbasins were selected manually based on the largest tributaries of the main river, intending to have an equal distribution of the subbasins and a representative location of the available meteorological stations, in the end 13 subbasins were created (see Figure 2-10).

13


Figure 2-10: Subbasins distribution of Abbay catchment, based on an automatic delineation

2.3.5 Research Outline The research is basically focused on the SWAT water resources module. In order to achieve the desired goals several types of data are taken into consideration: meteorological data (relative humidity, precipitation, temperature, solar radiation and wind speed), landuse maps (agriculture, forest, and deserts), hydrological data (rivers discharge flow, groundwater levels, water level of lakes and reservoirs) and some other socio-economic data (cities distribution and population density). The following are the objectives that are expected to be accomplished at the end of research period: - Creation of a SWAT model (Soil and Water Assessment Tool) that simulates the quantity of available surface and ground water, and will help to predict the environmental impact of land use, land management, climate change, soil erosion prevention and control in watersheds. This model will also propose an equilibrium point to optimize irrigation, agricultural practices and food security in the area. Projection of the future hydrological conditions of Abbay basin. The SWAT model will give a simulation of the current conditions and a seasonal aggregated forecasted rainfall data will be used as input to do a representation of the future hydrological conditions of the basin. - Simulation and analysis of the land use changes and their impacts on the hydrology, geomorphology and soils of the Blue Nile. Afterwards a projection of the land use changes will be done to determine how the condition of the basin in the near future is going to be. This allows to propose an ecologically and economically effective land use distribution. Analysing the agricultural productivity and its sustainability. A projection of the agricultural productivity on the basis of land use changes and hydrological data will be done to identify the critical role and choices of the farmers regarding agricultural yield and economic benefits. Op14

2 Hydro-Environmental Models and their Use 2.4 Further Reading

timization of the irrigation management practices of agricultural lands of the Blue Nile Basin based on the results of agricultural and hydrological droughts. This will be assessed using the SWAT model considering the status of agricultural management practices.

2.4 Further Reading Read about recent trends and improvements in the delineation of hydrological response units (HRU): Sanzana, P. et al.: Computer-assisted mesh generation based on hydrological response units for distributed hydrological modeling, Computers & Geosciences 57 (2013) 32–43 http://dx.doi.org/10.1016/j.cageo.2013.02.006

15

3 Climate Input 3.1 Precipitation

3 Climate Input The success of a hydrological model depends on the data availability. Typically the choice of data acquisition depends on time and other resources available. Data can be gathered for different scales with multiple methods. Data sets at a global scale serve as a starting point in hydrological modeling via basic climatology. A large variability of this broad- scale information is available to download at no or little cost, for example contour lines from SRTM data to create a DEM. In many watershed predictions, data has to be more detailed with catchment characteristics and a better temporal and spatial solution. So, the hydrologist will acquire hydrological data from national hydrological network or local field visits. This is mostly associated with more effort and costs and requires expert knowledge. The best option to predict catchment responses is the collection of some short-term measurements or even installation of an stream gauge and other hydrological equipment.

3.1 Precipitation Precipitation in the water balance equation contributes as the main input of water to a surface. It is therefore generally a sensitive parameter and needs careful assessment in hydrological modelling. The term ‘precipitation’ includes all condensation products which are released by the atmosphere to the surface. Very good info can be found on the online courses: “Precipitation Estimates” Part I & II (UCAR/COMET) on www.meted.ucar.edu.

3.1.1 Measurement equipment Since it is essential for hydrological modelling to know the amount and the time of precipitation, the basic measurement techniques and inaccuracies are shortly explained. As a result of the relatively easy application of measurement techniques, precipitation records exist for longer observation periods and denser measurements networks than discharge records. It is quite difficult to determine the exact precipitation on a catchment scale using the point-based gauging stations, because the amounts vary a lot even over a small terrain. The quality of the rain gauge measurement is affected by the location and the type of measurement. In order to reduce the effects of the environment to a minimum, the siting has to be in a certain distance from possible obstacles, like buildings or trees. Moreover, the location has to be representative of a far greater area and should act as a surrogate for the whole catchment. Rainfall depth is referred to the water layer that would accumulate on the horizontal projection of the earth surface and is measured by millimeters depth. The observation of precipitation can be executed with different designs of rain gauges. As an instrument for measuring rainfall serve cylindrical container with horizontal surfaces in variable size. Due to inaccuracies caused by splash from the surrounding area, the rain gauge should be delineated by a sharp rim or it can be raised up above the ground. The raising leads to air turbulence around the gauge, which can at least be reduced with a windshield. Another source of error is the loss caused by evapo16


ration, which can be diminished with a funnel shaped rain gauge. The funnel has to have steep sides to lessen the wetting loss for small rainfall events on sunny days. Moreover, the instrument itself causes eddies that reduce the catch of smaller raindrops or snowflakes. According to literature measurement errors range between 10% and 30%. Three measurement instruments that are currently in use are shown in Figure 3-1.

Hellman

Thies Ombrometer

Ott Pluvio²

Figure 3-1 Measurement equipment for precipitation

The Hellman rain gauge is a non- electric device for continuous recording of precipitation. The floating measurement system would drain independently after every 10 mm of precipitation height. The marker produces a sum curve of the amount of rainfall. The Thies Ombrometer is used to record readings of precipitation volume and intensity for digital transmission. The tipping actions of the tipping bucket are recorded by a contact and read out via the electronics as a pulse for further processing using intensity-related linearization. Related errors are caused by turbulences, fine drops, wind, evapotranspiration and interception. The Ott Pluvio precipitation gauge collects the water and weights the amount to recalculate the rainfall height. Advantages are that there are no evapotranspiration losses and the measurement of snow and hail is possible. Moreover there is no underestimation of intense rain. On the other hand it is more expensive and needs more maintenance.

3.1.2 Snow Snow Cover SWAT uses a user-defined temperature threshold Ts-r to differentiate between rainfall and snowfall. Snowfall is stored at the ground surface in the form of a snow pack. The amount of water stored in the snow pack is reported as snow water equivalent. Due to sublimation, melting or further snowfall, a change in size may occur. The mass balance of the snowpack is as follows:

17


𝑆𝑁𝑂𝑛𝑒𝑤 = 𝑆𝑁𝑂 + 𝑅𝑑𝑎𝑦 − 𝐸𝑠𝑢𝑏 − 𝑆𝑁𝑂𝑚𝑙𝑡

(3-1)

where 𝑆𝑁𝑂 is the water content of the snow pack on a given day [𝑚𝑚 𝐻2 𝑂], 𝑅𝑑𝑎𝑦 is the amount of precipitation on a given day [𝑚𝑚 𝐻2 𝑂], 𝐸𝑠𝑢𝑏 is the amount of sublimation on a given day [𝑚𝑚 𝐻2 𝑂] and 𝑆𝑁𝑂𝑚𝑙𝑡 is the amount of snow melt on a given day [𝑚𝑚 𝐻2 𝑂]. Figure 3-2 illustrates the phase transition of water.

Figure 3-2 Phase transition of water

A non-uniform distribution of the snow pack over the total area is caused by drifting, shading and aspects in topography. This results in a fraction of the subbasin area that is bare of snow. The influencing factors can be regarded similar from year to year. Figure 3-3 illustrates a snow distribution.

Figure 3-3 Snow distribution

18


SWAT determines the actual amount of snow present in the subbasin and the areal snow coverage with an aerial depletion curve. It is used to describe seasonal growth and recession of the snow pack as a function of present snow: −1

SNOcov

SNO SNO SNO = ⋅ ( + exp (cov1 − cov2 ⋅ )) SNO100 SNO100 SNO100

(3-2)

where SNOcov is a fraction of HRU covered by snow, SNO is the water content of the snow pack (real amount of snow water equivalent) [𝑚𝑚 𝐻2 𝑂] , SNO100 is the threshold depth of snow at 100% coverage [𝑚𝑚 𝐻2 𝑂] and cov1 , cov2 are the coefficients determining the curves shape. The threshold depth of snow at 100% coverage is a fixed, aerial factor.

Figure 3-4 illustrates an aeral depletion curve. The areal depletion curve affects snow melt only when the snow pack water content is between 0 and SNO100. So the influence of snow melt will be minimal when SNO100 is set to a very small value. As the value for SNO100 increases, the impact of the areal depletion curve will assume more importance of snow melt processes. The snow cover depends on vegetation distribution, wind loading or scouring of snow and interception. Figure 3-4 shows the areal depletion curve with 90% SNO100and 50% coverage.

19


Figure 3-4 Areal depletion curve

Snow melt The snow melt is controlled by air temperature, snow pack temperature, melting rate and the areal coverage of snow. Snow melt process is distinguished into two parts: One part affects the snow pack temperature and the other part accounts for the snow melt. The equation of the snow pack temperature is as follows:

𝑇𝑠𝑛𝑜𝑤(𝑑𝑛 ) = 𝑇𝑠𝑛𝑜𝑤(𝑑𝑛−1 ) ∗ (1 − 𝑙𝑠𝑛𝑜 ) + 𝑇𝑎𝑣 ∗ 𝑙𝑠𝑛𝑜

(3-3)

where Tsnow(dn) is the snow pack temperature on a given day [°𝐶], Tsnow(dn−1 ) is the snow pack temperature on the previous day [°𝐶], lsno is the snow temperature lag factor [0,1] and Tav is the mean air temperature on the current day [°𝐶]. It is a function of the mean daily temperature during preceding days and it is correlated to air temperature. The lagging factor 𝑙𝑠𝑛𝑜 controls the influence of the previous day’s snow pack temperature and accounts for snow pack density, snow pack depth and exposure. The larger 𝑙𝑠𝑛𝑜 , the smaller the influence of 𝑇𝑠𝑛𝑜𝑤(𝑑𝑛−1 ) . The snow pack will not melt until the temperature exceeds a user defined threshold value. The snow melt equation is as follows:

𝑆𝑁𝑂𝑚𝑙𝑡 = 𝑏𝑚𝑙𝑡 ∗ 𝑠𝑛𝑜𝑐𝑜𝑣 ∗ [

20

𝑇𝑠𝑛𝑜𝑤 + 𝑇𝑚𝑥 − 𝑇𝑚𝑙𝑡 ] 2

(3-4)

3 Climate Input 3.2 Temperature

where 𝑆𝑁𝑂𝑚𝑙𝑡 is the amount of snow melt on a given day [𝑚𝑚 𝐻2 𝑂], 𝑏𝑚𝑙𝑡 is the melt factor for the day [𝑚𝑚 𝐻2 𝑂⁄𝑑𝑎𝑦 °𝐶 ], 𝑠𝑛𝑜𝑐𝑜𝑣 is the fraction of the HRU covered by snow, 𝑇𝑠𝑛𝑜𝑤 is the snow pack temperature on a given day [°𝐶], 𝑇𝑚𝑥 is the maximum air temperature on a given day [°𝐶] and 𝑇𝑚𝑙𝑡 is the base temperature above which snow melt is allowed [°𝐶]. The melt factor is calculated using the equation:

𝑏𝑚𝑙𝑡 =

𝑏𝑚𝑙𝑡6 + 𝑏𝑚𝑙𝑡12 𝑏𝑚𝑙𝑡6 − 𝑏𝑚𝑙𝑡12 2∗𝜋 + ∗ 𝑠𝑖𝑛 ( ∗ (𝑑𝑛 − 81)) 2 2 365

(3-5)

where 𝑏𝑚𝑙𝑡6 is the melt factor for June 21 [𝑚𝑚 𝐻2 𝑂⁄𝑑𝑎𝑦 °𝐶 ], 𝑏𝑚𝑙𝑡12 is the melt factor for December 21 [𝑚𝑚 𝐻2 𝑂⁄𝑑𝑎𝑦 °𝐶 ] and 𝑑𝑛 is the day of the year [𝐽𝑢𝑙𝑖𝑎𝑛 𝐷𝑎𝑦]. Typical ranges for the snow melt factor are: Rural areas

1.4 -6.9

𝑚𝑚𝐻2 𝑂 𝑑𝑎𝑦 °𝐶

Urban areas

3.0- 8.0


Asphalt

1.7- 6.5


Melting water is assumed to have an energy of 0 when calculating erosion and the water is assumed to be evenly distributed over the 24h of the day.

3.2 Temperature 3.2.1 Air temperature The air temperature influences the physical, chemical and biological processes on the earth. Especially, it has a strong impact on plant production, decomposition of organic matter and mineralization.The soil and water temperature are derived from air temperature. SWAT needs daily maximum and minimum air temperature values (min. 03:00 am; max. 03:00 pm) as an input. Therefore the accuracy of the model is significantly improved by the use of measured data (in contrast to data generated with a weather generator). The conversion of daily to hourly data assumes a sinusoidal interpolation function:

𝑇ℎ𝑟 = 𝑇𝑎𝑣 +

(𝑇𝑚𝑥 − 𝑇𝑚𝑛 ) ∗ cos(0.2618 ∗ (ℎ𝑟 − 15)) 2

(3-6)

where 𝑇ℎ𝑟 is the air temperature for a certain hour of the day [°𝐶], 𝑇𝑎𝑣 is the average air temperature [°𝐶], 𝑇𝑚𝑥 is the daily max. temperature, 𝑇𝑚𝑛 is the daily min. temperature and ℎ𝑟 is the hour of the day. Figure 3-5 shows the diurnal variation of air temperature.

21


Diurnal variation of air temperature for August 2013 (Osterseeon) 35

Air temperature

30 25 20 15 10

5 0

12

24

36

48

hr

Figure 3-5 Diurnal variation of air temperature for August 2013 over two days (Osterseeon)

3.2.2 Soil temperature The soil temperature fluctuates due to seasonal and diurnal variations of the temperature at the surface. The annual variation in soil temperature follows a sinusoidal function and the amplitude decreases with depth. Furthermore the timing of the temperature peak varies with depth. The fluctuations of the temperature are shown in Figure 3-6.

Figure 3-6 Four-year average air and soil temperature at College Station, Texas (SWAT Technical Manual)

The soil temperature is described as follows:

−𝑧 𝑧 𝑇𝑠𝑜𝑖𝑙 (𝑧, 𝑑𝑛 ) = 𝑇𝐴𝐴 + 𝐴𝑠𝑢𝑟𝑓 ∗ 𝑒𝑥𝑝 ( ) ∗ 𝑠𝑖𝑛 (𝜔𝑡𝑚𝑝 ∗ 𝑑𝑛 − ) 𝑑𝑑 𝑑𝑑 22

(3-7)


where 𝑇𝑠𝑜𝑖𝑙 (𝑧, 𝑑𝑛 ) is the soil temperature [°𝐶] at depth 𝑧 and of the day 𝑑𝑛 of the year, 𝑇𝐴𝐴 is the average annual soil temperature [°𝐶], 𝐴𝑠𝑢𝑟𝑓 is the amplitude of the soil fluctuations [°𝐶], 𝑧 is 0 at the surface [𝑚𝑚], 𝑑𝑑 is the damping depth [𝑚𝑚] and 𝜔𝑡𝑚𝑝 is the angular frequency. There are two special cases: 1.  𝑧 = 0: no damping occurs and therefore:

𝑇𝑠𝑜𝑖𝑙 (𝑧, 𝑑𝑛 ) = 𝑇𝐴𝐴 + 𝐴𝑠𝑢𝑟𝑓 ∗ 𝑠𝑖𝑛(𝜔𝑡𝑚𝑝 ∗ 𝑑𝑛 )

(3-8)

2.  𝑧 = ∞: no fluctuations occur:

𝑇𝑠𝑜𝑖𝑙 (𝑧, 𝑑𝑛 ) = 𝑇𝐴𝐴

(3-9)

To calculate the layer specific soil temperatures the heat capacity and thermal conductivity must be known, which are not commonly measured and thus can be calculated as: 𝑇𝑠𝑜𝑖𝑙 (𝑧, 𝑑𝑛 ) = 𝑙 ∗ 𝑇𝑠𝑜𝑖𝑙 (𝑧, 𝑑𝑛 − 1) + (1 − 𝑙) ∗ (𝑑𝑓 ∗ (𝑇𝐴𝐴𝑖𝑟 − 𝑇𝑠𝑠𝑢𝑟𝑓 ) + 𝑇𝑠𝑠𝑢𝑟𝑓 )

(3-10)

where 𝑇𝑠𝑜𝑖𝑙 (𝑧, 𝑑𝑛 ) is the soil temperature at depth 𝑧 [°𝐶], 𝑙 is the lag coefficient (0-1), default = 0.8 [−], 𝑇𝑠𝑜𝑖𝑙 (𝑧, 𝑑𝑛 − 1) is the soil temperature for the day 𝑑𝑛 of the year in the layer from the previous day [°𝐶], 𝑑𝑓 is the depth factor quantifying influence of depth [−], 𝑇𝐴𝐴𝑖𝑟 is the average annual air temperature [°𝐶] and 𝑇𝑠𝑠𝑢𝑟𝑓 is the soil surface temperature [°𝐶]. For depths close to the soil surface, soil temperature is a function of the soil surface temperature. As depth increases, soil temperature is increasingly influenced by the average annual air temperature, until the damping depth. Then the soil temperature is within 5% of the average annual air temperature.

Calculation of the depth factor 𝑑𝑓 is done with the equation:

𝑑𝑓 =

𝑧𝑑 𝑧𝑑 + 𝑒𝑥𝑝[−0.867 − 2.078 ∗ 𝑧𝑑]

(3-11)

with

23


𝑧𝑑 = 𝑧/𝑑𝑑

(3-12)

where 𝑧𝑑 is the ratio in depth to dumping depth [−], 𝑧 is the depth at the center of the soil layer [𝑚𝑚] and 𝑑𝑑 is the damping depth [𝑚𝑚]. The calculation of the damping depth is expressed by combining the maximum dumping depth and the scaling factor: 500 1−𝜑 2 𝑑𝑑 = 𝑑𝑑𝑚𝑎𝑥 ∗ 𝑒𝑥𝑝 [𝑙𝑛 ( )∗( ) ] 𝑑𝑑𝑚𝑎𝑥 1+𝜑

(3-13)

where 𝑑𝑑𝑚𝑎𝑥 is the maximum damping depth [𝑚𝑚] and 𝜑 is the scaling factor for soil water [−]. The dumping depth 𝑑𝑑 itself is the dampened version of a maximum 𝑑𝑑 factor, which is influenced by the bulk density. The maximum dumping depth is calculated as follows:

𝑑𝑑𝑚𝑎𝑥 = 1000 +

2500 ∗ 𝜌𝑏 𝜌𝑏 + 686 ∗ 𝑒𝑥𝑝(−5.63 ∗ 𝜌𝑏 )

(3-14)

where 𝑑𝑑𝑚𝑎𝑥 is the maximum damping depth [𝑚𝑚], 𝜌𝑏 is the soil bulk density [𝑀𝑔⁄𝑚3 ].

Moreover, the soil water content plays a role and is included as s scaling factor:

𝜑=

𝑆𝑊 (0.356 − 0.144 ∗ 𝜌𝑏 ∗ 𝑧𝑡𝑜𝑡 )

(3-15)

where 𝜑 is the scaling factor (– ), 𝑆𝑊 is the amount of water in the soil profile [𝑚𝑚𝐻2 𝑂] and 𝑧𝑡𝑜𝑡 is the depth from the soil surface to the bottom of the soil profile [𝑚𝑚]. The surface temperature of the soil 𝑇𝑠𝑠𝑢𝑟𝑓 is dependent on the previous day‘s temperature, the amount of ground cover and the temperature of ground cover without any cover and it can be expressed like this:

𝑇𝑠𝑠𝑢𝑟𝑓 = 𝑏𝑐𝑣 ∗ 𝑇𝑠𝑜𝑖𝑙 (1, 𝑑𝑛 − 1) + (1 − 𝑏𝑐𝑣) ∗ 𝑇𝑏𝑎𝑟𝑒

(3-16)

where 𝑇𝑠𝑠𝑢𝑟𝑓 is the soil surface temperature for the curent day [°𝐶], 𝑏𝑐𝑣 is the weighting factor [−], 𝑇𝑠𝑜𝑖𝑙 (1, 𝑑𝑛 − 1) is the soil temperature of the first soil layer on the previous day and 𝑇𝑏𝑎𝑟𝑒 is the temperature of the bare soil:

24


𝑇𝑏𝑎𝑟𝑒 = 𝑇𝑎𝑣 + 𝜀𝑆𝑅 ∗

𝑇𝑚𝑎𝑥 − 𝑇𝑚𝑖𝑛 2

(3-17)

with the radiation term for bare soil surface temperature calculation:

𝜀𝑆𝑅 =

𝐻𝑑𝑎𝑦 ∗ (1 − 𝛼) − 14 20

(3-18)

Where is the 𝐻𝑑𝑎𝑦 is the solar radiation on the current day [𝑀𝐽⁄𝑚2 𝑑𝑎𝑦] and 𝛼 is the albedo for the day [−]. Any cover will significantly impact the soil surface temperature, this influence is incorporated with the weighting factor 𝑏𝑣𝑐. The influence of the plant or snow cover is expressed as:

𝐶𝑉 𝐶𝑉 + exp(7.563 − 1.297 ∗ 10−4 ∗ 𝐶𝑉) 𝑏𝑣𝑐 = 𝑚𝑎𝑥 𝑆𝑁𝑂 { 𝑆𝑁𝑂 + exp(6.055 − 0.3002 ∗ 𝑆𝑁𝑂)

(3-19)

where 𝐶𝑉 is the total aboveground biomass [𝑘𝑔⁄ℎ𝑎 ] and 𝑆𝑁𝑂 is the water content of the snow cover [𝑚𝑚𝐻2 𝑂]. So SWAT distinguishes between bare soil (𝑏𝑣𝑐 = 0) and soil cover (𝑏𝑣𝑐 = 1).

Figure 3-7 Temperature fluctuations in soil [2]

25

3 Climate Input 3.3 Elevation bands

3.2.3 Water temperature The water temperature of the river is required to model in-stream biological and water quality processes. Since water is inert the response to a changing air temperature is for a well mixed stream 3-7 hours (< 1 day). The larger the river gets, the more inert water reacts and the larger the delay time for a response on a change in air temperature.

𝑇𝑤𝑎𝑡𝑒𝑟 = 5 + 0.75 ∗ 𝑇𝑎𝑣

(3-20)

where 𝑇𝑤𝑎𝑡𝑒𝑟 is the water temperature [°𝐶] and 𝑇𝑎𝑣 the average air temperature [°𝐶]. In reality the water temperature is also influenced by solar radiation, relative humidity, wind speed, water depth, ground water inflow, artificial heat inputs, thermal conductivity of the sediments and impoundments along the stream network. But SWAT works with simplifications and neglects these aspects.

3.3 Elevation bands Orographic precipitation is a significant phenomenon in certain areas of the world. To account for orographic effects on both precipitation and temperature, SWAT allows up to 10 elevation bands to be defined in each subbasin. Precipitation and maximum and minimum temperatures are calculated for each band as a function of the respective lapse rate and the difference between the gauge elevation and the average elevation specified for the band. For precipitation, 𝑅𝑏𝑎𝑛𝑑 = 𝑅𝑑𝑎𝑦 + (𝐸𝐿𝑏𝑎𝑛𝑑 − 𝐸𝐿𝑔𝑎𝑢𝑔𝑒 ) ∗

𝑝𝑙𝑎𝑝𝑠 𝑤ℎ𝑒𝑛 𝑅𝑑𝑎𝑦 > 0.01 𝑑𝑎𝑦𝑠𝑝𝑐𝑝,𝑦𝑟 ∗ 1000

(3-21)

where 𝑅𝑏𝑎𝑛𝑑 is the precipitation falling in the elevation band [𝑚𝑚 𝐻2 𝑂], 𝑅𝑑𝑎𝑦 is the precipitation recorded at the gauge or generated from gauge data [𝑚𝑚 𝐻2 𝑂], 𝐸𝐿𝑏𝑎𝑛𝑑 is the mean elevation in the elevation band [𝑚], 𝐸𝐿𝑔𝑎𝑔𝑒 is the elevation at the recording gauge [𝑚], 𝑝𝑙𝑎𝑝𝑠 is the precipitation lapse rate [𝑚𝑚 𝐻2 𝑂/𝑘𝑚], 𝑑𝑎𝑦𝑠𝑝𝑐𝑝,𝑦𝑟 is the average number of days of precipitation in the subbasin in a year and the factor 1000 is needed to convert meters to kilometers. For temperature,

26

3 Climate Input 3.3 Elevation bands

𝑇𝑚𝑥,𝑏𝑎𝑛𝑑 = 𝑇𝑚𝑥 + (𝐸𝐿𝑏𝑎𝑛𝑑 − 𝐸𝐿𝑔𝑎𝑢𝑔𝑒 ) ∗

𝑡𝑙𝑎𝑝𝑠 1000

(3-22)

𝑇𝑚𝑛,𝑏𝑎𝑛𝑑 = 𝑇𝑚𝑛 + (𝐸𝐿𝑏𝑎𝑛𝑑 − 𝐸𝐿𝑔𝑎𝑢𝑔𝑒 ) ∗


(3-23)


(3-24)

𝑇𝑎𝑣,𝑏𝑎𝑛𝑑 = 𝑇𝑎𝑣 + (𝐸𝐿𝑏𝑎𝑛𝑑 − 𝐸𝐿𝑔𝑢𝑎𝑔𝑒 ) ∗

where 𝑇𝑚𝑥,𝑏𝑎𝑛𝑑 is the maximum daily temperature in the elevation band [°𝐶], 𝑇𝑚𝑛,𝑏𝑎𝑛𝑑 is the minimum daily temperature in the elevation band [°𝐶], 𝑇𝑎𝑣,𝑏𝑎𝑛𝑑 is the mean daily temperature in the elevation band [°𝐶], 𝑇𝑚𝑥 is the maximum daily temperature recorded at the gauge or generated from gauge data [°𝐶], 𝑇𝑚𝑛 is the minimum daily temperature recorded at the gauge or generated from gauge data [°𝐶], 𝑇𝑎𝑣 is the mean daily temperature recorded at the gauge or generated from gauge data [°𝐶], 𝐸𝐿𝑏𝑎𝑛𝑑 is the mean elevation in the elevation band [𝑚], 𝐸𝐿𝑔𝑎𝑔𝑒 is the elevation at the recording gauge [𝑚], 𝑡𝑙𝑎𝑝𝑠 is the temperature lapse rate [°𝐶/𝑘𝑚] and the factor 1000 is needed to convert meters to kilometers. Once the precipitation and temperature values have been calculated for each elevation band in the subbasin, new average subbasin precipitation and temperature values are calculated: 𝑏

𝑅𝑑𝑎𝑦 = ∑ 𝑅𝑏𝑎𝑛𝑑 ∗ 𝑓𝑟𝑏𝑛𝑑

(3-25)

𝑏𝑛𝑑−1

𝑏

𝑇𝑚𝑥 = ∑ 𝑇𝑚𝑥,𝑏𝑎𝑛𝑑 ∗ 𝑓𝑟𝑏𝑛𝑑

(3-26)

𝑏𝑛𝑑−1

𝑏

𝑇𝑚𝑛 = ∑ 𝑇𝑚𝑛,𝑏𝑎𝑛𝑑 ∗ 𝑓𝑟𝑏𝑛𝑑

(3-27)

𝑏𝑛𝑑−1

𝑏

𝑇𝑎𝑣 = ∑ 𝑇𝑎𝑣,𝑏𝑎𝑛𝑑 ∗ 𝑓𝑟𝑏𝑛𝑑

(3-28)

𝑏𝑛𝑑−1

where 𝑅𝑑𝑎𝑦 is the daily average precipitation adjusted for orographic effects [𝑚𝑚 𝐻2 𝑂], 𝑇𝑚𝑥 is the daily maximum temperature adjusted for orographic effects [°𝐶], 𝑇𝑚𝑛 is the daily minimum temperature adjusted for orographic effects [°𝐶], 𝑇𝑎𝑣 is the daily mean temperature adjusted for orographic effects [°𝐶], 𝑅𝑏𝑎𝑛𝑑 is the precipitation falling in the elevation band [𝑚𝑚 𝐻2 𝑂], 𝑇𝑚𝑥,𝑏𝑎𝑛𝑑 is the maximum daily temperature in the elevation band [°𝐶], 𝑇𝑚𝑛,𝑏𝑎𝑛𝑑 is the minimum daily temperature in the elevation band [°𝐶], 𝑇𝑎𝑣,𝑏𝑎𝑛𝑑 is the mean daily temperature in the ele27

3 Climate Input 3.4 Water Vapour and Relative Humidity

vation band [°𝐶], 𝑓𝑟𝑏𝑛𝑑 is the fraction of subbasin area within the elevation band and 𝑏 is the total number of elevation bands in the subbasin. The only processes modeled separately for each individual elevation band are the accumulation, sublimation and melting of snow. As with the initial precipitation and temperature data, after amounts of sublimation and snow melt are determined for each elevation band, subbasin average values are calculated. These average values are the values that are used in the remainder of the simulation and reported in the output files.

3.4 Water Vapour and Relative Humidity Relative humidity is required by SWAT if the Penman- Monteith or Pristley- Taylor equation is used to estimate potential evapotranspiration. It is also used to calculate the vapor pressure deficit on plant growth. Relative humidity is the ratio of an air volume‘s actual vapor pressure to its saturation vapor pressure:

𝑅ℎ =

𝑒 𝑒0

(3-29)

where 𝑅ℎ is the relative humidity [%], 𝑒 is the actual vapor pressure [𝑘𝑃𝑎] and 𝑒0 is the saturation vapor pressure[𝑘𝑃𝑎]. Saturation vapor pressure is the maximum vapor pressure that is thermodynamically stable and is a function of the air temperature:

𝑒0 = 𝑒𝑥𝑝 [

16.78 ∗ 𝑇𝑎𝑣 − 116.9 ] 𝑇𝑎𝑣 + 273.3

(3-30)

where 𝑒0 is the saturation vapor pressure[𝑘𝑃𝑎] and 𝑇𝑎𝑣 is the mean daily air temperature [°𝐶]. When relative humidity is known (it is typically measured) the actual vapor pressure can be calculated by rearranging the first equation:

𝑒 = 𝑒0 ∗ 𝑅ℎ

(3-31)

The saturation vapor pressure curve is obtained by plotting the 2nd equation. The slope is calculated by differentiating the 2nd equation:

∆=

28

4098 ⋅ 𝑒0 (𝑇𝑎𝑣 +

237.3)2

(3-32)

3 Climate Input 3.4 Water Vapour and Relative Humidity

where ∆ is the slope of the saturation vapor pressure curve [𝑘𝑃𝑎⁄°𝐶 ], 𝑒0 is the saturation vapor pressure [𝑘𝑃𝑎] and 𝑇𝑎𝑣 is the mean daily air temperature [°𝐶]. The rate of evaporation is proportional to the difference between the water vapor pressure of the surface layer and the vapor pressure of the overlying air. This water vapor pressure deficit is as follows:

𝑣𝑝𝑑 = 𝑒0 − 𝑒

(3-33)

where 𝑣𝑝𝑑 is the vapor pressure deficit [𝑘𝑃𝑎], 𝑒0 is the saturation vapor pressure[𝑘𝑃𝑎] and 𝑒 is the actual vapor pressure [𝑘𝑃𝑎]. The amount of water which can be held from the air depends on the temperature, as showed in Figure 3-8.

Figure 3-8 Water vapor pressure vs. air temperature vs. relative humidity

Figure 3-9 shows a psychrometer, a measurement device for relative humidity. A psychrometer is actually a rather simple type of hygrometer, an instrument that is used to measure the amount of humidity that is present in the atmosphere. A psychrometer measures the relative humidity in the atmosphere through the use of two thermometers. The first, a dry bulb thermometer, is used to measure the temperature by being exposed to the air. The second, a wet bulb thermometer, measures temperature by having the bulb dipped in a liquid. Through the comparison of both temperatures, individuals determine the relative humidity of the surrounding area by calculating the difference between the temperatures.

29

3 Climate Input 3.5 Wind Speed

Figure 3-9 Psychrometer

3.5 Wind Speed The estimation of the wind speed is required for the calculation of the evapotranspiration with the Penman-Monteith equation. SWAT estimates the variation of wind speed with elevation near the ground surface:

𝑢𝑧2

𝑧2 𝑎𝑎 = 𝑢𝑧1 ∗ ( ) 𝑧1

(3-34)

where 𝑢𝑧2 is the wind speed at height 𝑧2 [𝑚/𝑠] , 𝑢𝑧1 is the wind speed at height 𝑧1 [𝑚/𝑠] and 𝑎𝑎 is a factor which ranges from 0 to 1 and is set to 0.2.

𝑧𝑤 = ℎ𝑐 + 100

(3-35)

where 𝑧𝑤 is the height of the wind speed measurement [𝑐𝑚] and ℎ𝑐 is the canopy height [𝑐𝑚]. SWAT assumes that measurements were taken 1.7 m above the ground surface (DWD standard is 2.0 m). Moreover SWAT specifies a minimum difference of 1 m between measurement 30

3 Climate Input 3.6 Nomenclature

point 𝑧𝑤 and canopy height ℎ𝑐 . When canopy height exceeds 1m, SWAT adjusts the original measurement height ℎ𝑐 . The factor 𝑎𝑎 varies with atmospheric stability and surface roughness. Figure 3-10 shows a wind measurement device. It consists of hemispherical cups, each mounted on one end of horizontal arms, which in turn were mounted at equal angles to each other on a vertical shaft. The air flow past the cups in any horizontal direction turned the shaft in a manner that was proportional to the wind speed. Therefore, counting the turns of the shaft over a set time period produces the average wind speed for a wide range of speeds.

Figure 3-10 Wind sensor

3.6 Nomenclature 𝐴𝑠𝑢𝑟𝑓 𝐶𝑉

Amplitude of the soil fluctuations [°𝐶] Total aboveground biomass and residue on current day [𝑘𝑔⁄ℎ𝑎 ]

𝐸𝐿𝑏𝑎𝑛𝑑

Mean elevation in the elevation band [𝑚]

𝐸𝐿𝑔𝑎𝑔𝑒

Elevation at the recording gauge [𝑚]

𝐸𝑠𝑢𝑏

Amount of sublimation on a given day [𝑚𝑚 𝐻2 𝑂]

𝐻𝑑𝑎𝑦

Solar radiation reaching ground on the current day [𝑀𝐽⁄𝑚2 𝑑𝑎𝑦]

𝑅ℎ

Relative humidity [%]

𝑅𝑏𝑎𝑛𝑑

Precipitation falling in the elevation band [𝑚𝑚 𝐻2 𝑂]

𝑅𝑑𝑎𝑦

Amount of precipitation on a given day [𝑚𝑚 𝐻2 𝑂],

𝑅𝑑𝑎𝑦

Amount of precipitation recorded on a given day [𝑚𝑚 𝐻2 𝑂]

𝑆𝑁𝑂𝑚𝑙𝑡 𝑆𝑁𝑂 SNO100 𝑆𝑊

Amount of snow melt on a given day [𝑚𝑚 𝐻2 𝑂]. Water content of the snow pack on a given day [𝑚𝑚 𝐻2 𝑂] Threshold depth of snow at 100% coverage [𝑚𝑚 𝐻2 𝑂] Amount of water in the soil profile [𝑚𝑚𝐻2 𝑂] 31


𝑇ℎ𝑟

Air temperature for a certain hour of the day [°𝐶]

𝑇𝐴𝐴

Average annual soil temperature [°𝐶]

𝑇𝐴𝐴𝑖𝑟

Average annual air temperature [°𝐶]

Tav 𝑇𝑎𝑣,𝑏𝑎𝑛𝑑

Mean daily temperature in the elevation band [°𝐶]

𝑇𝑏𝑎𝑟𝑒

Temperature of the bare soil [°𝐶]

𝑇𝑚𝑙𝑡

Base temperature above which snow melt is allowed [°𝐶].

𝑇𝑚𝑛,𝑏𝑎𝑛𝑑 𝑇𝑚𝑛 𝑇𝑚𝑥,𝑏𝑎𝑛𝑑 𝑇𝑚𝑥 Tsnow(dn)

Minimum daily temperature in the elevation band [°𝐶] Minimum air temperature on a given day [°𝐶] Maximum daily temperature in the elevation band [°𝐶] Maximum air temperature on a given day [°𝐶] Snow pack temperature on a given day [°𝐶]

Tsnow(dn−1 )

Snow pack temperature on the previous day [°𝐶]

𝑇𝑠𝑜𝑖𝑙 (𝑧, 𝑑𝑛 )

Soil temperature [°𝐶] at depth 𝑧 and of the year 𝑑𝑛

𝑇𝑠𝑜𝑖𝑙 (𝑧, 𝑑𝑛 − 1)

Soil temperature in the layer from the previous day [°𝐶],

𝑇𝑠𝑠𝑢𝑟𝑓

Soil surface temperature[°𝐶]

𝑇𝑤𝑎𝑡𝑒𝑟

Water temperature [°𝐶]

𝑎𝑎

Exponent between 0 and 1 that varies with atmospheric stability and surface roughness that is used for calculating wind speed [−]

𝑏

Total number of elevation bands in the subbasin [−]

𝑏𝑚𝑙𝑡

Melt factor for the day [𝑚𝑚 𝐻2 𝑂⁄𝑑𝑎𝑦 °𝐶 ]

𝑏𝑚𝑙𝑡12

Melt factor for December 21 [𝑚𝑚 𝐻2 𝑂⁄𝑑𝑎𝑦 °𝐶 ]

𝑏𝑚𝑙𝑡6

Melt factor for June 21 [𝑚𝑚 𝐻2 𝑂⁄𝑑𝑎𝑦 °𝐶 ],

𝑏𝑐𝑣 cov1 , cov2 𝑑𝑛 𝑑𝑎𝑦𝑠𝑝𝑐𝑝,𝑦𝑟 𝑑𝑑 𝑑𝑑𝑚𝑎𝑥 𝑑𝑓

32

Mean air temperature on the current day [°𝐶]

Weighting factor for impact of ground cover on soil surface temperature [−] Coefficients determining the curves shape [−] Number of the day [𝐽𝑢𝑙𝑖𝑎𝑛 𝐷𝑎𝑦] Average number of days of precipitation in the subbasin in a year [−] Damping depth [𝑚𝑚] Maximum damping depth [𝑚𝑚] Depth factor quantifying influence of depth [−]

𝑒

Actual vapor pressure [𝑘𝑃𝑎]

𝑒0

Saturation vapor pressure[𝑘𝑃𝑎]


𝑓𝑟𝑏𝑛𝑑

Fraction of subbasin area within the elevation band [−]

ℎ𝑐

Canopy height [𝑐𝑚]

ℎ𝑟

Hour of the day [−]

𝑙 lsno

Lag coefficient (0-1), default = 0.8 [−] Snow temperature lag factor [−]

𝑝𝑙𝑎𝑝𝑠

Precipitation lapse rate [𝑚𝑚 𝐻2 𝑂/𝑘𝑚]

snocov

Fraction of HRU covered by snow [−]

𝑡𝑙𝑎𝑝𝑠

Temperature lapse rate [°𝐶/𝑘𝑚]

𝑢𝑧1

Wind speed at height 𝑧1 [𝑚/𝑠]

𝑢𝑧2

Wind speed at height 𝑧2 [𝑚/𝑠]

𝑣𝑝𝑑

Vapor pressure deficit [𝑘𝑃𝑎]

𝑧

Depth below soil surface [𝑚𝑚]

𝑧𝑡𝑜𝑡

Depth to the bottom of soil profile [𝑚𝑚].

𝑧𝑤

Height of the wind speed measurement[𝑐𝑚]

𝑧𝑑

Ratio in depth to dumping depth [−]

𝛼

Short wave reflectance or albedo for the day [−]

∆

Slope of the saturation vapor pressure curve [𝑘𝑃𝑎⁄°𝐶 ]

𝜀𝑆𝑅

Radiation term for bare soil surface temperature calculation [−]

𝜌𝑏

Soil bulk density [𝑀𝑔⁄𝑚3 ]

𝜑

Scaling factor for impact of soil water on damping depth [−]

𝜔𝑡𝑚𝑝

Angular frequency in soil temperature variation [−]

33

4 Evapotranspiration 3.6 Nomenclature

4 Evapotranspiration Evaporation is the conversion of liquid water into a gaseous state and its diffusion into the atmosphere. The evaporation rate is dependent on available energy from the sun and the amount of water present. These factors are determined by the region’s climate. The term evaporation accounts for water evaporating over soils and open water bodies whereas the term transpiration stands for water delivery of plants. Moreover water which is converted to vapor by sublimation is also subsumed. Since it is difficult to quantify the terms alone, the processes in total are called „Evapotranspiration“. The potential evapotranspiration (PE) is defined as water vapor amount which can be maximal transferred into the atmosphere from a wet land’s surface under given atmospheric conditions if the water supply were unrestricted. It is determined as a conceptual entity from measured data and expresses the meteorological control on evaporation. Two different definitions are available: According to Thornthwaite, 1948: The rate at which evapotranspiration would occur from a large area uniformly covered with growing vegetation that has access to an unlimited supply of soil water and that was not exposed to advection or heat storage effects. According to Penman, 1956: The amount of water which is transpired by a short crop, completely shading the ground with a uniform height and which never gets short of water.

There is no equilibrium between the solid and the gaseous state and it takes more energy to evaporate water than to vaporize it (compared to Figure 4-1).

Figure 4-1 Energy for evaporation

The energy for the phase transition is influenced by direct solar radiation, temperature, relative humidity and wind speed. The water vapor gradient is the driving force behind the process and expresses the difference of water vapor pressure of surface and atmosphere. Figure 4-2 illustrates the most important processes in evapotranspiration. 34


The evapotranspiration rate is normally expressed in mm per unit time, which can be hours, days or months. Roughly 62% of worldwide precipitation that falls over land is evaporated. Therefore it is the primary mechanism by which water is removed from the watershed.

Figure 4-2 Graphical representation of processes of evaporation

An exact quantification of evapotranspiration is crucial for the assessment of water resources, for the quantification of the impact of climate change on resources as well as for the quantification of the impact of land use change on resources. A direct estimation of the evapotranspiration is possible with a “Class A Pan”, as shown in Figure 4-3. It is essentially useful for farmers to determine how much water their crops will need. The measurement device combines temperature, humidity, rainfall, drought, dispersion, solar radiation and wind.

Figure 4-3 Measurement device “Class A Pan”

35

4 Evapotranspiration 4.1 Penman Monteith

For the calculation of the potential evapotranspiration different methods are available. The methods differ according to their parameter input.

4.1 Penman Monteith The Penman- Monteith equation is the most precise one, so it is advisable to use this method if the required input data are available. As an input the air temperature, the wind speed, the relative humidity and the solar radiation are needed.

𝜆𝐸 =

∆ ∗ (𝑅𝑛 − 𝐺) + 𝜌𝑎𝑖𝑟 ∗ 𝑐𝑃 ∗ 1+𝑟 ∆ + 𝛾 ∗ ( 𝑟 𝑐) 𝑎

(𝑒𝑧0 − 𝑒𝑧 ) 𝑟𝑎

[

𝑀𝐽 ] 𝑚²𝑑𝑎𝑦

(4-1)

Where 𝜆𝐸 is the latent heat flux density [𝑀𝐽⁄𝑚2 𝑑𝑎𝑦], 𝐸 is the depth rate evaporation [𝑚𝑚⁄𝑑𝑎𝑦], ∆ is the slope of the vapor pressure curve [𝑘𝑃𝑎⁄°𝐶 ], 𝑅𝑛 is net radiation at the crop surface [𝑀𝐽⁄𝑚2 𝑑𝑎𝑦], 𝐺 is soil heat flux density [𝑀𝐽⁄𝑚2 𝑑𝑎𝑦], 𝜌𝑎𝑖𝑟 is the air density in [𝑘𝑔⁄𝑚3 ], 𝑐𝑃 is the specific heat at constant pressure [𝑀𝐽⁄𝑘𝑔°𝐶 ] , 𝑒𝑧0 is the saturation vapor pressure of air at height z [𝑘𝑃𝑎], 𝛾 is the psychrometric constant [𝑘𝑃𝑎⁄°𝐶 ], 𝑟𝑎 is the aerodynamic resistance [𝑠⁄𝑚], and 𝑟𝑐 is the plant canopy resistance [𝑠⁄𝑚].

1) Firstly, the required physical constants are listed: o

σ = 5.6704 ∗ 10−8 [W⁄m2 K 4 ]

Stefan- Boltzmann constant

o

𝜌𝑎𝑖𝑟 = 1.293 [𝑘𝑔⁄𝑚3 ]

Air density

o

𝑐𝑃 = 1.013 ∗ 10−3 [𝑀𝐽⁄𝑘𝑔 °𝐶 ]

Specific heat at constant pressure

o

GSC = 4.921 [𝑚2 ℎ] = 0.082 [𝑚2 𝑚𝑖𝑛]

𝑀𝐽

𝑀𝐽

Solar constant

The air density 𝜌𝑎𝑖𝑟 could vary with the altitude and for more precise calculations a function for 𝜌𝑎𝑖𝑟 is needed.

36


Figure 4-4 Energy budget of the earth

Figure 4-4 illustrates the earth’s energy budget. The incoming radiation is also termed solar radiation: This is the short-wave radiation and ranges from 0.3 to 5 micrometers. Some parts are reflected by the surface and clouds and some parts are absorbed by water vapor, gases and the earth itself. For surfaces, the albedo is a steering parameter for reflection and absorption. The outgoing radiation is termed net longwave radiation. The emission depends on the emissive power of the body 𝜀 and according to the Stefan Boltzmann law also on the temperature of the emitting body T4. Outgoing longwave radiation can again be scattered back by the atmosphere and / or by clouds. Especially solar radiation exerts a major control of the movement of water since solar radiation determines the temperature. Both quantities consequently influence snow fall, snow melt and evaporation. 2) Sun- earth relationships Additionally it is obligatory for deriving the vectors for the sun-earth relationships, which have an annual distribution. Because of the daily average approach the Julian Day JD is needed for the calculations. It is a vector that reaches from 1 to 365 or 366, in case it’s a leap year, and represents the number of the day in the year. It gives information about annual distribution when calculating quantities. The solar radiation is inversely proportional to the square of its distance to the sun. The distance varies during the year since the earth‘s orbit is elliptical (compare Figure 4-5). The mean distance is called an Astronomic Unit (AU).

37


o the inverse distance of the sun:

𝑑𝑖𝑠𝑡 = 1 + 0.033 cos [

2𝜋 𝐽𝐷] [−] 365

(4-2)

Figure 4-5 Distance between sun and earth

o the solar declination:

d𝑒𝑐𝑙 = 0.40954 ∗ 𝑠𝑖𝑛[0.0172 ∗ (𝐽𝐷 − 79.35)] [𝑟𝑎𝑑]

(4-3)

The solar declination is the earth‘s latitude at which incoming solar rays are normal to the earth‘s surface and it also varies throughout the year (compare Figure 4-6).

38


Figure 4-6 Declination of the earth

To fully describe the position of a point at the celestial sphere for a given time, the hour angle is paired with the declination. The latitude of the area of interest must be known. o the sunset hour angle:

𝑤𝑠 = arccos [−tan(lat) tan(decl)] [𝑟𝑎𝑑]

(4-4)

3) The water vapor characteristics describe one part of the Penman- Monteith equation. Therefore the mean saturation vapour pressure, the actual water vapor pressure and the slope of the water vapour pressure have to be determined. o

mean saturation vapour pressure:

𝑒𝑧0 = 𝑒 (

o

16.78⋅𝑇−116.9 ) 𝑇+237.3

[𝑘𝑃𝑎]

(4-5)

the actual water vapour pressure:

𝑒𝑎 = 𝑒𝑧0 ⋅ 𝑅𝐻𝑚𝑒𝑎𝑛 [𝑘𝑃𝑎]

(4-6) 39


o

the slope of the water vapour pressure:

4098 ⋅ 𝑒𝑧0 𝑘𝑃𝑎 ∆= [ ] 2 (𝑇 + 237.3) °𝐶

(4-7)

4) The Radiation Balance is the most complex part of the Penman- Monteith equation. The net radiation provides energy for the evaporation:

𝑅𝑛 = (1 − 𝛼) ⋅ 𝑅𝑔 − 𝑅𝑙

(4-8)

where 𝑅𝑛 is the net radiation[𝑀𝐽/𝑚2 𝑑𝑎𝑦], 𝑅𝑔 is the global radiation [𝑀𝐽/𝑚2 𝑑𝑎𝑦], 𝑅𝑙 is the longwave emission [𝑀𝐽/𝑚2 𝑑𝑎𝑦] and 𝛼 is the albedo [−] determined as follows:

𝛼=

𝜃𝑑 𝜃𝑂

(4-9)

where 𝜃𝑑 is the reflected radiation and 𝜃𝑂 is the incoming solar radiation. Figure 4-7 shows some example values for the albedo 𝛼. Surface

Albedo 𝜶 [%]

Clouds

60-90

Snow

75-95

Old Snow

40-70

Glacier Ice

30-45

Sandy Soils

15-40

Farmland

7-17

Deciduous Forest

15-20

Evergreen Forest

5-12

Meadows

12-30

Concrete

14-22

Deep Water

3-10

Deep Water (high inclination angle)

80

Figure 4-7 Albedo values

40


Figure 4-8 shows a pyranometer, a measurement device to estimate global solar radiation 𝑅𝑔 on a planar surface. It is comprised of direct solar radiation and diffuse radiation. Both types of measured radiation are short- wave (ca. 290-4000nm) and the unit of measurement is 𝑊/𝑚2. The direct radiation casts shadows whereas the diffuse radiation creates sky light. The response varies with the cosine of the angle of incidence (zenith, 0°, full response whereas horizon, 90°, no response). A pyranometer is usually used at climate stations.

Figure 4-8 Kipp & Zonen Pyranometer; Rolf Gegenbach Messtechnik

In the following, the further steps to calculate the long wave emission 𝑅𝑙 are shown. On any given day, the extraterrestrial irradiance (incoming rate of energy) on a surface normal to the rays of the sun, 𝑅𝑎𝑛 , is: o extraterrestrial irradiance: 𝑅𝑎𝑛 = 𝐺𝑆𝐶 ⋅ 𝑑𝑖𝑠𝑡 [

𝑀𝐽 ] 𝑚² ⋅ 𝑑

(4-10)

The extraterrestrial radiation (clear sky) depends on the sunset hour angle 𝜛𝑠, the solar declination 𝑑𝑒𝑐𝑙, and the latitude 𝑙𝑎𝑡. o extraterrestric radiation:

𝑅𝑎 =

24 ∗ 60 ∗ 𝐺𝑆𝐶 ∗ 𝑑𝑖𝑠𝑡 𝜋 ∗ [(𝑤𝑠 sin(lat) sin(decl)) + (cos( lat) cos( decl) sin( 𝑤𝑠 ))] [

(4-11) 𝑚2

𝑀𝐽 ] ∗ 𝑑𝑎𝑦

Now it is possible to calculate the maximum global radiation for clear sky conditions: 41


o maximum global radiation:

𝑅𝑔𝑜 = (0.75 + 2 ⋅ 10−5 ⋅ 𝑒𝑙𝑒𝑣) ⋅ 𝑅𝑎

(4-12)

Together with the measured Rg and Rgo in combination with the actual vapor pressure and Stefan Boltzmann law (𝜎 ∗ 𝑇 4 ), the effective longwave emission can be estimated: o longwave emission:

𝑅𝑙 = σ ⋅ T 4 (0.34 − 0.14 √𝑒𝑎 ) [1.35

𝑅𝑔 𝑀𝐽 − 0.35] [ 2 ] 𝑅𝑔𝑜 𝑚 𝑑𝑎𝑦

(4-13)

5) The soil heat storage can be significant over a few hours, but is usually small from day to day because heat stored as the soil warms early in the day is lost when the soil cools late in the day or at night. Since the magnitude of daily soil heat flux over a 10- to 30- day period is small when the soil is under a crop cover, it can normally be ignored for most balance estimates. The soil heat flux is ignored by SWAT:

𝐺=0 [

𝑀𝐽 ] 𝑚2 𝑑𝑎𝑦

(4-14)

6) The psychometric constant 𝜸 represents a balance between the sensible heat gained from air flowing past a wet bulb thermometer and the sensible heat converted to latent heat.

𝛾=

𝑐𝑃 ⋅ 𝑃 [−] 0.622 ⋅ 𝜆

(4-15)

with latent heat of vaporization 𝜆:

𝜆 = 2.501 − 2.361 ⋅ 10

−3

⋅ 𝑇𝑎𝑣 [

𝑀𝐽 ] 𝑘𝑔

(4-16)

and atmospheric pressure 𝑃:

𝑃 = 101.3 − 0.01152 ⋅ 𝑒𝑙𝑒𝑣 + 0.544 ⋅ 10−6 ⋅ 𝑒𝑙𝑒𝑣 2 [𝑘𝑃𝑎]

(4-17)

7) The aerodynamic resistance 𝒓𝒂 determines the transfer of heat and water vapor from the evaporating surface to the atmosphere. Hence the wind regimes are influenced in plant cano-

42


pies. The aerodynamic resistance is dependent on the plant type and therefore also for displacement heights [𝑚] and roughness length [𝑚].

The derivation starts with the logarithmic wind profile after Prandtl:

𝑢(𝑧) =

𝑢∗ 𝑧 ⋅ 𝑙𝑛 ( ) 𝐾 𝑧𝑜𝑚

(4-18)

solved for the shear velocity 𝑢∗ :

𝑢∗ =

𝑢(𝑧) ∗ 𝐾 ln(𝑧) − ln(𝑧𝑜𝑚 )

(4-19)

and including a displacement height:

𝑢(𝑧) =

𝑢∗ 𝑧−𝑑 ∗ 𝑙𝑛 ( ) 𝐾 𝑧𝑜𝑚

(4-20)

Aerodynamic resistance is then calculated as the inverse of the shear velocity:

1 𝑟𝑎 = = 𝑢∗

𝑙𝑛(𝑧𝑚 − 𝑑) 𝑧 −𝑑 ∗ 𝑙𝑛 ( ℎ𝑧 ) 𝑧𝑜𝑚 𝑜𝑣 𝐾 2 ∗ 𝑢𝑧

(4-21)

where 𝑧𝑚 is the height of wind measurement [𝑚], 𝑑 is the zero plane displacement height [𝑚], 𝑧ℎ is the height of humidity measurement [𝑚], 𝑧𝑜𝑚 is the roughness length parameter (momentum transfer) [𝑚], 𝑧𝑜𝑣 is the roughness length parameter(heat and vapor) [𝑚] and 𝐾 is the vonKarmans constant. The surface roughness parameter is related to the mean height (ℎ𝐶 ) of the plant canopy by this relationship: Surface roughness parameter for air:

𝑧𝑜𝑚 = 0.123 ∗ ℎ𝑐 𝑧𝑜𝑚 = 0.058 ∗

ℎ1.19 𝑐

≤ 200𝑐𝑚 (4-22) ≥ 200𝑐𝑚

43


Surface roughness parameter for vapor transfer:

𝑧𝑜𝑣 = 0.1 ∗ 𝑧𝑜𝑚

(4-23)

Figure 4-9 represents the wind profile without (a) and with displacement height (b).

Figure 4-9 Wind profile

8) The plant canopy resistance 𝒓𝒄 is determined with the following equation:

𝑟𝑐 =

𝑟𝑙 0.5 ∗ 𝐿𝐴𝐼

(4-24)

where 𝑟𝑐 is the canopy resistance (𝑚 𝑠 −1 ), 𝑟𝑙 is the minimum effective stomatal resistance of a single leaf (𝑚 𝑠 −1 ) and 𝐿𝐴𝐼 is the leaf area index of the canopy. SWAT uses tabulated values for leaf conductance, as an inverse of the leaf resistance 𝑟𝑙 :

𝑔𝑙 =

1 𝑟𝑙

(4-25)

where 𝑔𝑙 is the maximum effective lead conductance in (𝑚 𝑠 −1 ). When the canopy resistnace is expressed as a function of leaf conductance instead of leaf resistance, then the equation becomes:

𝑟𝑐 =

44

1 0.5 ∗ 𝑔𝑙 ∗ 𝐿𝐴𝐼

(4-26)

4 Evapotranspiration 4.2 Priestley Taylor

where 𝑟𝑐 is the canopy resistance (𝑚 𝑠 −1 ), 𝑔𝑙 is the maximum effective lead conductance in (𝑚 𝑠 −1 ) and 𝐿𝐴𝐼 is the leaf area index of the canopy. LAI values depending on the development stage of the plant. The stomatal resistance is determined as follows:

𝑟𝑠 =

𝑟𝑙

(4-27)

𝐿𝐴𝐼𝑎𝑐𝑡𝑖𝑣𝑒

where 𝑟𝑠 is the stomatal resistance (𝑚 𝑠 −1 ), , 𝑟𝑙 is the minimum effective stomatal resistance of a single leaf (𝑚 𝑠 −1 ) and 𝐿𝐴𝐼𝑎𝑐𝑡𝑖𝑣𝑒 is the active leaf area index:

𝐿𝐴𝐼𝑎𝑐𝑡𝑖𝑣𝑒 = 𝐿𝐴𝐼 ∗ 0.5

(4-28)

4.2 Priestley Taylor The Priestley–Taylor equation is a simplified form of the Penman Monteith equation, where the aerodynamic component is left out of the equation. It is especially usable for humid conditions and wet surfaces. It provides estimations for potential evapotranspiration values 𝐸0 for low advective conditions. On the other hand it underestimates potential evapotranspiration 𝐸0 in semiarid/ arid regions, when advection is significant.

𝜆𝐸0 = 𝛼𝑝𝑒𝑡 ⋅

∆ 𝑀𝐽 ⋅ (𝐻𝑛𝑒𝑡 − 𝐺) [ ] ∆+ 𝛾 𝑚²𝑑𝑎𝑦

(4-29)

where 𝜆 is the latent heat of vaporization [𝑀𝐽⁄𝑘𝑔], 𝐸0 is the potential evapotranspiration [𝑚𝑚⁄𝑑𝑎𝑦], 𝛼𝑝𝑒𝑡 is a coefficient, ∆ is the slope of the vapor pressure curve [𝑘𝑃𝑎/°𝐶] 𝐻𝑛 is net radiation at the crop surface [𝑀𝐽⁄𝑚2 𝑑𝑎𝑦], 𝐺 is the soil heat flux density to the ground [𝑀𝐽⁄𝑚2 𝑑𝑎𝑦], 𝛾 is the psychrometric constant [𝑘𝑃𝑎/°𝐶].

4.3 Hargreaves The Hargreaves–Samani method is not truly a temperature-based method because it has a radiation term in it. Just because there is no measurement needed for the extraterrestrial radiation (RA), this method may be classified as a temperature-based method. The Hargreaves method is originally an equation developed from empirical lysimeter measurements and the equation was iteratively refined. The Hargreaves equation is given by: 𝑀𝐽

𝜆𝐸0 = 0.0023 ⋅ 𝑅𝑎 ⋅ (𝑇𝑚𝑎𝑥 − 𝑇𝑚𝑖𝑛 )0.5 ⋅ (𝑇𝑎𝑣 + 17.8) [𝑚²𝑑𝑎𝑦]

(4-30)

45

4 Evapotranspiration 4.4 Actual Evapotranspiration in SWAT

where 𝜆 is the latent heat of vaporization [𝑀𝐽/𝑘𝑔], 𝐸0 is the potential evapotranspiration [𝑚𝑚/𝑑𝑎𝑦], 𝑅𝑎 is extraterrestrial radiation [𝑀𝐽/𝑚2 𝑑𝑎𝑦] and 𝑇 is the temperature.

4.4 Actual Evapotranspiration in SWAT Once total potential evapotranspiration is determined, actual evaporation must be calculated. SWAT firstly evaporates any rainfall intercepted by the plant canopy. Next, SWAT calculates the maximum amount of transpiration and following the maximum amount of sublimation and soil evaporation. If snow is present in the HRU, sublimation will occur and only then evaporation from the soil will take place.

4.4.1 Evaporation of intercepted rainfall Any free water present in the canopy is readily available for removal by evapotranspiration. The amount of actual evapotranspiration contributed by intercepted rainfall is especially significant in forests where in some instances evaporation of intercepted rainfall is greater than transpiration. SWAT removes as much water as possible from canopy storage when calculating actual evaporation. If potential evapotranspiration, 𝐸0 , is less than the amount of free water held in the canopy, 𝑅𝐼𝑁𝑇(𝑓) ,then:

𝐸𝑎 = 𝐸𝑐𝑎𝑛 = 𝐸0

(4-31)

𝑅𝐼𝑁𝑇(𝑓) = 𝑅𝐼𝑁𝑇(𝑖) − 𝐸𝑐𝑎𝑛

(4-32)

where 𝐸𝑎 is the actual amount of evapotranspiration occurring in the watershed on a given day [𝑚𝑚𝐻2 𝑂], 𝐸𝑐𝑎𝑛 is the amount of evaporation from free water in the canopy on a given day [𝑚𝑚𝐻2 𝑂], 𝐸0 is the potential evapotranspiration on a given day [𝑚𝑚𝐻2 𝑂], 𝑅𝐼𝑁𝑇(𝑖) is the initial amount of free water held in the canopy on a given day [𝑚𝑚𝐻2 𝑂] and 𝑅𝐼𝑁𝑇(𝑓) is the final amount of free water held in the canopy on a given day [𝑚𝑚𝐻2 𝑂]. If potential evapotranspiration, 𝐸0 , is greater than the amount of free water held in the canopy, 𝑅𝐼𝑁𝑇 , then:

𝐸𝑐𝑎𝑛 = 𝑅𝐼𝑁𝑇(𝑖)

(4-33)

𝑅𝐼𝑁𝑇(𝑓) = 0

(4-34)

Once any free water in the canopy has been evaporated, the remaining evaporative water demand (𝐸0′ = 𝐸0 − 𝐸𝑐𝑎𝑛 ) is partitioned between vegetation and snow or soil.

4.4.2 Transpiration If the Penman- Monteith equation is selected as the potential evapotranspiration method, transpiration is also calculated. All other methods calculate the transpiration as follows:

46


𝐸𝑡 =

𝐸0′ ⋅ 𝐿𝐴𝐼 3.0

𝐸𝑡 = 𝐸0′

𝑖𝑓 0 ≤ 𝐿𝐴𝐼 ≤ 3.0 𝑖𝑓 𝐿𝐴𝐼 > 3.0

(4-35)

(4-36)

where 𝐸𝑡 is the maximum transpiration on a given day [𝑚𝑚𝐻2 𝑂], 𝐸0′ is the potential evapotranspiration adjusted for evaporation of free water in the canopy [𝑚𝑚𝐻2 𝑂] and 𝐿𝐴𝐼 is the leaf area index. The value for transpiration is the amount of transpiration that will occur on a given day when the plant is growing under ideal conditions. This implies that there is no water stress. The actual amount of transpiration may be less than this due to lack of available water in the soil profile. Figure 4-10 shows LAI values for different vegetation types.

Semi-arid shrubland; LAI= 0.6

Overgrazed tropical woodland; LAI=1.5

Temperate shrubland ; LAI=3.0

Tall prairie grass; LAI=2.5

Forest; LAI= 5.0

.0 Figure 4-10 LAI values for different vegetation types

4.4.3 Sublimation and evaporation from the soil The amount of sublimation and soil evaporation will be impacted by the degree of shading. The maximum amount of sublimation or soil evaporation on a given day is calculated as:

𝐸𝑆 = 𝐸0′ ∗ 𝑐𝑜𝑣𝑠𝑜𝑙

(4-37)

47


where 𝐸𝑆 is the maximum sublimation or soil evaporation on a given day [𝑚𝑚𝐻2 𝑂], 𝐸0′ is the potential evapotranspiration adjusted for evaporation of free water in the canopy [𝑚𝑚𝐻2 𝑂] and 𝑐𝑜𝑣𝑠𝑜𝑙 is the soil cover index. The soil cover index is calculated as:

𝑐𝑜𝑣𝑠𝑜𝑙 = 𝑒𝑥𝑝(−5.0 ∗ 10−5 ∗ 𝐶𝑉)

(4-38)

where 𝐶𝑉 is the aboveground biomass and residue [𝑘𝑔 ℎ𝑎−1 ]. It is modeled throughout the year using the plant growth model in SWAT. If the snow water content is greater than 0.5 mm H 2O, the soil cover index is set to 0.5. The maximum amount of sublimation or soil evaporation is reduced during periods of high plant water use with the relationship:

𝐸𝑆′ = 𝑚𝑖𝑛 [𝐸𝑆 ,

𝐸𝑆 ⋅ 𝐸0′ ] 𝐸𝑆 + 𝐸𝑡

(4-39)

where 𝐸𝑆′ is the the maximum sublimation or soil evaporation adjusted for plant water use on a given day [𝑚𝑚𝐻2 𝑂], 𝐸𝑆 is the maximum sublimation or soil evaporation on a given day [𝑚𝑚 𝐻2 𝑂], 𝐸0′ is the potential evapotranspiration adjusted for evaporation of free water in the canopy [𝑚𝑚 𝐻2 𝑂] and 𝐸𝑡 is the transpiration on a given day [𝑚𝑚 𝐻2 𝑂]. When 𝐸𝑡 is low 𝐸𝑆′ 𝐸𝑆 . 𝐸𝑆 However, as 𝐸𝑡 approaches 𝐸0′ , 𝐸𝑆′  1+𝑐𝑜𝑣 𝑠𝑜𝑙

Sublimation Once the maximum amount of sublimation or soil evaporation for the day is calculated, SWAT will first remove water from the snow pack to meet evaporative demand. If the water content of the snow pack is greater than the maximum sublimation or soil evaporation demand, then:

𝐸𝑠𝑢𝑏 = 𝐸𝑆′

(4-40)

𝑆𝑁𝑂(𝑓) = 𝑆𝑁𝑂(𝑖) − 𝐸𝑆′

(4-41)

𝐸𝑆′′ = 0

(4-42)

where 𝐸𝑠𝑢𝑏 is the amount of sublimation on a given day [𝑚𝑚𝐻2 𝑂], 𝐸𝑆′ is the maximum sublimation or soil evaporation adjusted for plant water use on a given day [𝑚𝑚𝐻2 𝑂], 𝑆𝑁𝑂(𝑖) is the amount of water in the snow pack on a given day prior to accounting for sublimation [𝑚𝑚𝐻2 𝑂], 𝑆𝑁𝑂(𝑓) is the amount of water in the snow pack on a given day after accounting for sublimation [𝑚𝑚𝐻2 𝑂] and 𝐸𝑆′′ is the maximum soil water evaporation on a given day [𝑚𝑚𝐻2 𝑂]. If the water content of the snow pack is less than the maximum sublimation or soil evaporation demand, then:

48


𝐸𝑠𝑢𝑏 = 𝑆𝑁𝑂(𝑖)

(4-43)

𝑆𝑁𝑂(𝑓) = 0

(4-44)

𝐸𝑆′′ = 𝐸𝑆′ − 𝐸𝑠𝑢𝑏

(4-45)

Soil water evaporation When an evaporation demand for soil water exists, SWAT must first partition the evaporative demand between the different layers. The depth distribution used to determine the maximum amount of water allowed to be evaporated is:

𝐸𝑠𝑜𝑖𝑙,𝑧 = 𝐸𝑆′′ ∗

𝑧 𝑧 + 𝑒𝑥𝑝(2.374 − 0.00713 ∗ 𝑧)

(4-46)

where 𝐸𝑠𝑜𝑖𝑙,𝑧 is the evaporative demand at depth 𝑧 [𝑚𝑚𝐻2 𝑂], 𝐸𝑆′′ is the maximum soil water evaporation on a given day [𝑚𝑚𝐻2 𝑂] and 𝑧 is the depth below the surface [𝑚𝑚]. The coefficients in this equation were selected so that 50% of the evaporative demand is extracted from the top 10mm of soil and 95% of the evaporative demand is extracted from the top 100mm of soil. The amount of evaporative demand for a soil layer is determined by taking the difference between the evaporative demands calculated at the upper and lower boundaries of the soil layer:

𝐸𝑠𝑜𝑖𝑙,𝑙𝑦 = 𝐸𝑠𝑜𝑖𝑙,𝑧𝑙 − 𝐸𝑠𝑜𝑖𝑙,𝑧𝑢

(4-47)

where 𝐸𝑠𝑜𝑖𝑙,𝑙𝑦 is the evaporative demand for layer 𝑙𝑦 [𝑚𝑚𝐻2 𝑂], 𝐸𝑠𝑜𝑖𝑙,𝑧𝑙 is the evaporative demand at the lower boundary of the soil layer [𝑚𝑚𝐻2 𝑂] and 𝐸𝑠𝑜𝑖𝑙,𝑧𝑢 is the evaporative demand at the upper boundary of the soil layer [𝑚𝑚 𝐻2 𝑂]. Figure 4-11 graphs the depth distribution of the evaporative demand for a soil that has been partitioned into 1mm layers assuming a total soil evaporation demand of 100mm. As already mentioned, the depth distribution assumes 50% of the evaporative demand is met by soil water stored in the top 10mm of the soil profile. With our example of a 100mm total evaporative demand, 50mm of water is 50%. This is a demand that the top layer cannot satisfy. SWAT does not allow a different layer to compensate for the inability of another layer to meet its evaporative demand. The evaporative demand not met by a soil layer results in a reduction in actual evapotranspiration for the HRU.

49


Figure 4-11 Soil evaporative demand distribution with depth

SWAT uses a parameter to allow the user to modify the depth distribution used to meet evaporative demand. The modified equation is:

𝐸𝑠𝑜𝑖𝑙,𝑙𝑦 = 𝐸𝑠𝑜𝑖𝑙,𝑧𝑙 − 𝐸𝑠𝑜𝑖𝑙,𝑧𝑢 ⋅ 𝑒𝑠𝑐𝑜

(4-48)

where 𝐸𝑠𝑜𝑖𝑙,𝑙𝑦 is the evaporative demand for layer 𝑙𝑦 [𝑚𝑚 𝐻2 𝑂], 𝐸𝑠𝑜𝑖𝑙,𝑧𝑙 is the evaporative demand at the lower boundary of the soil layer [𝑚𝑚 𝐻2 𝑂], 𝐸𝑠𝑜𝑖𝑙,𝑧𝑢 is the evaporative demand at the upper boundary of the soil layer [𝑚𝑚 𝐻2 𝑂] and 𝑒𝑠𝑐𝑜 is the soil evaporation compensation coefficient. Solutions to this equation for different values of 𝑒𝑠𝑐𝑜 are graphed in Figure 4-12. As the value for 𝑒𝑠𝑐𝑜 is reduced, the model is able to extract more water from lower levels to meet evaporative demand.

50


Figure 4-12 Soil evaporative demand distribution with depth (esco factor)

When the water content of a soil layer is below field capacity, the evaporative demand for the layer is reduced according to the following equations:

𝐸𝑠𝑜𝑖𝑙,𝑙𝑦 ′ = 𝐸𝑠𝑜𝑖𝑙,𝑙𝑦

𝑤ℎ𝑒𝑛 𝑆𝑊𝑙𝑦 < 𝐹𝐶𝑙𝑦

(4-49)

2.5 ⋅ (𝑆𝑊𝑙𝑦 − 𝐹𝐶𝑙𝑦 ) 𝐸𝑠𝑜𝑖𝑙,𝑙𝑦 ′ = 𝐸𝑠𝑜𝑖𝑙,𝑙𝑦 ⋅ exp ( ) 𝑤ℎ𝑒𝑛 𝑆𝑊𝑙𝑦 > 𝐹𝐶𝑙𝑦 𝐹𝐶𝑙𝑦 − 𝑊𝑃𝑙𝑦 where 𝐸𝑠𝑜𝑖𝑙,𝑙𝑦 ′ is the evaporative demand of layer 𝑙𝑦 [𝑚𝑚 𝐻2 𝑂] adjusted for the water content, 𝐹𝐶𝑙𝑦 is the water content of layer 𝑙𝑦 at field capacity [𝑚𝑚 𝐻2 𝑂] and 𝑊𝑃𝑙𝑦 is the water content of layer 𝑙𝑦 at wilting point [𝑚𝑚 𝐻2 𝑂]. In addition to limiting the amount of water removed by evaporation in dry conditions, SWAT defines a maximum value of water that can be removed at any time. This maximum value is 80% of the plant available water on a given day where the plant available water is defined as the total water content of the soil layer minus the water content of the soil layer at wilting point (-1.5 𝑀𝑃𝑎).

𝐸𝑠𝑜𝑖𝑙,𝑙𝑦 ′′ = min(𝐸𝑠𝑜𝑖𝑙,𝑙𝑦 ′

0.8 ⋅ (𝑆𝑊𝑙𝑦 − 𝑊𝑃𝑙𝑦 ))

(4-50)

51

4 Evapotranspiration 4.5 Further Reading

where 𝐸𝑠𝑜𝑖𝑙,𝑙𝑦 ′′ is the amount of water removed from layer 𝑙𝑦 by evaporation [𝑚𝑚 𝐻2 𝑂], 𝐸𝑠𝑜𝑖𝑙,𝑙𝑦 ′ is the evaporative demand for layer 𝑙𝑦 adjusted for water content [𝑚𝑚 𝐻2 𝑂], 𝑆𝑊𝑙𝑦 is the soil water content of layer 𝑙𝑦 [𝑚𝑚 𝐻2 𝑂] and 𝑊𝑃𝑙𝑦 is the water content of layer 𝑙𝑦 at wilting point [𝑚𝑚 𝐻2 𝑂].

4.5 Further Reading Read more about evapotranspiration in the FAO-56 manual (Allen et al., 1998). There are more details about the methods for calculating inputs to the Penman Monteith method: http://www.fao.org/docrep/x0490e/x0490e00.htm

4.6 Nomenclature 𝐶𝑉

Aboveground biomass and residue [𝑘𝑔 ℎ𝑎−1 ]

𝐸

Depth rate evaporation [𝑚𝑚⁄𝑑𝑎𝑦]

𝐸𝑠𝑜𝑖𝑙,𝑙𝑦 ′ 𝐸𝑠𝑜𝑖𝑙,𝑙𝑦

′′

Amount of water removed from layer 𝑙𝑦 by evaporation [𝑚𝑚 𝐻2 𝑂]

𝐸0

Potential evapotranspiration [𝑚𝑚⁄𝑑𝑎𝑦]

𝐸0′

Potential evapotranspiration adjusted for evaporation of free water in the canopy [𝑚𝑚 𝐻2 𝑂]

𝐸𝑆

Maximum sublimation or soil evaporation on a given day [𝑚𝑚𝐻2 𝑂]

𝐸𝑆′

Maximum sublimation or soil evaporation adjusted for plant water use on a given day [𝑚𝑚𝐻2 𝑂]

𝐸𝑆′′

Maximum soil water evaporation on a given day [𝑚𝑚𝐻2 𝑂]

𝐸𝑎

Actual amount of evapotranspiration on a given day [𝑚𝑚𝐻2 𝑂]

𝐸𝑐𝑎𝑛

Amount of evaporation from free water in the canopy on a given day [𝑚𝑚𝐻2 𝑂]

𝐸𝑠𝑜𝑖𝑙,𝑙𝑦

Evaporative demand for layer 𝑙𝑦 [𝑚𝑚𝐻2 𝑂]

𝐸𝑠𝑜𝑖𝑙,𝑧

Evaporative demand at depth 𝑧 [𝑚𝑚𝐻2 𝑂]

𝐸𝑠𝑜𝑖𝑙,𝑧𝑙

Evaporative demand at the lower boundary of the soil layer [𝑚𝑚𝐻2 𝑂]

𝐸𝑠𝑜𝑖𝑙,𝑧𝑢

Evaporative demand at the upper boundary of the soil layer [𝑚𝑚 𝐻2 𝑂]

𝐸𝑠𝑢𝑏

Amount of sublimation on a given day [𝑚𝑚𝐻2 𝑂]

𝐸𝑡

Maximum transpiration on a given day [𝑚𝑚𝐻2 𝑂]

𝐹𝐶𝑙𝑦

52

Evaporative demand for layer adjusted for water content 𝑙𝑦 [𝑚𝑚𝐻2 𝑂]

Water content of layer 𝑙𝑦 at field capacity [𝑚𝑚 𝐻2 𝑂]

𝐺

Soil heat flux density [𝑀𝐽⁄𝑚2 𝑑𝑎𝑦]

𝐻0

Extraterrestrial daily irradiation [𝑀𝐽/𝑚2 𝑑𝑎𝑦]

𝐻𝑛

Net radiation at the crop surface [𝑀𝐽⁄𝑚2 𝑑𝑎𝑦]


𝐾 𝐿𝐴𝐼 𝐿𝐴𝐼𝑎𝑐𝑡𝑖𝑣𝑒 𝑃

Von- Karmans constant [−] Leaf area index of the canopy [−] Active leaf area index [−] Atmospheric pressure [𝑘𝑃𝑎]

𝑅𝐼𝑁𝑇(𝑓)

Final amount of free water held in the canopy on a given day [𝑚𝑚𝐻2 𝑂]

𝑅𝐼𝑁𝑇(𝑖)

Initial amount of free water held in the canopy on a given day [𝑚𝑚𝐻2 𝑂]

𝑅𝑎

𝑀𝐽

Extraterrestrial radiation [𝑚2 ∗𝑑𝑎𝑦]

𝑅𝑔

Global radiation [𝑀𝐽/𝑚2 𝑑𝑎𝑦],

𝑅𝑔𝑜

Maximum global radiation [𝑚∗𝑑𝑎𝑦]

𝑅𝑙

Longwave emission [𝑀𝐽/𝑚2 𝑑𝑎𝑦]

𝑅𝑛

Net radiation at the crop surface [𝑀𝐽⁄𝑚2 𝑑𝑎𝑦]

𝑆𝑐

Solar constant [

𝑀𝐽

𝑀𝐽 𝑚2 ⋅𝑚𝑖𝑛

]

𝑆𝑁𝑂(𝑓)

amount of water in the snow pack on a given day after accounting for sublimation [𝑚𝑚𝐻2 𝑂]

𝑆𝑁𝑂(𝑖)

Amount of water in the snow pack on a given day prior to accounting for sublimation [𝑚𝑚𝐻2 𝑂]

𝑆𝑊𝑙𝑦

Soil water content of layer 𝑙𝑦 [𝑚𝑚 𝐻2 𝑂]

𝑊𝑃𝑙𝑦

Water content of layer 𝑙𝑦 at wilting point [𝑚𝑚 𝐻2 𝑂]

𝑐𝑃

Specific heat at constant pressure [𝑀𝐽⁄𝑘𝑔°𝐶 ]

𝑐𝑃

Specific heat at constant pressure [ 1.013 ∗ 10−3 𝑀𝐽⁄𝑘𝑔 °𝐶 ]

𝑐𝑜𝑣𝑠𝑜𝑙 𝑑

Soil cover index [−] Zero plane displacement height [𝑚]

𝑑𝑖𝑠𝑡

Inverse distance of the sun

𝑑𝑒𝑐𝑙

Solar declination [𝑟𝑎𝑑]

𝑒𝑧

Actual water vapour pressure [𝑘𝑃𝑎]

𝑒𝑧0

Saturation vapor pressure of air at height z [𝑘𝑃𝑎]

𝑒𝑧0

Mean saturation vapour pressure [𝑘𝑃𝑎]

𝑒𝑠𝑐𝑜

Soil evaporation compensation coefficient [−]

𝑔𝑙

Maximum effective lead conductance in (𝑚 𝑠 −1 )

𝑙𝑎𝑡

Latitude [𝑟𝑎𝑑]

ℎ𝑐

Mean height of the plant canopy [𝑚]

𝑟𝑎

Aerodynamic resistance [𝑠⁄𝑚]

53


𝑟𝑐

Plant canopy resistance [𝑠⁄𝑚]

𝑟𝑙

Minimum effective stomatal resistance of a single leaf (𝑚 𝑠 −1 )

𝑟𝑠

Stomatal resistance (𝑚 𝑠 −1 )

𝑢∗

Shear velocity [ ] 𝑠

𝑢(𝑧)

Displacement height [𝑚]

𝑤𝑠

Sunset hour angle [𝑟𝑎𝑑]

𝑧

depth below the surface 𝑧 [𝑚𝑚]

𝑧ℎ

Height of humidity measurement [𝑚],

𝑧𝑚

Height of wind measurement [𝑚]

𝑧𝑜𝑚

Roughness length parameter (momentum transfer) [𝑚]

𝑧𝑜𝑣

Roughness length parameter (heat and vapor) [𝑚]

𝛼 𝛼𝑝𝑒𝑡 ∆ 𝜌𝑎𝑖𝑟

54

𝑚

Albedo [−] Coefficient in Pristley Taylor equation [−] Slope of the vapor pressure curve [𝑘𝑃𝑎⁄°𝐶 ] Air density [ 1.293 𝑘𝑔⁄𝑚3 ]

𝛾

Psychrometric constant [𝑘𝑃𝑎⁄°𝐶 ]

𝜆

Latent heat of vaporization [

𝑀𝐽 𝑘𝑔

]

𝜆𝐸

Latent heat flux density [𝑀𝐽⁄𝑚2 𝑑𝑎𝑦]

𝜃𝑂

Incoming solar radiation

𝜃𝑑

Reflected radiation

σ

Stefan- Boltzmann constant [ 5.6704 ∗ 10−8 W⁄m2 K 4 ]

5 Infiltration 5.1 Matrix Potential and Soil Hydraulic Characteristics

5 Infiltration Transport of soil water affects heat and solute transport in soils, defines rates of biological processes in soil and water supply to plants, governs transpiration and ground water replenishment, controls runoff, and has many other important functions in the environment. Therefore, simulations of water transport in soil have many applications in hydrology, meteorology, agronomy, environmental protection, and other soil-related disciplines. Success of a multitude of projects depends on the correctness of the model of soil water transport. In this chapter different models are presented.

5.1 Matrix Potential and Soil Hydraulic Characteristics 5.1.1 Matrix potential The water potential quantifies the ability of water to travel in a certain medium from one place to another. This might occur due to osmosis, gravity, (mechanical) pressure, or capillary effects in, such as in the soil matrix. Mostly the matrix potential 𝜓 can be very large compared to the other parts of the water potential. The matrix potential reduces the energy state of water in the vicinity of the particle surfaces, this is why the water attracted by the soil matrix has an energy state lower than that of pure water; hence the matrix potential is always negative. Matrix potential only occurs in unsaturated soils, if it gets close to zero then the pores of the soil are almost completely saturated. So, the matrix potential not only depends on the composition of the soils and its physical and chemical properties, but also on the level of saturation of the soil. Although the capillary rise of water due to matrix potential may occur very slow, it is very important for the supply of plant roots with water.

5.1.2 Soil Hydraulic Characteristics The soil water retention curve, 𝜃(ℎ), describes the relationship between the water content, 𝜃, and the energy status of water at a given location in the soil. Knowledge of 𝜃(ℎ) is essential for the hydraulic characterization of a soil, and bears information about the matrix potential of a soil (layer). Figure 5-1 shows that coarse textured soils lose their water relatively quickly (at small negative pressure heads) and abruptly above the water table, while fine-textured soils lose their water much more gradually. This reflects the particle or pore-size distribution of the medium involved.

55

5 Infiltration 5.2 Darcy Richards Approach

Figure 5-1: Typical soil water retention curves for relatively coarse- (solid line), medium- (dashed line), and finetextured (dotted line) soils.

The hydraulic conductivity characterizes the ability of a soil to transmit water. Its value depends on many factors such as the pore-size distribution of the medium, and the tortuosity, shape, roughness, and degree of interconnectedness of the pores. The hydraulic conductivity decreases considerably as soil becomes unsaturated since less pore space is filled with water, the flow paths become increasingly tortuous, and drag forces between the fluid and the solid phases increase. Figure 5-2 presents examples of typical 𝐾(𝜃) and 𝐾(ℎ) functions for relatively coarse, medium-, and fine-textured soils. Notice that the hydraulic conductivity at saturation is significantly larger for coarse-textured soils than fine-textured soils.

Figure 5-2: Typical curves of the hydraulic conductivity K, as a function of the pressure head (left) and water content (right) for coarse- (solid line), medium- (dashed line), and fine-textured (dotted line) soils.

5.2 Darcy Richards Approach Hagen and Poiseuille conducted in 1841 experiments on laminar flow in pipes, with the postulation that water, while flowing from a point of higher energy head to a point of lower energy loses energy along the flow path due to friction in proportion to the bulk velocity. Based on their work 56

5 Infiltration 5.3 SCS-CN Method

Darcy recognized a similarity between water flow through a saturated soil and laminar flow in pipes. His experiments in sand columns indicated that the flow rate was proportional to the head loss, with a constant proportionality factor. This findings are expressed as Darcy´s Law:

𝑞 = −𝐾

𝑑𝐻 𝑑𝑧

(5-1)

𝑚3

Where 𝑞 is the specific discharge [𝑠𝑚2 ], also known as Darcy flux, 𝐻 is the hydraulic head [𝑚] (𝐻 = ℎ + 𝑧 with ℎ is the soil water pressure head), 𝑧 is the elevation [𝑚]and 𝐾 is the proportion𝑚 ality constant [ 𝑠 ], the hydraulic conductivity. As Darcy´s Law only applies to water flow through saturated soil, Richards presented in 1931 a modified equation for unsaturated flow in porous media. In 1-D form the change of soil water 𝑚3

content 𝜃 [𝑚3 ] over time 𝑡 [s] is:

𝑑𝜃 𝑑 =− 𝑞 𝑑𝑡 𝑑𝑧

(5-2)

𝑑𝜃 𝑑 𝑑𝐻 = [𝐾 ] 𝑑𝑡 𝑑𝑧 𝑑𝑧

(5-3)

𝑑𝜃 𝑑 𝑑ℎ 𝑑𝑧 𝑑 𝑑ℎ = [𝐾 ( + )] = [𝐾 ( + 1)] 𝑑𝑡 𝑑𝑧 𝑑𝑧 𝑑𝑧 𝑑𝑧 𝑑𝑧

(5-4)

Subsitituting with Darcy´s Law

Substituting with 𝐻 = ℎ + 𝑧 yields

Note: This approaches and equations are here only presented for the sake of completeness, none of them are used in the SWAT model!

5.3 SCS-CN Method 5.3.1 Background Simple methods for predicting runoff from watersheds are particularly important in hydrologic engineering and hydrological modelling and they are used in many hydrologic applications, 57


such as flood design and water balance calculation models. The Soil Conservation Service Curve Number (SCS-CN) method was originally developed by the SCS (US Department of Agriculture), to predict direct runoff volumes for given rainfall events in small rural water sheds across the USA It soon became one of the most popular techniques among engineers and practitioners, because it is a simple but well-established method, it features easy to obtain and well-documented environmental inputs, and it accounts for many of the factors affecting runoff generation, incorporating them in a single CN parameter, is based on the area's hydrologic soil group, land use, treatment and hydrologic condition. References, such as from USDA indicate the runoff curve numbers for characteristic land cover descriptions and a hydrologic soil group. The main weaknesses reported are that the SCS-CN method does not consider the impact of rainfall intensity, it does not address the effects of spatial scale, it is highly sensitive to changes in values of its single parameter, CN. The SCS-CN method was soon adopted for various regions, land uses and climate conditions. It was also evolved well beyond its original scope and it became an integral part of continuous simulation models. It is now widely used and an efficient method for determining the approximate amount of direct runoff from a rainfall event in a particular area under varying land use, slope and soil types.

5.3.2 Theoretical Model

Figure 5-3: Relationship between rainfall, Infiltration and Runoff

Figure 5-3 illustrates the basic principle of runoff generation. Runoff occurs when rainfall exceeds the infiltration capacity of the soil. The surface runoff 𝑄𝑠𝑢𝑟𝑓 can then be expressed by the relation between effective rainfall 𝑅𝑒 and infiltration 𝐹:

𝑄𝑠𝑢𝑟𝑓 = 𝑅𝑒 − 𝐹 . 58

(5-5)


𝑑𝜃 𝑑 𝑑ℎ 𝑑𝑧 𝑑 𝑑ℎ = [𝐾 ( + )] = [𝐾 ( + 1)] 𝑑𝑡 𝑑𝑧 𝑑𝑧 𝑑𝑧 𝑑𝑧 𝑑𝑧

(5-6)

The basic assumption between potential retention 𝑆 of the soil and infiltration is:

𝐹 𝑄 = 𝑆 𝑅𝑒

(5-7)

The potential retention is the value 𝐼𝑎 and 𝐹 would reach after a very long event of rain. Using the relationship of 𝐹 = 𝑅𝐸 − 𝑄𝑠𝑢𝑟𝑓 yields to the SCS curve number equation:

𝑄𝑠𝑢𝑟𝑓 =

(𝑅𝑑𝑎𝑦 − 𝐼𝑎 )

2

(5-8)

(𝑅𝑑𝑎𝑦 − 𝐼𝑎 + 𝑆)

Where 𝑄𝑠𝑢𝑟𝑓 is the accumulated surface runoff [𝑚𝑚 𝐻2 𝑂], 𝑅𝑑𝑎𝑦 is the rainfall depth for the day [𝑚𝑚 𝐻2 𝑂], 𝐼𝑎 is the initial abstraction which includes surface storage, interception and infiltration prior to runoff [𝑚𝑚 𝐻2 𝑂]. 𝑆 is the retention parameter [𝑚𝑚 𝐻2 𝑂], which varies spatially due to changes in soils, land use management and slope and in time due to changes in soil water content. It is expressed in terms of the dimensionless curve number (CN) through the relationship:

𝑆 = 25.4 ∙ ( 𝐶𝑁 =

1000 − 10) 𝐶𝑁

25400 𝑆 + 254

(5-9)

(5-10)

59


The initial abstraction 𝐼𝑎 is commonly approximated as 0.2 𝑆, so Eq. (5-8) becomes:

𝑄𝑠𝑢𝑟𝑓 =

(𝑅𝑑𝑎𝑦 − 0.2 ⋅ 𝑆)² 𝑅𝑑𝑎𝑦 + 0.8 ⋅ 𝑆

(5-11)

This is now a very handy equation, as it only requires the CN parameter describing the soil and surfaces. E.g. for paved and sealed areas S will be 0, thus CN will be 100 and all the precipitation becomes directly runoff. Vice versa, if the soil is very permeable S can get very large and no runoff will occur. So, the higher the CN value, the higher the runoff potential is. Figure 5-4 shows how to graphically determine the surface runoff 𝑄𝑠𝑢𝑟𝑓 in relation of the rainfall 𝑅𝑑𝑎𝑦 and different 𝐶𝑁 values.

Figure 5-4: CN curve sheet

5.3.3 SCS Curve Number (CN) in SWAT In the previous section it has been shown that the surface runoff 𝑄𝑠𝑢𝑟𝑓 is only dependent on the curve number CN of the soil (and of the precipitation, which should be a known parameter). In this section it will be shown how to properly determine CN - which is a function of the soils permeability, the land use and antecedent soil water conditions - and how it is used in SWAT.

60


5.3.3.1 Hydrologic Soil Groups The U.S. Natural Resource Conservation Service (NRCS) classifies soils into four different Hydrologic Soil Groups A, B, C & D. The classification is based on similar infiltration characteristics of the soil and similar runoff potential under storm conditions. The definition of the classes are the following.

A:

The soils have a high infiltration rate even when completely wetted. They chiefly consist of deep, well drained to excessively drained sands or gravels. They have a high rate of water transmission.

B:

The soils have a moderate infiltration rate when completely wetted. They chiefly are moderately deep to deep, moderately well well-drained to well-drained soils that have moderately fine to moderately coarse textures. They have a moderate rate of water transmission.

C:

The soils have a slow infiltration rate when completely wetted. They chiefly have a layer that impends downward movement of water or have moderately fine to fine texture. They have a slow rate of water transmission.

D:

The soils have a very slow infiltration rate when completely wetted. They chiefly consist of clay soils that have a high swelling potential, soils that have a permanent water table, soils that have a claypan or clay layer at or near the surface, and shallow soils over nearly impervious material. They have a very slow rate of water transmission.

.

Table 5-1 shows a table with different curve numbers in the case of different cover types and curve numbers. Assumed is a slope of 5%. The SWAT data base usually assumes ‘fair’ hydrologic conditions.

61


Table 5-1: Curve numbers for different cover types and hydrologic soil groups

5.3.3.2 Antecedent soil moisture conditions (hydraulic conditions) The SCS defines three antecedent moisture conditions: I) II) III)

DRY (wilting point) AVERAGE MOISTURE WET (field capacity)

The moisture condition I curve number is the lowest value the daily curve number can assume in dry conditions. The curve numbers for moisture conditions I and III are calculated with equations:

𝐶𝑁𝐼 = 𝐶𝑁𝐼𝐼 −

20 ∙ (100 − 𝐶𝑁𝐼𝐼 )

(5-12)

(100 − 𝐶𝑁𝐼𝐼 + exp(2.533 − 0.0636 ∙ (100 − 𝐶𝑁𝐼𝐼 )))

𝐶𝑁𝐼𝐼𝐼 = 𝐶𝑁𝐼𝐼 ∙ exp(0.00673 ∙ (100 − 𝐶𝑁𝐼𝐼 ))

(5-13)

Where 𝐶𝑁𝐼 is the moisture condition I curve number, 𝐶𝑁𝐼𝐼 is the moisture condition II

curve number, and 𝐶𝑁𝐼𝐼𝐼 is the moisture condition III curve number. Figure 5-5 shows the different CN curves.

62


Figure 5-5: CN values for different moisture conditions

5.3.3.3 Retention Parameter In SWAT the user may choose between two methods for calculating the retention parameter. The traditional method is to allow the retention parameter to vary with soil profile water content. An alternative method allows the retention parameter to vary with accumulated plant evapotranspiration. Calculation of the daily CN value as a function of plant evapotranspiration was added because the soil moisture method was predicting too much runoff in shallow soils. By calculating daily CN as a function of plant evapotranspiration, the value is less dependent on soil storage and more dependent on antecedent climate. When the retention parameter varies with soil profile water content (traditional method), the following equation is used:

𝑆 = 𝑆𝑚𝑎𝑥 ∙ (1 −

𝑆𝑊 ) (𝑆𝑊 + exp(𝑤1 − 𝑤2 ∙ 𝑆𝑊))

(5-14)

𝑆 is the retention parameter for a given day (mm). 𝑆𝑚𝑎𝑥 is the maximum retention value for 𝑆, it is calculated using 𝐶𝑁𝐼 in Eq. (5-9). 𝑆𝑊 is the soil water content [mm H2O] of the entire profile excluding the amount of water held in the profile at wilting point; 𝑤1 and 𝑤2 are the shape coefficients. The shape coefficients themselves are determined using the following assumptions:

63


1. 𝑆𝑚𝑎𝑥 is calculated using 𝐶𝑁𝐼 and original equation for S – SW corresponds to wilting point. 2. The retention parameter for moisture condition III curve number corresponds to field capacity soil profile water content. 3. Soil has the curve number of 99 (S = 2.54) when completely saturated.

𝑤1 = 𝑙𝑛 ⌊

𝐹𝐶 − 𝐹𝐶⌋ + 𝑤2 ∙ 𝐹𝐶 −1 1 − 𝑆3 ∙ 𝑆𝑚𝑎𝑥

(𝑙𝑛 ⌊ 𝑤2 =

𝐹𝐶 𝑆𝐴𝑇 − 𝐹𝐶⌋ − 𝑙𝑛 ⌊ −1 −1 − 𝑆𝐴𝑇⌋) 1 − 𝑆3 ∙ 𝑆𝑚𝑎𝑥 1 − 2.54 ∙ 𝑆𝑚𝑎𝑥 (𝑆𝐴𝑇 − 𝐹𝐶)

(5-15)

(5-16)

With 𝐹𝐶 as the amount of water in the soil profile at field capacity [𝑚𝑚 𝐻2 𝑂], 𝑆3 Retention parameter for moisture condition CN III number [𝑚𝑚], 𝑆𝑚𝑎𝑥 is the retention parameter for moisture condition number CN I [𝑚𝑚], 𝑆𝐴𝑇 is the amount of water in the soil profile when completely saturated [𝑚𝑚 𝐻2 𝑂], 2.54 is the retention parameter value for a curve number of 99. When the retention parameter varies with plant evapotranspiration (alternative method), the following equation is used to update the retention parameter at the end of every day:

−𝑐𝑛𝑐𝑜𝑒𝑓 ∙ 𝑆𝑝𝑟𝑒𝑣 𝑆 = 𝑆𝑝𝑟𝑒𝑣 + 𝐸0 ∙ 𝑒𝑥𝑝 ( ) − 𝑅𝑑𝑎𝑦 + 𝑄𝑠𝑢𝑟𝑓 𝑆𝑚𝑎𝑥

(5-17)

where S is the retention parameter for a given day [𝑚𝑚], 𝑆𝑝𝑟𝑒𝑣 is the retention parameter for the previous day [𝑚𝑚], 𝐸0 is the potential evapotranspiration for the day (𝑚𝑚 𝑑−1), 𝑐𝑛𝑐𝑜𝑒𝑓 is the weighting coefficient used to calculate the retention coefficient for daily curve number calculations dependent on plant evapotranspiration, 𝑆𝑚𝑎𝑥 is the maximum value the retention parameter can achieve on any given day [𝑚𝑚], 𝑅𝑑𝑎𝑦 is the rainfall depth for the day [𝑚𝑚 𝐻2 𝑂], and 𝑄𝑠𝑢𝑟𝑓 is the surface runoff. The initial value of the retention parameter is defined as 𝑆 = 0.9 ⋅ 𝑆𝑚𝑎𝑥 .

5.3.3.4 Slope Adjustment The moisture condition II curve numbers, which are used in SWAT are assumed to be appropriate for 5% slopes. In case of higher slopes the curve number can be adjusted with the following equation:

𝑆𝐶𝑁𝐼𝐼𝑠 =

64

𝐶𝑁𝐼𝐼𝐼 − 𝐶𝑁𝐼𝐼 ∙ [1 − 2 ∙ exp(−13.86 ∙ 𝑠𝑙𝑝)] + 𝐶𝑁𝐼𝐼 3

(5-18)

5 Infiltration 5.4 Green and Ampt

𝑆𝐶𝑁𝐼𝐼𝑠 is the moisture condition II curve number, which is adjusted. 𝐶𝑁𝐼𝐼 is the moisture condition II curve number for default 5 % slope, 𝐶𝑁𝐼𝐼𝐼 is the moisture condition III curve number for the default 5 % slope and 𝑠𝑙𝑝 is the average slope for the sub basin [𝑚⁄𝑚]. Caution: SWAT does not adjust Curve Numbers for the slope automatically. This has to be done prior to entering the curve number values in the management input file.

Figure 5-6: Different curve numbers for different slopes

5.4 Green and Ampt 5.4.1 Approach The Green and Ampt equation was developed to predict infiltration assuming excess water at the surface at all times (Green and Ampt, 1911) and is based on Darcy´s law:

𝑓 = −𝐾𝑒

(𝜓𝑓 + 𝑧𝑓 ) − 𝐻 𝜓𝑓 + 𝑧𝑓 − 𝐻 𝑑ℎ ℎ2 − ℎ1 = −𝐾𝑒 = −𝐾𝑒 = −𝐾𝑒 𝑑𝑧 𝑧2 − 𝑧1 𝑧𝑓 − 0 𝑧𝑓

(5-19)

Figure 5-7 illustrates the meaning of the variables.𝑓 is the infiltration flux [𝑚𝑚/ℎ] 𝐾𝑒 is the effective hydraulic conductivity [𝑚𝑚/ℎ], 𝑧𝑓 is the depth of the wetting front [𝑚𝑚], 𝜓𝑓 is the wetting front matrix potential [𝑚𝑚] (effective suction, also called 𝑆𝑓 ), and 𝐻 is the ponding depth [𝑚𝑚].

65


Figure 5-7: Green and Ampt based on Darcy

For the equation the following assumptions are made: I) II) III)

As rain continues to fall and water infiltrates, the wetting front advances at the same rate with depth, which produces a well-defined wetting front. The volumetric water contents remain constant above and below the wetting front as it advances. The soil-water suction immediately below the wetting front remains constant with both time and location as the wetting front advances.

The model assumes the soil above the wetting front to be completely saturated and that there is a sharp break in moisture content at the wetting front (piston flow). Figure 5-8 illustrates the difference between the moisture distribution in the depth modelled by Green and Ampt and in reality.

66


Figure 5-8: Comparison of moisture content distribution modeled by Green and Ampt and natural observed distribution.

The depth of the wetting front can be related to the cumulative amount of infiltrated water, 𝐹 [𝑚𝑚], by: 𝐹 = 𝑧𝑓 ⋅ (θs − θi ) = 𝑧𝑓 ⋅ Δ𝜃𝑣

(5-20)

Assuming surface ponding occurs, but neglecting the ponding depth (𝐻 → 0) and rearranging Eq. (5-20) to solve for 𝑧𝑓 and substituting it into Eq. (5-19) the infiltration rate into the surface at time 𝑡 𝑓𝑖𝑛𝑓,𝑡 [𝑚𝑚/ℎ] can be written as:

𝑓𝑖𝑛𝑓,𝑡 = 𝐾𝑒 ⋅ (1 +

𝜓𝑓 ⋅ Δ𝜃𝑣 ) 𝐹𝑖𝑛𝑓,𝑡

(5-21)

𝐾𝑒 is the effective hydraulic conductivity [𝑚𝑚/ℎ], 𝜓𝑓 is the wetting front matrix potential [𝑚𝑚] (effective suction, also called 𝑆𝑓 ), Δ𝜃𝑣 is the change in volumetric moisture content across the wetting front [𝑚𝑚/𝑚𝑚], which is the difference between initial condition and saturation (note: volumetric moisture content at saturation is equal to the soil porosity 𝜙 [𝑚𝑚/𝑚𝑚]), and 𝐹𝑖𝑛𝑓,𝑡 [𝑚𝑚 𝐻2 𝑂] is the cumulative infiltration at time step 𝑡. So, the Infiltration rate 𝑓𝑖𝑛𝑓,𝑡 itself is a function of the infiltrated volume 𝐹𝑖𝑛𝑓,𝑡 , which in turn is a function of the infiltration rates in previous time steps. To avoid numerical errors over long time steps, 𝑓𝑖𝑛𝑓,𝑡 is replaced by 𝑑𝐹𝑖𝑛𝑓 ⁄𝑑𝑡. This yields to:

67


𝐹𝑖𝑛𝑓,𝑡 + |𝜓𝑓 | ⋅ Δ𝜃𝑣 𝐹𝑖𝑛𝑓,𝑡 = 𝐹𝑖𝑛𝑓,𝑡−1 + 𝐾𝑒 ⋅ Δ𝑡 + |𝜓𝑓 | ⋅ Δ𝜃𝑣 ⋅ ln [ ] 𝐹𝑖𝑛𝑓,𝑡−1 + |𝜓𝑓 | ⋅ Δ𝜃𝑣

(5-22)

Eq. (5-22) must be solved iteratively in SWAT, in order to obtain 𝐹𝑖𝑛𝑓,𝑡 . When the rainfall intensity is less than the infiltration rate, all the rainfall will infiltrate during the time period and the cumulative infiltration for that time period is calculated:

𝐹𝑖𝑛𝑓,𝑡 = 𝐹𝑖𝑛𝑓,𝑡−1 + 𝑅Δ𝑡

(5-23)

𝑅Δ𝑡 is the amount of rain falling during the time step [𝑚𝑚 𝐻2 𝑂]. If rainfall intensity is constant and eventually exceeds infiltration rate 𝑓𝑖𝑛𝑓,𝑡 , then at some moment the surface will become saturated and ponding when the infiltration rate equals the precipitation rate 𝑤 [𝑚𝑚/ℎ]. The depth infiltrated at that moment (𝐹𝑖𝑛𝑓,𝑠 ) is given by setting 𝑓 = 𝑤 in Eq. (5-21) and solving for 𝐹𝑖𝑛𝑓,𝑠 :

𝜓𝑓 ⋅ Δ𝜃𝑣 𝐹𝑖𝑛𝑓,𝑠 = 𝑤 𝐾𝑒 − 1

(5-24)

The ponding time 𝑡𝑝 [ℎ], the time from the beginning of the rainfall event until ponding occurs is given by:

𝑡𝑝 =

𝐹𝑖𝑛𝑓,𝑠 𝑤

(5-25)

5.4.2 Parameter Estimation To apply the Green and Ampt model, the effective hydraulic conductivity 𝐾𝑒 , the wetting front matrix potential 𝜓𝑓 , the porosity of the soil 𝜙 [mm/mm], and the change in volumetric moisture content Δ𝜃𝑣 across the wetting front need to be measured or estimated. The effective hydraulic conductivity 𝐾𝑒 can be approximated with the saturated hydraulic conductivity of the soil 𝐾𝑠𝑎𝑡 and the curve number CN: 0.286 56.82 ⋅ 𝐾𝑠𝑎𝑡 𝐾𝑒 = −2 1 + 0.051 ⋅ exp(0.062 ⋅ 𝐶𝑁)

68

(5-26)

5 Infiltration 5.5 Nomenclature

The wetting front matrix potential 𝜓𝑓 is calculated as a function of porosity, and the percentage of sand 𝑚𝑠 and clay 𝑚𝑐 in the soil:

𝜓𝑓 = 10 ⋅ exp[6.5309 − 7.32561 ⋅ 𝜙 + 0.001583 ⋅ 𝑚𝑐2 + 3.808479 ⋅ 𝜙 2 + 0.000344 ⋅ 𝑚𝑠 ⋅ 𝑚𝑐 − 0.049837 ⋅ 𝑚𝑠 ⋅ 𝜙 + 0.001608 ⋅ 𝑚𝑠2 ⋅ 𝜙 2 + 0.001602 ⋅ 𝑚𝑐2 ⋅ 𝜙 2 − 0.0000136 ⋅ 𝑚𝑠2 ⋅ 𝑚𝑐 − 0.003479 ⋅ 𝑚𝑐2 ⋅ 𝜙 − 0.000799 ⋅ 𝑚𝑠2 ⋅ 𝜙]

(5-27)

As the initial moisture content 𝜃𝑖 is not given and difficult to determine, the change in volumetric moisture content Δ𝜃𝑣 across the wetting front is calculated at the beginning of each day as:

Δ𝜃𝑣 = (1 −

𝑆𝑊 ) ⋅ (0.95 ⋅ 𝜙) 𝐹𝐶

(5-28)

SW is the soil water content of the entire profile excluding the amount of water held in the profile at the wilting point [mm H2O]. FC is the amount of water in the soil at field capacity [mm H2O]. In case rainfall is in progress at the beginning of a time step Δ𝜃𝑣 is calculated as:

Δ𝜃𝑣 = 0.001 ⋅ (0.95 ⋅ 𝜙)

(5-29)

For each time step, SWAT calculates the amount of water entering the soil. The water that does not infiltrate into the soil becomes surface runoff.

5.5 Nomenclature 𝐶𝑁

Curve number [−]

𝐶𝑁𝐼

Moisture condition I curve number [−]

𝐶𝑁𝐼𝐼

Moisture condition II curve number [−]

𝐶𝑁𝐼𝐼𝐼

Moisture condition III curve number [−]

𝐸0 𝐹𝑖𝑛𝑓,𝑡 𝐹

potential evapotranspiration for the day (𝑚𝑚 𝑑−1 ) Cumulative infiltration at time step t [𝑚𝑚 𝐻2 𝑂] Amount of infiltration [𝑚𝑚 𝐻2 𝑂]

𝐹𝐶

Amount of water in the soil profile at field capacity [𝑚𝑚 𝐻2 𝑂]

𝐻

Ponding depth [𝑚𝑚]

𝐼𝑎

Initial abstraction which includes surface storage, interception and infiltration prior to runoff [𝑚𝑚 𝐻2 𝑂]

𝐾𝑒

Effective hydraulic conductivity [𝑚𝑚/ℎ] 69


Saturated hydraulic conductivity [ 𝑠 ]

𝑄𝑠𝑢𝑟𝑓

Accumulated surface runoff [𝑚𝑚 𝐻2 𝑂]

𝑅Δ𝑡

Amount of rain falling during the time step [𝑚𝑚 𝐻2 𝑂]

𝑅𝑒

Effective rainfall [𝑚𝑚 𝐻2 𝑂]

𝑆

Retention parameter in SCS curve number equation [𝑚𝑚]

𝑆3

Retention parameter for moisture condition CN III number [𝑚𝑚]

𝑆𝑚𝑎𝑥

Maximum retention value for 𝑆, it is calculated using 𝐶𝑁𝐼

𝑆𝑝𝑟𝑒𝑣

Retention parameter from SCS curve number equation for the previous day [𝑚𝑚]

𝑆𝐴𝑇

Amount of water in the soil profile when completely saturated [𝑚𝑚 𝐻2 𝑂]

𝑆𝐶𝑁𝐼𝐼𝑠 𝑆𝑊 𝑐𝑛𝑐𝑜𝑒𝑓

𝑑𝐻 𝑓𝑖𝑛𝑓,𝑡

Moisture condition II curve number, which is adjusted Amount of water in soil profile [𝑚𝑚 𝐻2 𝑂] Weighting coefficient used to calculate the retention coefficient for daily curve number calculations dependent on plant evapotranspiration Hydraulic head [𝑚] infiltration rate into the surface at time 𝑡 [𝑚𝑚/ℎ]

𝑚𝑐

Percentage of clay in the soil [%]

𝑚𝑠

Percentage of sand in the soil [%]

𝑞

Specific discharge (Darcy flux) [

𝑚3 ] 𝑠𝑚2 𝑚

𝑠𝑙𝑝

Average slope for the sub basin [𝑚 𝑜𝑟 % ]

𝑡𝑝

Ponding time [ℎ]

𝑤

Precipitation rate [𝑚𝑚/ℎ]

𝑤1 , 𝑤2

70

𝑚

𝐾𝑠𝑎𝑡

Shape coefficients in retention parameter adjustment for soil moisture content [−]

𝑧𝑓

Depth of the wetting front [𝑚𝑚]

𝑧

Elevation [𝑚]

𝜙

Soil porosity [𝑚𝑚/𝑚𝑚]

𝜓𝑓

wetting front matrix potential [𝑚𝑚]

𝜓𝑓

Wetting front matrix potential [𝑚𝑚]

𝜃

Soil water content [𝑚3 ]

𝜃𝑖

Initial moisture content [𝑚3 ]

𝑚3

𝑚3


Δ𝜃𝑣

Change in volumetric moisture content across the wetting front [𝑚𝑚/𝑚𝑚]

71

6 Vertical Soil Water Movement 6.1 Soil Characteristics and Soil Water Content in SWAT

6 Vertical Soil Water Movement Water that enters the soil may move along one of several different pathways. The water may be removed from the soil by plant uptake or evaporation. It can percolate past the bottom of the soil profile and ultimately become aquifer recharge. A final option is that water may move laterally in the profile and contribute to stream flow, which will be explained in Chapter 7. In this Chapter the vertical pathways are described, as plant uptake of water removes the majority of water that enters the soil profile.

6.1 Soil Characteristics and Soil Water Content in SWAT 6.1.1 Soil Charcteristics The soil’s bulk density defines the relative amounts of pore space and soil matrix. Bulk density is calculated:

𝜌𝑏 =

𝑀𝑠

(6-1)

𝑉𝑇

𝜌𝑏 is the bulk density [𝑡𝑜𝑛/𝑚3 ], 𝑀𝑠 is the mass of the solids [𝑡𝑜𝑛], and 𝑉𝑇 is the total volume [𝑚3 ], which itself is the sum of water, air and solid volume. The relationship between soil porosity 𝜙𝑠𝑜𝑖𝑙 and soil bulk density is:

𝜙𝑠𝑜𝑖𝑙 = 1 −

𝜌𝑏 𝜌𝑠

(6-2)

𝜌𝑠 is the density of the solid fraction [𝑡𝑜𝑛/𝑚3 ]. It is a function of the mineral composition of the soil matrix. Based on research, a default value of 2.65 𝑡𝑜𝑛/𝑚3 can be used for the particle density. Soil pores vary in size and shape due to textural and structural arrangement. Based on the diameter of the pore at the narrowest point, the pores may be classified as macropores (narrowest diameter > 100 𝜇𝑚) mesopores (narrowest diameter 30-100 𝜇𝑚), and micropores (narrowest diameter < 30 𝜇𝑚). Macropores conduct water only during flooding or ponding rain and drainage of water from these pores is complete soon after cessation of the water supply. Macropores control aeration and drainage processes in the soil. Mesopores conduct water even after macropores have emptied, e.g. during non-ponding rain and redistribution. Micropores retain soil solution or conduct it very slowly. When comparing soils of different texture, clay soils contain a greater fraction of mesopores and micropores while sand soils contain mostly macropores. This is evident when the hydraulic conductivities of clay and sand soils are compared. The conductivity of a sand soil can be several orders of magnitude greater than that for a clay soil. The water content of a soil can range from zero when the soil is oven dried to a maximum value (𝜙𝑠𝑜𝑖𝑙 ) when the soil is saturated. For plant-soil interactions, two intermediate stages are recognized: field capacity and permanent 72

6 Vertical Soil Water Movement 6.1 Soil Characteristics and Soil Water Content in SWAT

wilting point. Field capacity is the water content found when a thoroughly wetted soil has drained for approximately two days. Permanent wilting point is the water content found when plants growing in the soil wilt and do not recover if their leaves are kept in a humid atmosphere overnight. To allow these two stages to be quantified more easily, they have been redefined in terms of tensions at which water is held by the soil. Field capacity is the amount of water held in the soil at a tension of 0.033 𝑀𝑃𝑎 and the permanent wilting point is the amount of water held in the soil at a tension of 1.5 𝑀𝑃𝑎. The amount of water held in the soil between field capacity and permanent wilting point is considered to be the water available for plant extraction.

Table 6-1: Water contents for various soils at different moisture conditions

Water content (fraction of total water volume) Texture

Clay Content (%)

Saturation

Field Capacity

Permanent Wilting Point

Sand

3

0.40

0.06

0.02

Loam

22

0.50

0.29

0.05

Clay

47

0.60

0.41

0.2

A sand soil drains more quickly than loam and clay. Only 15% of the water present in the sand soil at saturation remains at field capacity. 58% of the water present at saturation in the loam remains at field capacity while 68% of the water present at saturation in the clay soil remains at field capacity. The reduction of water loss with increase in clay content is cause by two factors. As mentioned previously, clay soils contain more mesopores and micropores than sand soils. Also, unlike sand and silt particles, clay particles possess a net negative charge. Due to the polar nature of water molecules, clay particles are able to attract and retain water molecules. The higher water retention of clay soils is also seen in the fraction of water present at permanent wilting point. In the soils listed in Table 6-1, the volumetric water content of the clay is 0.20 at the wilting point while the sand and loam have a volumetric water content of 0.02 and 0.05 respectively.

6.1.2 Soil Water Content The plant available water, also referred to as the available water capacity, is calculated by subtracting the fraction of water present at permanent wilting point from that present at field capacity.

𝐴𝑊𝐶 = 𝐹𝐶 − 𝑊𝑃

(6-3)

where 𝐴𝑊𝐶 is the plant available water content, 𝐹𝐶 is the water content at field capacity, and 𝑊𝑃 is the water content at permanent wilting point. For the three soil textures listed in Table 6-1, the sand has an available water capacity of 0.04, the loam has an available water capacity of 0.24 and the clay has an available water capacity of 0.21. Even though the clay contains a 73

6 Vertical Soil Water Movement 6.2 Water Uptake by Plants

greater amount of water than the loam at all three tensions, the loam has a larger amount of water available for plant uptake than the clay. This characteristic is true in general. SWAT estimates the permanent wilting point volumetric water content for each soil layer as:

𝑊𝑃𝑙𝑦 = 0.40 ⋅

𝑚𝑐 ⋅ 𝜌𝑏 100

(6-4)

where 𝑊𝑃𝑙𝑦 is the water content at wilting point expressed as a fraction of the total soil volume, 𝑚𝑐 is the percent clay content of the layer (%), and 𝜌𝑏 is the bulk density for the soil layer [𝑡𝑜𝑛/𝑚3]. Field capacity water content is estimated as:

𝐹𝐶𝑙𝑦 = 𝑊𝑃𝑙𝑦 + 𝐴𝑊𝐶𝑙𝑦

(6-5)

𝐹𝐶𝑙𝑦 is the water content at field capacity expressed as a fraction of the total soil volume, 𝑊𝑃𝑙𝑦 is the water content at wilting point expressed as a fraction of the total soil volume, and 𝐴𝑊𝐶𝑙𝑦 is the available water capacity of the soil layer expressed as a fraction of the total soil volume. 𝐴𝑊𝐶𝑙𝑦 is defined by the user. Water in the soil can flow under saturated or unsaturated conditions. In saturated soils, flow is driven by gravity and usually occurs in the downward direction. Unsaturated flow is caused by gradients arising due to adjacent areas of high and low water content. Unsaturated flow may occur in any direction. SWAT directly simulates saturated flow only. The model records the water contents of the different soil layers but assumes that the water is uniformly distributed within a given layer. This assumption eliminates the need to model unsaturated flow in the horizontal direction. Saturated flow occurs when the water content of a soil layer surpasses the field capacity for the layer.

6.2 Water Uptake by Plants The potential water uptake from the soil surface to any depth in the root zone is estimated with the function:

𝑤𝑢𝑝,𝑧 =

𝐸𝑡 𝑧 ∙ [1 − 𝑒𝑥𝑝 (−𝛽𝑤 ∙ )] [1 − 𝑒𝑥𝑝(−𝛽𝑤 )] 𝑧𝑟𝑜𝑜𝑡

(6-6)

𝑤𝑢𝑝,𝑧 is the potential water uptake from the soil surface to a specific depth, 𝑧, on a given day [𝑚𝑚𝐻2 𝑂], 𝐸𝑡 is the maximum transpiration rate [𝑚𝑚/𝑑], 𝛽𝑤 is the water use distribution parameter, 𝑧 is the depth below soil surface [𝑚𝑚], 𝑧𝑟𝑜𝑜𝑡 is the depth of root development in the soil [𝑚𝑚], 𝑤𝑢𝑝,𝑧𝑙 is the potential water uptake for the profile to the lower boundary [𝑚𝑚𝐻2 𝑂], 𝑤𝑢𝑝,𝑧𝑢 potential water uptake for the upper boundary of the soil layer [𝑚𝑚𝐻2 𝑂], and 𝑤𝑢𝑝,𝑙𝑦 is the potential water uptake for the specific layer 𝑙𝑦 [𝑚𝑚𝐻2 𝑂] :

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6 Vertical Soil Water Movement 6.2 Water Uptake by Plants

𝑤𝑢𝑝,𝑙𝑦 = 𝑤𝑢𝑝,𝑧𝑙 − 𝑤𝑢𝑝,𝑧𝑢

(6-7)

As the root network is much denser at the soil surface than in the depth, the water uptake from the upper layers is assumed to be much greater than that in the lower layers. The water use distribution parameter 𝛽𝑤 , which describes the water uptake over depth is set to 10 in SWAT. Within this value, 50% of the water uptake will occur in the upper 6% of the root zone. Figure 6-1 illustrates the uptake of water at different depths in the root zone.

Figure 6-1: Depth distribution of water uptake

If the upper layers in the soil profile do not contain enough water to meet the potential water uptake calculated, one can access the water content of lower layers to compensate. The adjusted potential water uptake is:

𝑤𝑢𝑝,𝑙𝑦 ′ = 𝑤𝑢𝑝,𝑙𝑦 + 𝑤𝑑𝑒𝑚𝑎𝑛𝑑 ∙ 𝑒𝑝𝑐𝑜

(6-8)

𝑤𝑢𝑝,𝑙𝑦 ′ is the adjusted potential water uptake [𝑚𝑚𝐻2 𝑂] for layer 𝐿𝑦 , 𝑤𝑢𝑝,𝑙𝑦 is the potential water uptake [𝑚𝑚𝐻2 𝑂], 𝑤𝑑𝑒𝑚𝑎𝑛𝑑 is the water uptake demand not met by overlying soil layers [𝑚𝑚𝐻2 𝑂] and 𝑒𝑝𝑐𝑜 is the plant water uptake parameter, which is chosen between 0.01 and 1. The bigger it is the more water can be taken from lower layers. As the water content of the soil gets lower, the water in the soil is retained more and more by the soil particles and it becomes more and more difficult for the plant to extract water from the

75

6 Vertical Soil Water Movement 6.3 Percolation

soil. To reflect the decrease in the efficiency of the plant in extracting water from dryer soil layers, the potential water uptake is modified with the following equations:

𝑆𝑊𝑙𝑦 𝑤 ′′ 𝑢𝑝,𝑙𝑦 = 𝑤 ′ 𝑢𝑝,𝑙𝑦 ∙ 𝑒𝑥𝑝 [5 ∙ ( − 1)] 𝑤ℎ𝑒𝑛 𝑆𝑊𝑙𝑦 < 0.25 ∙ 𝐴𝑊𝐶𝑙𝑦 (0.25 ∙ 𝐴𝑊𝐶𝑙𝑦 )

(6-9)

𝑤 ′′ 𝑢𝑝,𝑙𝑦 = 𝑤 ′ 𝑢𝑝,𝑙𝑦

(6-10)

𝑤ℎ𝑒𝑛 𝑆𝑊𝑙𝑦 ≥ 0.25 ∙ 𝐴𝑊𝐶𝑙𝑦

𝑤 ′′ 𝑢𝑝,𝑙𝑦 is the potential water uptake adjusted for initial soil water content [𝑚𝑚𝐻2 𝑂], 𝑤′𝑢𝑝,𝑙𝑦 is the adjusted potential water uptake for layer 𝑙𝑦 [𝑚𝑚𝐻2 𝑂], 𝑆𝑊𝑙𝑦 is the soil water content of layer 𝑙𝑦 [𝑚𝑚𝐻2 𝑂], and 𝐴𝑊𝐶𝑙𝑦 is the available water capacity for the layer 𝑙𝑦 [𝑚𝑚𝐻2 𝑂]. The actual water uptake 𝑤𝑎𝑐𝑐𝑡𝑢𝑎𝑙𝑢𝑝,𝑙𝑦 [𝑚𝑚𝐻2 𝑂] for the layer 𝑙𝑦 can be determined as:

𝑤𝑎𝑐𝑐𝑡𝑢𝑎𝑙𝑢𝑝,𝑙𝑦 = 𝑚𝑖𝑛[𝑤𝑢𝑝,𝑙𝑦 ′′ , (𝑆𝑊𝑙𝑦 − 𝑊𝑃𝑙𝑦 )]

(6-11)

The total water uptake 𝑤𝑎𝑐𝑐𝑡𝑢𝑎𝑙𝑢𝑝 [𝑚𝑚𝐻2 𝑂], which is equal to the actual transpiration 𝐸𝑡,𝑎𝑐𝑡 [𝑚𝑚𝐻2 𝑂] for a given day is: 𝑛

𝑤𝑎𝑐𝑐𝑡𝑢𝑎𝑙𝑢𝑝 = ∑ 𝑤𝑎𝑐𝑐𝑡𝑢𝑎𝑙𝑢𝑝,𝑙𝑦 = 𝐸𝑡,𝑎𝑐𝑡

(6-12)

𝑙𝑦=1

6.3 Percolation Percolation is calculated for each soil layer in the profile. Water is allowed to percolate if the water content exceeds the field capacity water content for that layer and the layer below is not saturated. When the soil layer is frozen, no water flow out of the layer is calculated. The volume of water available for percolation in the soil layer is calculated:

𝑆𝑊𝑙𝑦,𝑒𝑥𝑐𝑒𝑠𝑠 = 𝑆𝑊𝑙𝑦 − 𝐹𝐶𝑙𝑦

𝑖𝑓 𝑆𝑊𝑙𝑦 > 𝐹𝐶𝑙𝑦

(6-13)

𝑆𝑊𝑙𝑦,𝑒𝑥𝑐𝑒𝑠𝑠 = 0

𝑖𝑓 𝑆𝑊𝑙𝑦 < 𝐹𝐶𝑙𝑦

(6-14)

𝑆𝑊𝑙𝑦,𝑒𝑥𝑐𝑒𝑠𝑠 is the drainable volume of water in the soil layer on a given day [𝑚𝑚 𝐻2 𝑂], 𝑆𝑊𝑙𝑦 is the water content of the soil layer for a given day [𝑚𝑚 𝐻2 𝑂] and 𝐹𝐶𝑙𝑦 is the is the water content of the soil layer at field capacity [𝑚𝑚 𝐻2 𝑂].

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6 Vertical Soil Water Movement 6.4 Bypass Flow

The amount of water that moves from one layer to the underlying layer is calculated using storage routing methodology. The equation used to calculate the amount of water that percolates to the next layer 𝑤𝑝𝑒𝑟𝑐,𝑙𝑦 [𝑚𝑚 𝐻2 𝑂] is:

−∆𝑡 𝑤𝑝𝑒𝑟𝑐,𝑙𝑦 = 𝑆𝑊𝑙𝑦,𝑒𝑥𝑐𝑒𝑠𝑠 ∙ (1 − 𝑒𝑥𝑝 ⌊ ⌋) 𝑇𝑇𝑝𝑒𝑟𝑐

(6-15)

Δ𝑡 is the length of the time step [ℎ], and 𝑇𝑇𝑝𝑒𝑟𝑐 is the travel time for percolation [ℎ], which can be calculated as:

𝑇𝑇𝑝𝑒𝑟𝑐 =

𝑆𝐴𝑇𝑙𝑦 − 𝐹𝐶𝑙𝑦 𝐾𝑠𝑎𝑡

(6-16)

𝑆𝐴𝑇𝑙𝑦 is the amount of water in the soil layer when completely saturated [𝑚𝑚 𝐻2 𝑂], 𝐹𝐶𝑙𝑦 is the water content of the soil layer at field capacity [𝑚𝑚 𝐻2 𝑂], and 𝐾𝑠𝑎𝑡 is the saturated hydraulic conductivity for the layer [𝑚𝑚 ℎ−1]. If the HRU has a seasonal high water table, percolation is not allowed when

𝑆𝑊𝑙𝑦+1 ≤ 𝐹𝐶𝑙𝑦+1 + 0.5 ∙ (𝑆𝐴𝑇𝑙𝑦+1 − 𝐹𝐶𝑙𝑦+1 )

(6-17)

𝑆𝑊𝑙𝑦+1 is the water content of the underlying soil layer [𝑚𝑚 𝐻2 𝑂], 𝐹𝐶𝑙𝑦+1 is the water content of the underlying soil layer at field capacity [𝑚𝑚 𝐻2 𝑂], and 𝑆𝐴𝑇𝑙𝑦+1 is the amount of water in the underlying soil layer when completely saturated [𝑚𝑚 𝐻2 𝑂]. The water will instead stay ponded in the upper layer.

6.4 Bypass Flow Some soils, such as the Vertisols which appear in the tropics and subtropics, are characterized by a propensity to shrink when dried and swell when moistened. When the soil is dry, large cracks form at the soil surface. This behavior is a result of the type of soil material present and the climate. Vertisols contain at least 30% clay with the clay fraction dominated by smectitic mineralogy and occur in areas with cyclical wet and dry periods. One criteria used to classify a soil as a Vertisol is the formation of shrinkage cracks in the dry season that penetrate to a depth of more than 50 𝑐𝑚 and are at least 1 𝑐𝑚 wide at 50 𝑐𝑚 depth. The cracks can be considerably wider up to 30 𝑐𝑚 cracks at the surface. Although this is not usual and 6-15 𝑐𝑚 cracks are more typical. During dry season and at the beginning of the raining season vertisols have long vertical cracks as shown in Figure 6-2.

77


Figure 6-2: Dry vertisol with huge cracks (source: www.geography.hunter.cuny.edu)

To accurately predict surface runoff and infiltration in areas dominated by soils that have vertisol properties, the temporal change in soil volume must be quantified, as traditional models of infiltration are applicable to soils in which cracks have been closed by swelling and the soil acts as a relatively homogenous porous medium. When bypass flow is modelled, SWAT calculates the crack volume of the soil matrix for each day of simulation by layer. On days in which precipitation events occur, infiltration and surface runoff is first calculated for the soil peds using the curve number or Green & Ampt method (Section 5.4). If any surface runoff is generated, it is allowed to enter the cracks. A volume of water equivalent to the total crack volume for the soil profile may enter the profile as bypass flow. Surface runoff in excess of the crack volume remains overland flow. Water that enters the cracks fills the soil layers beginning with the lowest layer of crack development. After cracks in one layer are filled, the cracks in the overlying layer are allowed to fill. The crack volume initially estimated for a layer is calculated:

𝑐𝑟𝑘𝑙𝑦,𝑖 = 𝑐𝑟𝑘𝑚𝑎𝑥,𝑙𝑦 ⋅

𝑐𝑜𝑒𝑓𝑐𝑟𝑘 ⋅ 𝐹𝐶𝑙𝑦 − 𝑆𝑊𝑙𝑦 𝑐𝑜𝑒𝑓𝑐𝑟𝑘 ⋅ 𝐹𝐶𝑙𝑦

(6-18)

𝑐𝑟𝑘𝑙𝑦,𝑖 is the initial crack volume calculated for the soil layer on a given day expressed as a depth [𝑚𝑚], 𝑐𝑟𝑘𝑚𝑎𝑥,𝑙𝑦 is the maximum crack volume possible for the soil layer [𝑚𝑚], 𝑐𝑜𝑒𝑓𝑐𝑟𝑘 is an adjustment coefficient for crack flow, 𝐹𝐶𝑙𝑦 is the water content of the soil layer at field ca-

78


pacity [𝑚𝑚 𝐻2 𝑂], and 𝑆𝑊𝑙𝑦 is the water content of the soil layer on a given day [𝑚𝑚 𝐻2 𝑂]. The adjustment coefficient for crack flow, 𝑐𝑜𝑒𝑓𝑐𝑟𝑘 , is set to 0.10 as default value. When the soil is wetting and/or when the moisture content of the profile is above 90% of the field capacity water content, the crack volume for a given day is equal to the volume calculated with Eq. (6-18). 𝑓𝑜𝑟 𝑆𝑊 < 0.90 ⋅ 𝐹𝐶 𝑎𝑛𝑑 𝑐𝑟𝑘𝑙𝑦,𝑖 > 𝑐𝑟𝑘𝑙𝑦,𝑑−1 ∶

𝑐𝑟𝑘𝑙𝑦 = 𝑙𝑐𝑟𝑘 ⋅ 𝑐𝑟𝑘𝑙𝑦,𝑑−1 + (1.0 − 𝑙𝑐𝑟𝑘 ) ⋅ 𝑐𝑟𝑘𝑙𝑦,𝑖

(6-19)

𝑓𝑜𝑟 𝑆𝑊 > 0.90 ⋅ 𝐹𝐶 𝑜𝑟 𝑐𝑟𝑘𝑙𝑦,𝑖 < 𝑐𝑟𝑘𝑙𝑦,𝑑−1 ∶

𝑐𝑟𝑘𝑙𝑦 = 𝑐𝑟𝑘𝑙𝑦,𝑖

(6-20)

𝑐𝑟𝑘𝑙𝑦 is the crack volume for the soil layer on a given day expressed as a depth [𝑚𝑚], 𝑙𝑐𝑟𝑘 is the lag factor for crack development during drying, 𝑐𝑟𝑘𝑙𝑦,𝑑−1 is the crack volume for the soil layer on the previous day [𝑚𝑚], 𝑐𝑟𝑘𝑙𝑦,𝑖 is the initial crack volume calculated for the soil layer on a given day. As the tension at which water is held by the soil particles increases, the rate of water diffusion slows. Because the rate of water diffusion is analogous to the coefficient of consolidation in classical consolidation theory, the reduction in diffusion will affect crack formation. The lag factor is introduced during the drying stage to account for the change in moisture redistribution dynamics that occurs as the soil dries. The lag factor 𝑙𝑐𝑟𝑘 , is set to a value of 0.99. The maximum crack volume for the layer, 𝑐𝑟𝑘𝑚𝑎𝑥,𝑙𝑦 is calculated as:

𝑐𝑟𝑘𝑚𝑎𝑥,𝑙𝑦 = 0.916 ⋅ 𝑐𝑟𝑘𝑚𝑎𝑥 ⋅ exp[−0.0012 ⋅ 𝑧𝑙,𝑙𝑦 ] ⋅ 𝑑𝑒𝑝𝑡ℎ𝑙𝑦

(6-21)

𝑐𝑟𝑘𝑚𝑎𝑥,𝑙𝑦 is the maximum crack volume possible for the soil layer [𝑚𝑚], 𝑐𝑟𝑘𝑚𝑎𝑥 is the potential crack volume for the soil profile expressed as a fraction of the total volume, 𝑧𝑙,𝑙𝑦 is the depth from the soil surface to the bottom of the soil layer [𝑚𝑚], and 𝑑𝑒𝑝𝑡ℎ𝑙𝑦 is the depth of the soil layer [𝑚𝑚]. The potential crack volume for the soil profile, 𝑐𝑟𝑘𝑚𝑎𝑥 , is defined by the user and based on, e.g. measurements. When all crack volumes for all layers 𝑛 are calculated, the total crack volume 𝑐𝑟𝑘 - expressed as depth [𝑚𝑚] - for the soil profile is determined as: 𝑛

𝑐𝑟𝑘 = ∑ 𝑐𝑟𝑘𝑙𝑦

(6-22)

𝑙𝑦=1

In the next step the initial surface runoff 𝑄𝑠𝑢𝑟𝑓,𝑖 using the curve number or Green and Ampt method is calculated, the amount of runoff is reduced by the volume of cracks on the present day:

79

6 Vertical Soil Water Movement 6.5 Perched Water Table

𝑖𝑓 𝑄𝑠𝑢𝑟𝑓,𝑖 > 𝑐𝑟𝑘: 𝑄𝑠𝑢𝑟𝑓 = 𝑄𝑠𝑢𝑟𝑓,𝑖 − 𝑐𝑟𝑘 𝑖𝑓 𝑄𝑠𝑢𝑟𝑓,𝑖 < 𝑐𝑟𝑘: 𝑄𝑠𝑢𝑟𝑓 = 0

(6-23) (6-24)

Finally, the bypass flow 𝑤𝑐𝑟𝑘,𝑏𝑡𝑚 [𝑚𝑚 𝐻2 𝑂] through the soil profile can be calculated as:

𝑤𝑐𝑟𝑘,𝑏𝑡𝑚 = 0.5 ⋅ 𝑐𝑟𝑘 ⋅ (

𝑐𝑟𝑘𝑙𝑦,𝑒𝑛𝑑 ) 𝑑𝑒𝑝𝑡ℎ𝑙𝑦,𝑒𝑛𝑑

(6-25)

𝑐𝑟𝑘𝑙𝑦,𝑒𝑛𝑑 is the crack volume of the deepest layer [𝑚𝑚] and 𝑑𝑒𝑝𝑡ℎ𝑙𝑦,𝑒𝑛𝑑 is the depth of the deepest layer [𝑚𝑚]. When 𝑤𝑐𝑟𝑘,𝑏𝑡𝑚 is calculated, each soil layer is filled to field capacity with water, beginning with the lowest layer and moving upward until the cracks are fully filled with water.

6.5 Perched Water Table A perched water table (Figure 6-3) is an accumulation of groundwater that is above the water table in the unsaturated zone. The groundwater is usually trapped above an impermeable soil layer, such as clay, and actually forms a lens of saturated material in the unsaturated zone. For an HRU with a seasonal high water table, if the soil profile becomes saturated to the point that percolation for upper soil layers to lower soil layers is inhibited, water will pond in the soil profile and create a perched water table. SWAT allows the user to define the depth to an impervious layer for the HRU. If the depth to the impervious layer is in the soil profile, no water is allowed to percolate out of the soil profile (see Figure 6-3). If the impervious layer is defined below the soil profile, percolation out of the soil profile is adjusted from the value determined with Eq. (6-15) using:

𝑤𝑝𝑒𝑟𝑐,𝑏𝑚 = 𝑤𝑝𝑒𝑟𝑐,𝑏𝑡𝑚,𝑜𝑟𝑖𝑔 ∙

𝑑𝑒𝑝𝑡ℎ𝑑𝑖𝑓𝑓 𝑑𝑒𝑝𝑡ℎ𝑑𝑖𝑓𝑓 + 𝑒𝑥𝑝[8.833 − 2.598 ∙ 𝑑𝑒𝑝𝑡ℎ𝑑𝑖𝑓𝑓 ]

(6-26)

where 𝑤𝑝𝑒𝑟𝑐,𝑏𝑡𝑚 is the amount of water percolating out of the soil profile on a given day [𝑚𝑚 𝐻2 𝑂], 𝑤𝑝𝑒𝑟𝑐,𝑏𝑡𝑚,𝑜𝑟𝑖𝑔 is the amount of water percolating out of the soil profile on a given day calculated with Eq. (6-15) [𝑚𝑚 𝐻2 𝑂], and 𝑑𝑒𝑝𝑡ℎ𝑑𝑖𝑓𝑓 is the distance from the bottom of the soil profile to the impervious layer [𝑚]. Water builds up in the soil profile from the bottom of the profile. After the bottom layer of the profile reaches saturation, any water exceeding the storage capacity of the bottom layer is allowed to fill the overlying layer. This continues upward until all the excess water has been distributed. The height of the perched water table is calculated:

80

6 Vertical Soil Water Movement 6.6 Nomenclature

ℎ𝑤𝑡𝑏𝑙 =

𝑆𝑊 − 𝐹𝐶 ∙ 𝑑𝑒𝑝𝑡ℎ𝑖𝑚𝑝 (𝑃𝑂𝑅 − 𝐹𝐶) ∙ (1 − 𝜙𝑎𝑖𝑟 )

(6-27)

ℎ𝑤𝑡𝑏𝑙 is the height of the water table [𝑚𝑚], 𝑆𝑊 is the water content of the soil profile [𝑚𝑚 𝐻2 𝑂], FC is the water content of the soil profile at field capacity [𝑚𝑚 𝐻2 𝑂], 𝑃𝑂𝑅 is the porosity of the soil profile [𝑚𝑚], 𝜙𝑎𝑖𝑟 is the air-filled porosity expressed as a fraction, and 𝑑𝑒𝑝𝑡ℎ𝑖𝑚𝑝 is the depth to the impervious layer [𝑚𝑚].

Figure 6-3: Perched water table

6.6 Nomenclature 𝐴𝑊𝐶

Plant available water content [𝑚𝑚 𝐻2 𝑂]

𝐴𝑊𝐶𝑙𝑦

Available water capacity of the soil layer expressed as a fraction of the total soil volume [𝑚𝑚 𝐻2 𝑂]

𝑐𝑜𝑒𝑓𝑐𝑟𝑘

Adjustment coefficient for crack flow [−]

𝑐𝑟𝑘𝑙𝑦,𝑑−1

Crack volume for the soil layer on the previous day [𝑚𝑚]

𝑐𝑟𝑘𝑙𝑦,𝑒𝑛𝑑

Crack volume of the deepest layer [𝑚𝑚]

𝑐𝑟𝑘𝑙𝑦,𝑖 𝑐𝑟𝑘𝑚𝑎𝑥,𝑙𝑦 𝑐𝑟𝑘𝑚𝑎𝑥

Initial crack volume calculated for the soil layer [𝑚𝑚] Maximum crack volume possible for the soil layer [𝑚𝑚] potential crack volume for the soil profile [𝑚𝑚]

𝑑𝑒𝑝𝑡ℎ𝑑𝑖𝑓𝑓

Distance from the bottom of the soil profile to the impervious layer [𝑚]

𝑑𝑒𝑝𝑡ℎ𝑖𝑚𝑝

Depth to the impervious layer [𝑚𝑚]

𝑑𝑒𝑝𝑡ℎ𝑙𝑦,𝑒𝑛𝑑 𝑑𝑒𝑝𝑡ℎ𝑙𝑦 𝐸𝑡

Depth of the deepest layer [𝑚𝑚] Depth of the soil layer [𝑚𝑚] Maximum transpiration rate [𝑚𝑚/𝑑]

𝐸𝑡,𝑎𝑐𝑡

Actual transpiration [𝑚𝑚𝐻2 𝑂]

𝑒𝑝𝑐𝑜

Plant water uptake parameter

𝐹𝐶

Water content at field capacity 𝑚𝑚 𝐻2 𝑂]

81


𝐹𝐶𝑙𝑦 𝐹𝐶𝑙𝑦+1

Water content at field capacity expressed as a fraction of the total soil volume [𝑚𝑚 𝐻2 𝑂] Water content of the underlying soil layer at field capacity [𝑚𝑚 𝐻2 𝑂]

ℎ𝑤𝑡𝑏𝑙

Height of the water table [𝑚𝑚]

𝐾𝑠𝑎𝑡

Saturated hydraulic conductivity for the layer [𝑚𝑚 ℎ−1 ]

𝑙𝑐𝑟𝑘

Lag factor for crack development during drying [−]

𝑀𝑠

Mass of the solids in the soil [𝑡𝑜𝑛]

𝑚𝑐

Percent clay content of the layer [%]

𝑃𝑂𝑅

Porosity of the soil profile [𝑚𝑚]

𝑄𝑠𝑢𝑟𝑓,𝑖

Initial surface runoff [𝑚𝑚 𝐻2 𝑂]

𝑆𝐴𝑇𝑙𝑦

Amount of water in the soil layer when completely saturated [𝑚𝑚 𝐻2 𝑂]

𝑆𝐴𝑇𝑙𝑦+1

Amount of water in the underlying soil layer when completely saturated [𝑚𝑚 𝐻2 𝑂]

𝑆𝑊𝑙𝑦 𝑆𝑊𝑙𝑦+1 𝑆𝑊𝑙𝑦,𝑒𝑥𝑐𝑒𝑠𝑠

Soil water content of layer 𝑙𝑦 [𝑚𝑚𝐻2 𝑂] Water content of the underlying soil layer [𝑚𝑚 𝐻2 𝑂] Drainable volume of water in the soil layer on a given day [𝑚𝑚 𝐻2 𝑂]

𝑇𝑇𝑝𝑒𝑟𝑐

Travel time for percolation [ℎ]

𝑉𝑇

Total volume of the soil [𝑚3 ]

𝑤𝑎𝑐𝑐𝑡𝑢𝑎𝑙𝑢𝑝

Actual water uptake [𝑚𝑚𝐻2 𝑂]

𝑤𝑐𝑟𝑘,𝑏𝑡𝑚

Bypass flow [𝑚𝑚 𝐻2 𝑂]

𝑤𝑑𝑒𝑚𝑎𝑛𝑑

Water uptake demand not met by overlying soil layers [𝑚𝑚𝐻2 𝑂]

𝑤𝑝𝑒𝑟𝑐,𝑏𝑡𝑚,𝑜𝑟𝑖𝑔

Amount of water percolating out of the soil profile on a given day [𝑚𝑚 𝐻2 𝑂], calculated with Eq. (6-15)

𝑤𝑝𝑒𝑟𝑐,𝑏𝑡𝑚

Amount of water percolating out of the soil profile on a given day [𝑚𝑚 𝐻2 𝑂]

𝑤𝑝𝑒𝑟𝑐,𝑙𝑦 𝑤𝑢𝑝,𝑙𝑦

Potential water uptake for the specific layer 𝑙𝑦 [𝑚𝑚𝐻2 𝑂]

𝑤𝑢𝑝,𝑙𝑦 ′

Adjusted potential water uptake [𝑚𝑚𝐻2 𝑂] for layer 𝐿𝑦

𝑤 ′′ 𝑢𝑝,𝑙𝑦

Potential water uptake adjusted for initial soil water content [𝑚𝑚𝐻2 𝑂]

𝑤𝑢𝑝,𝑧

Potential water uptake from the soil surface to a specific depth z [𝑚𝑚𝐻2 𝑂]

𝑤𝑢𝑝,𝑧𝑙

Potential water uptake for the profile to the lower boundary [𝑚𝑚𝐻2 𝑂]

𝑤𝑢𝑝,𝑧𝑢

Potential water uptake for the upper boundary of the soil layer [𝑚𝑚𝐻2 𝑂]

𝑊𝑃 82

Amount of water that percolates to the next layer [𝑚𝑚 𝐻2 𝑂]

Water content at permanent wilting point


𝑊𝑃𝑙𝑦 𝑧

Water content at wilting point expressed as a fraction of the total soil volume Depth below soil surface [𝑚𝑚]

𝑧𝑙,𝑙𝑦

Depth from the soil surface to the bottom of the soil layer [𝑚𝑚]

𝑧𝑟𝑜𝑜𝑡

Depth of root development in the soil [𝑚𝑚]

Δ𝑡

Length of the time step [ℎ]

𝛽𝑤

Water use distribution parameter

𝜌𝑏

Soil’s bulk density [𝑡𝑜𝑛/𝑚3 ]

𝜌𝑠

Density of the solid fraction [𝑡𝑜𝑛/𝑚3]

𝜙𝑎𝑖𝑟

Air-filled porosity expressed as a fraction [−]

𝜙𝑠𝑜𝑖𝑙

Soil porosity [%]

83

7 Lateral Flows 7.1 Surface Runoff

7 Lateral Flows Understanding the mechanisms by which catchments route vertical water inputs laterally to stream channels is crucial to the development of accurate predictive models of watershed processes. It is commonly assumed that lateral redistribution occurs as overland or subsurface flow. In this chapter the most important forms of lateral water flows are presented.

7.1 Surface Runoff Surface runoff occurs whenever the rate of water application to the ground surface exceeds the rate of infiltration.

7.1.1 Peak Runoff Rate The peak runoff rate is the maximum runoff flow rate that occurs with a given rainfall event. The peak runoff rate is an indicator of the erosive power of a storm and is used to predict sediment loss. SWAT calculates the peak runoff rate with a modified rational method. The rational method is widely used in the design of ditches, channels and storm water control systems. The rational method is based on the assumption that if a rainfall of intensity 𝑖 begins at time 𝑡 = 0 and continues indefinitely, the rate of runoff will increase until the time of concentration, 𝑡 = 𝑡𝑐𝑜𝑛𝑐 [h]; when the entire sub basin area is contributing to flow at the outlet. The equation is:

𝑞𝑝𝑒𝑎𝑘 =

𝐶 ⋅ 𝑖 ⋅ 𝐴𝑟𝑒𝑎 3.6

(7-1)

Where 𝑞𝑝𝑒𝑎𝑘 is the peak runoff rate [m³/s], 𝐶 is the runoff coefficient [-], 𝑖 is the rainfall intensity [mm/hr], 𝐴𝑟𝑒𝑎 is the subbasin area [km²] and 3.6 is a unit conversion factor.

7.1.2 Time of concentration The time of concentration 𝑡𝑐𝑜𝑛𝑐 is the time from the beginning of a rainfall event until the entire sub basin area is contributing to flow at the outlet. This means, the time of concentration is the time for a drop of water to flow from the most remote point in the sub basin to the outlet. The time of concentration is calculated by summing the overland flow time and the channel flow time (time it takes for flow in the upstream channels to reach the outlet):

𝑡𝑐𝑜𝑛𝑐 = 𝑡𝑜𝑣 + 𝑡𝑐ℎ

84

(7-2)


7.1.2.1 Overland Flow Time of Concentration The overland flow time of concentration can be computed using the equation:

𝑡𝑜𝑣 =

𝐿𝑠𝑙𝑝 3600 ∙ 𝑣𝑜𝑣

(7-3)

Where 𝐿𝑠𝑙𝑝 is the subbasin slope length [m], 𝑣𝑜𝑣 is the overland flow velocity [m/s] and 3600 is a unit conversion factor. The overland flow velocity can be estimated from Manning´s equation by

𝑣𝑜𝑣

𝑞𝑜𝑣 0.4 ∙ 𝑠𝑙𝑝0.3 = 𝑛0.6

(7-4)

Where 𝑞𝑜𝑣 is the average overland flow rate [m³/s], 𝑠𝑙𝑝 is the average slope in the subbasin [m/m], and 𝑛 is Manning´s roughness coefficent. Assuming an average flow rate of 6.35 [mm/h] and converting units

𝑣𝑜𝑣

0.005 ∙ 𝐿𝑠𝑙𝑝 0.4 ∙ 𝑠𝑙𝑝0.3 = 𝑛0.6

(7-5)

Yields with substitution to:

𝑡𝑜𝑣 =

𝐿𝑠𝑙𝑝 0.6 ∙ 𝑛0.6 18 ∙ 𝑠𝑙𝑝0.3

(7-6)

Table 7-1 shows typical roughness coefficients 𝑛 for overland flow. Table 7-1: Values of Manning´s roughness coefficient for overland flow

85


7.1.2.2 Channel Flow Time of Concentration The channel flow time of concentration 𝑡𝑐ℎ [ℎ] can be calculated as:

𝑡𝑐ℎ =

𝐿𝑐 3.6 ∙ 𝑣𝑐

(7-7)

Where 𝐿𝑐 is the average flow channel length for the sub basin [𝑘𝑚], 𝑣𝑐 is the average channel velocity [𝑚/𝑠], and 3.6 is a unit conversion factor. The average channel flow length can be estimated as:

𝐿𝑐 = √𝐿 ⋅ 𝐿𝑐𝑒𝑛 (7-8) With 𝐿 as the channel length from the most distant point tot he subbasin outlet [𝑘𝑚], and 𝐿𝑐𝑒𝑛 is the distance along the channel to the subbasin centroid [𝑘𝑚]. Assuming 𝐿𝑐𝑒𝑛 = 0.5 ⋅ 𝐿, the average channel flow length is:

𝐿𝑐 = 0.71 ∙ 𝐿

(7-9)

The average velocity can be estimated from Manning´s equation under the assumption of a trapezoidal channel with 2:1 sides and a 10:1 bottom width-depth ratio.

0.489 ∙ 𝑞𝑐ℎ 0.25 ∙ 𝑠𝑙𝑝𝑐ℎ 0.375 𝑣𝑐 = 𝑛0.75

(7-10)

𝑣𝑐 is the average velocity [𝑚/𝑠], 𝑞𝑐ℎ is the average channel flow rate [𝑚³/𝑠], 𝑠𝑙𝑝𝑐ℎ is the channel slope [𝑚/𝑚], and 𝑛 is the roughness coefficient. To express the average channel flow rate in units of 𝑚𝑚/ℎ, the following expression is used:

𝑞𝑐ℎ =

𝑞𝑐ℎ ∗ ∙ 𝐴𝑟𝑒𝑎 3.6

(7-11)

𝑞𝑐ℎ ∗ accounts for the average channel flow rate in 𝑚𝑚/ℎ, 𝐴𝑟𝑒𝑎 is the sub basin area [𝑘𝑚²], and 3.6 is the unit conversion factor. The average channel flow rate is related to the unit source area flow rate:

𝑞𝑐ℎ ∗ = 𝑞0 ∗ ∙ (100 ∙ 𝐴𝑟𝑒𝑎)

−0.5

(7-12)

With 𝑞0 ∗ being the source area flow rate [𝑚𝑚/ℎ], and 100 as a conversion factor. Assuming the unit source area flow rate is 6.35 [𝑚𝑚/ℎ], rearranging the previous equations it yields:

86


0.317 ∙ 𝐴𝑟𝑒𝑎0.125 ∙ 𝑠𝑙𝑝𝑐ℎ 0.375 𝑣𝑐 = 𝑛0.75

(7-13)

And finally leads to the time of concentration for channel flow 𝑡𝑐ℎ [ℎ]:

𝑡𝑐ℎ =

0.62 ∙ 𝐿 ∙ 𝑛0.75 𝐴𝑟𝑒𝑎0.125 ∙ 𝑠𝑙𝑝𝑐ℎ 0.375

(7-14)

7.1.3 Runoff Coefficent The runoff coefficient is the ratio of the inflow rate (𝑖 ⋅ 𝑎𝑟𝑒𝑎) to the peak discharge rate (𝑞𝑝𝑒𝑎𝑘 ). It varies from storm to storm and is calculated with the equation:

𝐶=

𝑄𝑠𝑢𝑟𝑓 𝑅𝑑𝑎𝑦

(7-15)

Where 𝑄𝑠𝑢𝑟𝑓 is the surface runoff [𝑚𝑚 𝐻2 𝑂] and 𝑅𝑑𝑎𝑦 is the rainfall for the day [𝑚𝑚 𝐻2 𝑂].

7.1.4 Rainfall intesity The rainfall intensity 𝑖 [𝑚𝑚/ℎ] is the average rainfall rate during the time of concentration. Based on this definition, it can be calculated with the equation:

𝑖=

𝑅𝑡𝑐 𝑡𝑐𝑜𝑛𝑐

(7-16)

𝑅𝑡𝑐 [𝑚𝑚 𝐻2 𝑂] denotes the amount of rain falling during the time of concentration, 𝑡𝑐𝑜𝑛𝑐 is the time of concentration for the subbasin [ℎ]. The amount of rain falling during the time of concentration is proportional to the amount of rain falling during the 24h period:

𝑅𝑡𝑐 = 𝛼𝑡𝑐 ∙ 𝑅𝑑𝑎𝑦

(7-17)

With 𝑅𝑑𝑎𝑦 as the amount of rain falling during the day [𝑚𝑚 𝐻2 𝑂] and 𝛼𝑡𝑐 is the fraction of daily rainfall during the time of concentration. For short storm events most of the rain will fall during the time of concentration, causing 𝛼𝑡𝑐 to approach to 1.0. The minimum value of 𝛼𝑡𝑐 would appear in storms of uniform intensity (𝑖24 = 𝑖):

𝛼𝑡𝑐,𝑚𝑖𝑛 =

𝑅𝑡𝑐 𝑖 ∙ 𝑡𝑐𝑜𝑛𝑐 𝑡𝑐𝑜𝑛𝑐 = = 𝑅𝑑𝑎𝑦 𝑖24 ∙ 24 24

(7-18)

87


In SWAT the fractions of rain falling in the time of concentration as a function of the fraction of daily rain falling in the half-hour of highest intensity, are estimated as:

𝛼𝑡𝑐 = 1 − 𝑒𝑥𝑝 [2 ∙ 𝑡𝑐𝑜𝑛𝑐 ∙ ln(1 − 𝛼0.5 )]

(7-19)

With 𝛼0.5 as the fraction of daily rainfall that occurs during the time of concentration.

7.1.5 Modified Rational Formula The modified rational formula is used to estimate the peak flow 𝑞𝑝𝑒𝑎𝑘 as follows:

𝑞𝑝𝑒𝑎𝑘 =

𝛼𝑡𝑐 ∙ 𝑄𝑠𝑢𝑟𝑓 ∙ 𝐴𝑟𝑒𝑎 3.6 ∙ 𝑡𝑐𝑜𝑛𝑐

(7-20)

Finally it only requires knowledge on 𝛼𝑡𝑐 , 𝑄𝑠𝑢𝑟𝑓 , 𝐴𝑟𝑒𝑎 and 𝑡𝑐𝑜𝑛𝑐 to calculate the peak flow.

7.1.6 Surface Runoff Lag SWAT incorporates a surfacerunoff storage feature to lag a portion of the surface runoff release to the main channel, in case of large subbasins with a time of concentration greater than 1 day. The amount of surface runoff 𝑄′𝑠𝑢𝑟𝑓 is either calculated with Green and Ampt or the Curve Number, then the amount of surface runoff 𝑄𝑠𝑢𝑟𝑓 [𝑚𝑚 𝐻2 𝑂] discharged to the main channel is calculated as:

𝑄𝑠𝑢𝑟𝑓 = (𝑄′𝑠𝑢𝑟𝑓 + 𝑄𝑠𝑡𝑜𝑟,𝑖−1 ) ∙ (1 − 𝑒𝑥𝑝 ⌊

−𝑠𝑢𝑟𝑙𝑎𝑔 ⌋) 𝑡𝑐𝑜𝑛𝑐

(7-21)

𝑄𝑠𝑡𝑜𝑟,𝑖−1 is the surface runoff stored or lagged from the previous day, 𝑠𝑢𝑟𝑙𝑎𝑔 is the surface runoff lag coefficient, and 𝑡𝑐𝑜𝑛𝑐 is the time of concentration for the subbasin. The term

(1 − 𝑒𝑥𝑝 ⌊

−𝑠𝑢𝑟𝑙𝑎𝑔 ⌋) 𝑡𝑐𝑜𝑛𝑐

expresses the fraction of total available water that enters the stream on the day 𝑄𝑠𝑢𝑟𝑓 is generated. Figure 7-1 shows plots for this expression with different values for 𝑠𝑢𝑟𝑙𝑎𝑔. The more 𝑠𝑢𝑟𝑙𝑎𝑔 decreases in value, the more water is held in storage. The delay in release of surface runoff will smooth the stream flow hydrograph simulated in the stream.

88


Figure 7-1: Influence of 𝑠𝑢𝑟𝑙𝑎𝑔 and 𝑡𝑐𝑜𝑛𝑐 on fraction of surface runoff released

7.1.7 Transmission Losses In arid and semi-arid watersheds ephemeral channels abstract large amounts of the stream flow. This transmission losses reduce the volume of the runoff. SWAT takes account of this with the assumptions that there is no lateral inflow and no out-of-bank flow contribution to runoff. Prediction equation for runoff volume 𝑉𝑜𝑙𝑄𝑠𝑢𝑟𝑓,𝑓 after transmission losses

𝑉𝑜𝑙𝑄𝑠𝑢𝑟𝑓,𝑓 = {

𝑣𝑜𝑙𝑄𝑠𝑢𝑟𝑓,𝑖 𝑣𝑜𝑙𝑄𝑠𝑢𝑟𝑓,𝑖

0 𝑎𝑥 + 𝑏𝑥 ⋅ 𝑣𝑜𝑙𝑄𝑠𝑢𝑟𝑓,𝑖

≤ 𝑣𝑜𝑙𝑡ℎ𝑟 > 𝑣𝑜𝑙𝑡ℎ𝑟

(7-22)

With 𝑣𝑜𝑙𝑄𝑠𝑢𝑟𝑓,𝑖 as the runoff volume prior to the losses [𝑚³], 𝑎𝑥 is the regression intercept for channel of length L and width W [𝑚³], 𝑏𝑥 is the regression slope for channel of length L and width W, 𝑣𝑜𝑙𝑡ℎ𝑟 threshold volume for a channel of length L and width W [𝑚³]:

𝑣𝑜𝑙𝑡ℎ𝑟 = −

𝑎𝑥 𝑏𝑥

(7-23)

89

7 Lateral Flows 7.2 Subsurface Flow and Lag

The corresponding equation for peak

𝑞𝑝𝑒𝑎𝑘,𝑓 =

1 (3600 ∙ 𝑑𝑢𝑟𝑓𝑙𝑤 )

∙ [𝑎𝑥 − (1 − 𝑏𝑥 ) ∙ 𝑣𝑜𝑙𝑄𝑠𝑢𝑟𝑓,𝑖 ] + 𝑏𝑥 ∙ 𝑞𝑝𝑒𝑎𝑘,𝑖

(7-24)

Where 𝑞𝑝𝑒𝑎𝑘,𝑓 is again the peak rate after transmission losses [𝑚³/𝑠], 𝑄𝑠𝑢𝑟𝑓 is the accumulated runoff [𝑚𝑚 𝐻2 𝑂] 𝑑𝑢𝑟𝑓𝑙𝑤 is the flow duration [ℎ]:

𝑑𝑢𝑟𝑓𝑙𝑤 =

𝑄𝑠𝑢𝑟𝑓 ⋅ 𝐴𝑟𝑒𝑎 3.6 ∙ 𝑞𝑝𝑒𝑎𝑘

(7-25)

To calculate the regression parameters for channels of differing lengths and widths, the parameters of a unit channel are needed. A unit channel is defined as a channel of length L = 1 km and width W = 1 m. The unit channel parameters are calculated as the following:

𝑘𝑟 = −2.22 ∙ 𝑙𝑛 [1 − 2.6466 ∙

𝐾𝑐ℎ ∙ 𝑑𝑢𝑟𝑓𝑙𝑤 ] 𝑣𝑜𝑙𝑄𝑠𝑢𝑟𝑓,𝑖

(7-26)

𝑎𝑟 = −0.2258 ∙ 𝐾𝑐ℎ ∙ 𝑑𝑢𝑟𝑓𝑙𝑤

(7-27)

𝑏𝑟 = 𝑒𝑥𝑝[−0.4905 ∙ 𝑘𝑟 ]

(7-28)

𝑘𝑟 is the decay factor [1/𝑘 𝑘𝑚], 𝑎𝑟 is the unit channel regression intercept [𝑚³], 𝑏𝑟 is the unit channel regression slope, 𝐾𝑐ℎ is the effective hydraulic conductivity of the natural channale [𝑚𝑚/ℎ]. The regression parameters are:

𝑏𝑥 = 𝑒𝑥𝑝[−𝑘𝑟 ∙ 𝐿 ∙ 𝑊] 𝑎𝑥 =

𝑎𝑟 ∙ (1 − 𝑏𝑥 ) (1 − 𝑏𝑟 )

(7-29) (7-30)

With 𝑎𝑥 is the regression intercept [𝑚3 ] for a channel of length L and width W, 𝑏𝑥 is the regression slope for a channel of length 𝐿 and width 𝑊 , 𝑘𝑟 is the decay factor [1⁄𝑚 𝑘𝑚]. Transmission losses from the surface are assumed to percolate into the shallow aquifer.

7.2 Subsurface Flow and Lag 7.2.1 Subsurface Flow In soils with high hydraulic conductivity lateral flow is a significant transport process. Rainfall percolates vertically, at first, until it encounters an impermeable layer. The water then ponds upon the impermeable layer forming a saturated zone of water, i.e. a perched water table. This saturated zone is the source for water for lateral subsurface flow. 90


SWAT incorporates a kinematic storage model or subsurface flow, which simulates subsurface flow in a two-dimensional cross-section along a flow path down a steep hill slope. The kinematic approximation is used in its derivation. Figure 7-2 shows the setup of the hills lope segment. It has a permeable soil layer of depth 𝐷𝑝𝑒𝑟𝑚 and length 𝐿ℎ𝑖𝑙𝑙 with an impermeable soil layer or boundary beneath it.

Figure 7-2: Behavior of the water table as assumed in the kinematic storage model.

The kinematic wave approximation of saturated subsurface flow assumes the stream lines of flow in the saturated zone to be parallel to the impermeable boundary layer. Furthermore, the hydraulic gradient equals the slope of the bed. The drainable volume of water stored in the saturated zone of the hillslope segment 𝑆𝑊𝑙𝑦,𝑒𝑥𝑐𝑒𝑠𝑠 [𝑚𝑚 𝐻2 𝑂] is:

𝑆𝑊𝑙𝑦,𝑒𝑥𝑐𝑒𝑠𝑠 =

1000 ∙ 𝐻0 ∙ 𝜙𝑑 ∙ 𝐿ℎ𝑖𝑙𝑙 2

(7-31)

with 𝐻0 as the saturated thickness [𝑚𝑚/𝑚𝑚] normal to the hillslope at the outlet expressed as a fraction of the total thickness, 𝜙𝑑 as the drainable porosity of the soil [𝑚𝑚/𝑚𝑚], 𝐿ℎ𝑖𝑙𝑙 is the hillslope length [𝑚] and 1000 is a conversionfactor. To solve for 𝐻0 the equation can be rearranged as:

𝐻0 =

2 ∙ 𝑆𝑊𝑙𝑦,𝑒𝑥𝑐𝑒𝑠𝑠 1000 ∙ 𝜙𝑑 ∙ 𝐿ℎ𝑖𝑙𝑙

(7-32)

91


The drainable porosity of the soil layer can be calculated as:

𝜙𝑑 = 𝜙𝑠𝑜𝑖𝑙 − 𝜙𝑓𝑐

(7-33)

𝜙𝑑 is the drainable porosity of the soil layer [𝑚𝑚/𝑚𝑚], 𝜙𝑠𝑜𝑖𝑙 is the total porosity of the soil layer [𝑚𝑚/𝑚𝑚], and 𝜙𝑓𝑐 is the porosity of the soil layer filled with water when the layer is at field capacity water content [𝑚𝑚/𝑚𝑚]. The net discharge at the hill slope exit, 𝑄𝑙𝑎𝑡 [𝑚𝑚 𝐻2 𝑂/𝑑𝑎𝑦] is given as:

𝑄𝑙𝑎𝑡 = 24 ∙ 𝐻0 ∙ 𝑣𝑙𝑎𝑡

(7-34)

Where 24 is the factor to convert hours to day and 𝑣𝑙𝑎𝑡 is the velocity flow at the outlet [mm/h]: 𝑣𝑙𝑎𝑡 = 𝐾𝑠𝑎𝑡 ∙ sin(𝛼ℎ𝑖𝑙𝑙 )

(7-35)

𝛼ℎ𝑖𝑙𝑙 is the angle of the hill slope. All terms can now be combined to obtain the discharge 𝑄𝑙𝑎𝑡 :

2 ∙ 𝑆𝑊𝑙𝑦,𝑒𝑥𝑐𝑒𝑠𝑠 ∙ 𝐾𝑠𝑎𝑡 ∙ 𝑠𝑙𝑝 𝑄𝑙𝑎𝑡 = 0.024 ( ) 𝜙𝑑 ∙ 𝐿ℎ𝑖𝑙𝑙

(7-36)

7.2.2 Flow Lag In very large basins where the time of concentration is greater than 1 day, only a percentage of the lateral flow will exit the slope as discharge 𝑄𝑙𝑎𝑡 . SWAT takes the lag into account with a lateral flow storage feature. The amount of lateral flow released to the channel at a given day is then calculated as:

𝑄𝑙𝑎𝑡 = (𝑄𝑙𝑎𝑡 ′ ∙ 𝑄𝑙𝑎𝑡𝑠𝑡𝑜𝑟, 𝑖−1 ) ∙ (1 − 𝑒𝑥𝑝 ⌊

−1 ⌋) 𝑇𝑇𝑙𝑎𝑔

(7-37)

𝑄𝑙𝑎𝑡 ′ is the amount of lateral flow generated in the subbasin on a given day [𝑚𝑚 𝐻2 𝑂], 𝑄𝑙𝑎𝑡𝑠𝑡𝑜𝑟, 𝑖−1 is the lateral flow stored from the previous day [𝑚𝑚 𝐻2 𝑂 ], 𝑇𝑇𝑙𝑎𝑔 accounts for the lateral flow travel time in days and can be calculated as:

𝑇𝑇𝑙𝑎𝑔 =

𝑡𝑖𝑙𝑒𝑙𝑎𝑔 24

If tile drainages 𝑡𝑖𝑙𝑒𝑙𝑎𝑔 are present in the basin; if they are not present it is:

92

(7-38)


𝑇𝑇𝑙𝑎𝑔 = 10.4 ∙

𝐿ℎ𝑖𝑙𝑙 𝐾𝑠𝑎𝑡,𝑚𝑥

(7-39)

With 𝐾𝑠𝑎𝑡,𝑚𝑥 beeing the highest saturated hydraulic conductivity in the soil. The expression: −1 ⌋) (1 − 𝑒𝑥𝑝 ⌊ 𝑇𝑇𝑙𝑎𝑔

represents the fraction of the total available water that will be allowed to enter the stream. Figure 7-3 shows the influence of 𝑇𝑇𝑙𝑎𝑔 on the lateral flow.

Figure 7-3: Influence of 𝑇𝑇𝑙𝑎𝑔 on the fraction of lateral flow released.

93

7 Lateral Flows 7.3 Base Flow

7.3 Base Flow 7.3.1 Shallow Aquifer 7.3.1.1 Water Balance The water balance in the shallow aquifer can be described as:

𝑎𝑞𝑠ℎ,𝑖 = 𝑎𝑞𝑠ℎ,𝑖−1 + 𝑤𝑟𝑐ℎ𝑟𝑔,𝑠ℎ − 𝑄𝑔𝑤 − 𝑤𝑟𝑒𝑣𝑎𝑝 − 𝑤𝑝𝑢𝑚𝑝,𝑠ℎ

(7-40)

where 𝑎𝑞𝑠ℎ,𝑖 is the amount of water stored in the shallow aquifer on day 𝑖 [𝑚𝑚 𝐻2 𝑂], 𝑎𝑞𝑠ℎ,𝑖−1 is the amount of water stored in the shallow aquifer on day 𝑖 − 1 [𝑚𝑚 𝐻2 𝑂], 𝑤𝑟𝑐ℎ𝑟𝑔,𝑠ℎ is the amount of recharge entering the shallow aquifer on day 𝑖 [𝑚𝑚 𝐻2 𝑂], 𝑄𝑔𝑤 is the ground water flow, or base flow, into the main channel on day 𝑖 [𝑚𝑚 𝐻2 𝑂], 𝑤𝑟𝑒𝑣𝑎𝑝 is the amount of water moving into the soil zone in response to water deficiencies on day 𝑖 [𝑚𝑚 𝐻2 𝑂], and 𝑤𝑝𝑢𝑚𝑝,𝑠ℎ is the amount of water removed from the shallow aquifer by pumping on day 𝑖 [𝑚𝑚 𝐻2 𝑂]. 7.3.1.2 Ground Water Discharge The ground water (shallow aquifers only) contributes via the base flow to the main channel within the sub basin. Figure 7-4 illustrates the difference between subsurface flow and lateral ground water flow.

Figure 7-4: Difference between subsurface and ground water / base flow (source: criticalzone.org)

94


If the base flow exceeds the user specified threshold value 𝑎𝑞𝑠ℎ𝑡ℎ𝑟,𝑞 the base flow enters the stream. The steady state response of ground water flow to recharge is :

𝑄𝑔𝑤 =

8000 ∙ 𝐾𝑠𝑎𝑡 𝐿𝑔𝑤

2

∙ ℎ𝑤𝑡𝑏𝑙

(7-41)

𝑄𝑔𝑤 is the ground water flow, or base flow, into the main channel [𝑚𝑚 𝐻2 𝑂], 𝐾𝑠𝑎𝑡 is the hydraulic conductivity [𝑚𝑚/𝑑𝑎𝑦], 𝐿𝑔𝑤 is the distance from the ridge or subbasin divided for the groundwater system to the main channel [𝑚], and ℎ𝑤𝑡𝑏𝑙 is the water table height [𝑚]. Water table fluctuations, as a result of non-steady-state response of ground water flow is calculated as:

𝑑ℎ𝑤𝑡𝑏𝑙 𝑤𝑟𝑐ℎ𝑟𝑔,𝑠ℎ − 𝑄𝑔𝑤 = 𝑑𝑡 800 ∙ 𝜇

(7-42)

With 𝑑ℎ𝑤𝑡𝑏𝑙 ⁄𝑑𝑡 being the water table change with time [𝑚𝑚/𝑑𝑎𝑦], 𝑤𝑟𝑐ℎ𝑟𝑔,𝑠ℎ the amount of water that enters the aquifer via recharge [𝑚𝑚 𝐻2 𝑂], and 𝜇 is the specified yield of the shallow aquifer [𝑚/𝑚]. With the assumption of linearly related change between ground water flow and water table height, both equations can be combined as:

𝑑𝑄𝑔𝑤 𝐾𝑠𝑎𝑡 = 10 ∙ ∙ (𝑤𝑟𝑐ℎ𝑟𝑔,𝑠ℎ − 𝑄𝑔𝑤 ) = 𝛼𝑔𝑤 ∙ (𝑤𝑟𝑐ℎ𝑟𝑔,𝑠ℎ − 𝑄𝑔𝑤 ) 𝑑𝑡 𝜇 ∙ 𝐿𝑔𝑤 2

(7-43)

𝛼𝑔𝑤 is the base flow recession constant or constant of proportionality. With integration and rearranging to solve for 𝑄𝑔𝑤,𝑖 (ground water flow on day 𝑖) it yields:

𝑖𝑓 𝑎𝑞𝑠ℎ > 𝑎𝑞𝑠ℎ𝑡ℎ𝑟,𝑞 : 𝑄𝑔𝑤,𝑖 = 𝑄𝑔𝑤,𝑖−1 ∙ 𝑒𝑥𝑝[−𝛼𝑔𝑤 ∙ ∆𝑡] + 𝑤𝑟𝑐ℎ𝑟𝑔,𝑠ℎ ∙ (1 − 𝑒𝑥𝑝[−𝛼𝑔𝑤 ∙ ∆𝑡]) 𝑖𝑓 𝑎𝑞𝑠ℎ ≤ 𝑎𝑞𝑠ℎ𝑡ℎ𝑟,𝑞 : 𝑄𝑔𝑤,𝑖 = 0 For the special case that the aquifer doesn´t receive recharge (𝑤𝑟𝑐ℎ𝑟𝑔,𝑠ℎ = 0), the equations simplify to:

𝑖𝑓 𝑎𝑞𝑠ℎ > 𝑎𝑞𝑠ℎ𝑡ℎ𝑟,𝑞 : 𝑄𝑔𝑤 = 𝑄𝑔𝑤,0 ⋅ exp[−𝛼𝑔𝑤 ⋅ 𝑡]

(7-44)

𝑖𝑓 𝑎𝑞𝑠ℎ ≤ 𝑎𝑞𝑠ℎ𝑡ℎ𝑟,𝑞 : 𝑄𝑔𝑤,𝑖 = 0

95


The base flow recession constant 𝛼𝑔𝑤 can then be expressed as:

𝛼𝑔𝑤 =

𝑄𝑔𝑤,0 1 1 2.3 ⋅ ln ⌊ ⌋= ∙ ln(10) = 𝑁 𝑄𝑔𝑤,𝑁 𝐵𝐹𝐷 𝐵𝐹𝐷

(7-45)

With 𝐵𝐹𝐷 as the number of base flow days for the watershed.

7.3.1.3 Revap Water may move from the shallow aquifer into the overlying unsaturated zone. In periods when the material overlying the aquifer is dry, water in the capillary fringe that separates the saturated and unsaturated zones will evaporate and diffuse upward. As water is removed from the capillary fringe by evaporation, it is replaced by water from the underlying aquifer. Water may also be removed from the aquifer by deep-rooted plants which are able to uptake water directly from the aquifer.

Figure 7-5: Evaporation from Shalow Aquifer

SWAT models the movement of water into overlying unsaturated layers as a function of water demand for evapotranspiration. To avoid confusion with soil evaporation and transpiration, this process has been termed „𝑟𝑒𝑣𝑎𝑝‟. This process is significant in watersheds where the saturated zone is not very far below the surface or where deep-rooted plants are growing. Because 96


the type of plant cover will affect the importance of 𝑟𝑒𝑣𝑎𝑝 in the water balance, the parameters governing 𝑟𝑒𝑣𝑎𝑝 are usually varied by land use. Revap is allowed to occur only if the amount of water stored in the shallow aquifer exceeds a threshold value specified by the user, 𝑎𝑞𝑠ℎ𝑡ℎ𝑟,𝑟𝑣𝑝 . Maximum amount of water moving into the soil zone in response to water deficiencies

𝑤𝑟𝑒𝑣𝑎𝑝,𝑚𝑥

𝑤𝑟𝑒𝑣𝑎𝑝,𝑚𝑥 = 𝛽𝑟𝑒𝑣 ∙ 𝐸0

(7-46)

𝛽𝑟𝑒𝑣 is the 𝑟𝑒𝑣𝑎𝑝 coefficient, and 𝐸0 is the potential evapotranspiration for the day [𝑚𝑚 𝐻2 𝑂]. The actual amount of that will move into the soil zone 𝑤𝑟𝑒𝑣𝑎𝑝 [𝑚𝑚 𝐻2 𝑂] is calculated as:

𝑤𝑟𝑒𝑣𝑎𝑝

=

0

if

𝑎𝑞𝑠ℎ ≤


=

𝑤𝑟𝑒𝑣𝑎𝑝,𝑚𝑥 − 𝑎𝑞𝑠ℎ𝑡ℎ𝑟,𝑟𝑣𝑝

if

𝑎𝑞𝑠ℎ𝑡ℎ𝑟,𝑟𝑣𝑝 < 𝑎𝑞𝑠ℎ < 𝑎𝑞𝑠ℎ𝑡ℎ𝑟,𝑟𝑣𝑝 +𝑤𝑟𝑒𝑣𝑎𝑝,𝑚𝑥 )


=

if

𝑎𝑞𝑠ℎ

𝑤𝑟𝑒𝑣𝑎𝑝,𝑚𝑥

𝑎𝑞𝑠ℎ𝑡ℎ𝑟,𝑟𝑣𝑝

(7-47)

≥ (𝑎𝑞𝑠ℎ𝑡ℎ𝑟,𝑟𝑣𝑝 + 𝑤𝑟𝑒𝑣𝑎𝑝,𝑚𝑥 )

𝑤𝑟𝑒𝑣𝑎𝑝,𝑚𝑥 [𝑚𝑚 𝐻2 𝑂] is the maximum amount of water moving into the soil zone in response to water deficiencies and 𝑎𝑞𝑠ℎ𝑡ℎ𝑟,𝑟𝑣𝑝 is the threshold water level in the shallow aquifer for 𝑟𝑒𝑣𝑎𝑝 to occur [𝑚𝑚 𝐻2 𝑂]

7.3.1.4 Ground Water Height The ground water table height ℎ𝑤𝑡𝑏𝑙,𝑖 can easily be calculated, given the height at the previous day, recharge rate 𝑤𝑟𝑐ℎ𝑟𝑔,𝑠ℎ , the base flow recession constant 𝛼𝑔𝑤 , and the specific yield 𝜇 are known:

ℎ𝑤𝑡𝑏𝑙,𝑖 = ℎ𝑤𝑡𝑏𝑙,𝑖−1 ⋅ 𝑒𝑥𝑝[−𝛼𝑔𝑤 ⋅ ∆𝑡] +

𝑤𝑟𝑐ℎ𝑟𝑔 ⋅(1−𝑒𝑥𝑝[−𝛼𝑔𝑤 ⋅∆𝑡])

(7-48)

800⋅𝜇⋅𝛼𝑔𝑤

7.3.2 Deep Aquifer The water balance for the deep aquifer is:

𝑎𝑞𝑑𝑝,𝑖 = 𝑎𝑞𝑑𝑝,𝑖−1 + 𝑤𝑑𝑒𝑒𝑝 − 𝑤𝑝𝑢𝑚𝑝,𝑑𝑝

(7-49)

where 𝑎𝑞𝑑𝑝,𝑖 is the amount of water stored in the deep aquifer on day 𝑖 [𝑚𝑚 𝐻2 𝑂], 𝑎𝑞𝑑𝑝,𝑖−1 is the amount of water stored in the deep aquifer on day 𝑖 − 1 [𝑚𝑚 𝐻2 𝑂], 𝑤𝑑𝑒𝑒𝑝 is the amount of 97

7 Lateral Flows 7.4 River Routing

water percolating from the shallow aquifer into the deep aquifer on day 𝑖 [𝑚𝑚 𝐻2 𝑂], and 𝑤𝑝𝑢𝑚𝑝,𝑑𝑝 is the amount of water removed from the deep aquifer by pumping on day 𝑖 [𝑚𝑚 𝐻2 𝑂]. If the deep aquifer is specified as the source of irrigation water or water removed for use outside the watershed, the model will allow an amount of water up to the total volume of the deep aquifer to be removed on any given day. Water entering the deep aquifer is not considered in future water budget calculations and can be considered to be lost from the system

7.4 River Routing 7.4.1 Open Channel Flow and Characterisitcs Open channel flows are those that are not entirely included within rigid boundaries. The surface of the flow is called a free surface, because that flow boundary is freely deformable, in contrast to the solid boundaries. The boundary conditions at the free surface of an open-channel flow are always that both the pressure and the shear stress are zero everywhere. SWAT uses Manning´s equation to define the rate and velocity of flow: 2

𝑞𝑐ℎ

1

𝐴𝑐ℎ ⋅ 𝑅𝑐ℎ 3 ⋅ 𝑠𝑙𝑝𝑐ℎ 2 = 𝑛

(7-50)

𝑞𝑐ℎ is the average channel flow [𝑚³/s], 𝐴𝑐ℎ the cross sectional area of the channel [𝑚²], 𝑅𝑐ℎ is the hydraulic radius for a given depth of flow (in very shallow channels it is equal to the flow depth ℎ), 𝑛 is Manning´s roughness coefficient, and 𝑠𝑙𝑝𝑐ℎ is the average channel slope. SWAT assumes the channels or reaches to have a trapezoidal shape, as indicated in Figure 7-6.

Figure 7-6: Trapezoidal channel dimensions

98


The user is required to define certain characteristics of the stream, such as width 𝑊 [𝑚] and depth ℎ [𝑚]. 𝑧𝑐ℎ is set to 2. The bottom width 𝑊𝑏𝑡𝑚 is calculated with the top width of the channel when filled and the depth ℎ:

𝑊𝑏𝑡𝑚 = 𝑊 − 2 ∙ 𝑧𝑐ℎ ∙ ℎ

(7-51)

Due to the assumption 𝑧𝑐ℎ = 2, it is possible that the bottom width gets less than zero. In that case the model sets 𝑊𝑏𝑡𝑚 = 0.5 ⋅ 𝑊 and a new value for 𝑧𝑐ℎ is calculated:

𝑧𝑐ℎ =

(𝑊 − 𝑊𝑏𝑡𝑚 ) 2⋅ℎ

(7-52)

For a given depth of water in the channel, the width of the channel 𝑊 [𝑚] at water level is:

𝑊 = 𝑊𝑏𝑡𝑚 + 2 ∙ 𝑧𝑐ℎ ∙ ℎ

(7-53)

The cross section area of the channel flow 𝐴𝑐ℎ [𝑚] is:

𝐴𝑐ℎ = (𝑊𝑏𝑡𝑚 + 𝑧𝑐ℎ ∙ ℎ) ∙ ℎ

(7-54)

The wetted perimeter 𝑃𝑐ℎ [𝑚] for a given depth:

𝑃𝑐ℎ = 𝑊𝑏𝑡𝑚 + 2 ∙ ℎ ∙ √1 + 𝑧𝑐ℎ 2

(7-55)

The hydraulic radius 𝑅𝑐ℎ [𝑚]:

𝑅𝑐ℎ =

𝐴𝑐ℎ 𝑃𝑐ℎ

(7-56)

The volume of water held in the channel 𝑉𝑐ℎ [𝑚3 ]:

𝑉𝑐ℎ = 1000 ∙ 𝐿𝑐ℎ ∙ 𝐴𝑐ℎ

(7-57)

For the case that the water in the stream exceeds the channel depth, the excess water spreads across the flood plain. The flood plain dimensions which are used by SWAT are illustrated in Figure 7-7.

99


Figure 7-7: Flood plain dimension

The bottom width of the flood plain is calculated as:

𝑤𝑏𝑡𝑚,𝑓𝑙𝑑 = 5 ∙ 𝑊

(7-58)

The total water depth ℎ𝑡𝑜𝑡 is composed as:

ℎ𝑡𝑜𝑡 = ℎ + ℎ𝑓𝑙𝑑

(7-59)

The cross section area 𝐴𝑐ℎ,𝑓𝑙𝑑 , incl. the flood plain, with 𝑧𝑓𝑙𝑑 = 4:

𝐴𝑐ℎ,𝑓𝑙𝑑 = (𝑊𝑏𝑡𝑚 + 𝑧𝑐ℎ ⋅ ℎ) ⋅ ℎ + (𝑊𝑏𝑡𝑚,𝑓𝑙𝑑 + 𝑧𝑓𝑙𝑑 ⋅ ℎ𝑓𝑙𝑑 ) ⋅ ℎ𝑓𝑙𝑑

(7-60)

The wetted perimeter 𝑃𝑐ℎ,𝑓𝑙𝑑 :

𝑃𝑐ℎ,𝑓𝑙𝑑 = 𝑊𝑏𝑡𝑚 + 2 ⋅ ℎ ⋅ √1 + 𝑧𝑐ℎ 2 + 4 ⋅ 𝑊𝑏𝑛𝑘𝑓𝑢𝑙𝑙 + 2 ⋅ ℎ𝑓𝑙𝑑 ⋅ √1 + 𝑧𝑓𝑙𝑑 2

(7-61)

7.4.2 Flow Rate and Velocity SWAT uses Manning´s equation to calculate the flow rate 𝑞𝑐ℎ [𝑚3 /𝑠] and flow velocity 𝑣𝑐 [𝑚/ 𝑠] in the channel segment for a given time step: 2

𝑞𝑐ℎ

1

𝐴𝑐ℎ ⋅ 𝑅𝑐ℎ 3 ⋅ 𝑠𝑙𝑝𝑐ℎ 2 = 𝑛 2

(7-62)

1

𝑅𝑐ℎ 3 ⋅ 𝑠𝑙𝑝𝑐ℎ 2 𝑣𝑐 = 𝑛

(7-63)

With SWAT routes the water as volume. The daily for cross-sectional area of flow, 𝐴𝑐ℎ is calculated as:

𝐴𝑐ℎ =

100

𝑉𝑐ℎ 1000 ∙ 𝐿𝑐ℎ

(7-64)


This equation can be rearranged to solve for the flow depth ℎ [𝑚]:

ℎ=√

𝐴𝑐ℎ 𝑊𝑏𝑡𝑚 2 𝑊𝑏𝑡𝑚 +( ) − 𝑧𝑐ℎ 2 ∙ 𝑧𝑐ℎ 2 ∙ 𝑧𝑐ℎ

(7-65)

In the case of a water filled flood plain:

2

ℎ𝑡𝑜𝑡

=ℎ+√

(𝐴𝑐ℎ − 𝐴𝑐ℎ,𝑓𝑙𝑑 ) 𝑊𝑏𝑡𝑚,𝑓𝑙𝑑 𝑊𝑏𝑡𝑚,𝑓𝑙𝑑 +( ) − 𝑧𝑓𝑙𝑑 2 ∙ 𝑧𝑓𝑙𝑑 2 ∙ 𝑧𝑓𝑙𝑑

(7-66)

7.4.3 Variable Storage Routing Method For a given segment of the reach, the storage routing is based on the continuity equation:

∆𝑉𝑠𝑡𝑜𝑟𝑒𝑑 = 𝑉𝑖𝑛 − 𝑉𝑜𝑢𝑡

(7-67)

𝑞𝑖𝑛,1 + 𝑞𝑖𝑛,2 𝑞𝑜𝑢𝑡,1 + 𝑞𝑜𝑢𝑡,2 𝑣𝑠𝑡𝑜𝑟𝑒𝑑,2 − 𝑣𝑠𝑡𝑜𝑟𝑒𝑑,1 = 𝛥𝑡 ⋅ ( ) − 𝛥𝑡 ⋅ ( ) 2 2

(7-68)

It can also be written as:

Where 𝛥𝑡 is the time step, 𝑞𝑖𝑛,1 , 𝑞𝑖𝑛,2 is the inflow rate at the beginning (1) and the end (2) of the time step, 𝑞𝑜𝑢𝑡,1 , 𝑞𝑜𝑢𝑡,2 is the outflow rate at the beginning and the end of the time step. 𝑉𝑠𝑡𝑜𝑟𝑒𝑑 accounts for the water volume stored in the water body at the beginning and the end of a time step. With all known variables on the left side it is:

𝑞𝑖𝑛,𝑎𝑣𝑒 +

𝑉𝑠𝑡𝑜𝑟𝑒𝑑,1 𝑞𝑜𝑢𝑡,1 𝑉𝑠𝑡𝑜𝑟𝑒𝑑,2 𝑞𝑜𝑢𝑡,2 − = + ∆𝑡 2 ∆𝑡 2

(7-69)

𝑞𝑖𝑛,1 + 𝑞𝑖𝑛,2 2

(7-70)

And

𝑞𝑖𝑛,𝑎𝑣𝑒 =

The travel time 𝑇𝑇 [𝑠], which the water needs to travel through the channel is obtained by dividing the volume of water in the channel by the flow rate:

101


𝑇𝑇 =

𝑉𝑠𝑡𝑜𝑟𝑒𝑑 𝑉𝑠𝑡𝑜𝑟𝑒𝑑,1 𝑉𝑠𝑡𝑜𝑟𝑒𝑑,2 = = 𝑞𝑜𝑢𝑡 𝑞𝑜𝑢𝑡,1 𝑞𝑜𝑢𝑡,2

(7-71)

With the ravel time the storage coefficient 𝑆𝐶 can be determined:

𝑆𝐶 =

2 ⋅ Δ𝑡 2 ⋅ 𝑇𝑇 + Δ𝑡

(7-72)

The out flow from a given channel segment at the end of a time step 𝑞𝑜𝑢𝑡,2 which is the sought quantity can be calculated as:

2 ⋅ Δ𝑡 2 ⋅ Δ𝑡 𝑞𝑜𝑢𝑡,2 = ( ) ⋅ 𝑞𝑖𝑛,𝑎𝑣𝑒 + (1 − ) ⋅ 𝑞𝑜𝑢𝑡,1 2 ⋅ 𝑇𝑇 + Δ𝑡 2 ⋅ 𝑇𝑇 + Δ𝑡

(7-73)

Or with the storage coefficient:

𝑞𝑜𝑢𝑡,2 = (𝑆𝐶) ⋅ 𝑞𝑖𝑛,𝑎𝑣𝑒 + (1 − 𝑆𝐶) ⋅ 𝑞𝑜𝑢𝑡,1

(7-74)

Rearranging this equation and solving for the output volume 𝑉𝑜𝑢𝑡,2 it yields:

𝑉𝑜𝑢𝑡,2 = 𝑆𝐶 ⋅ (𝑉𝑖𝑛 + 𝑉𝑠𝑡𝑜𝑟𝑒𝑑,1 )

(7-75)

7.4.4 Muskingum Flood Routing Method A flood wave that advances into a reach segment produces a wedge of storage, as the inflow exceeds the out flow (Figure 7-8).

102


Figure 7-8: Prism and wedge storages in a reach segment

The cross sectional area of flow is assumed to be proportional to the discharge of a given stream segment. The volume of prism storage can be expressed as:

𝑉𝑠𝑡𝑜𝑟𝑒𝑑, 𝑝𝑟𝑖𝑠𝑚 = 𝐾 ∙ 𝑞𝑜𝑢𝑡

(7-76)

With 𝐾 as the ratio of storage to discharge [𝑠]. The volume of the wedge is:

𝑉𝑠𝑡𝑜𝑟𝑒𝑑, 𝑤𝑒𝑑𝑔𝑒 = 𝐾 ∙ 𝑋 ∙ (𝑞𝑖𝑛 − 𝑞𝑜𝑢𝑡 )

(7-77)

With 𝑋 as a weighting factor between 0.0 and 0.5. It is dependent on the type of storage, e.g. for a reservoir it is 0.0, for rivers it is between 0.0 and 0.3, with a mean of 0.2. The combined volume 𝑉𝑠𝑡𝑜𝑟𝑒𝑑 is:

𝑉𝑠𝑡𝑜𝑟𝑒𝑑 = K ⋅ (𝑋 ⋅ 𝑞𝑖𝑛 + (1 − 𝑋) ⋅ 𝑞𝑜𝑢𝑡 )

(7-78)

This can be combined with the continuity equation:

𝑉𝑠𝑡𝑜𝑟𝑒𝑑,2 − 𝑉𝑠𝑡𝑜𝑟𝑒𝑑,1 = Δ𝑡 ⋅ (

𝑞𝑖𝑛,1 + 𝑞𝑖𝑛,2 𝑞𝑜𝑢𝑡,1 + 𝑞𝑜𝑢𝑡,2 ) − Δ𝑡 ⋅ ( ) 2 2

(7-79)

Solved for the out flow rate and the out flow volume at the end of the time step it yields:

103


𝑞𝑜𝑢𝑡,2 = 𝐶1 ∙ 𝑞𝑖𝑛,2 + 𝐶2 ∙ 𝑞𝑖𝑛,1 + 𝐶3 ∙ 𝑞𝑜𝑢𝑡,1

(7-80)

𝑉𝑜𝑢𝑡,2 = 𝐶1 ∙ 𝑉𝑖𝑛,2 + 𝐶2 ∙ 𝑉𝑖𝑛,1 + 𝐶3 ∙ 𝑉𝑜𝑢𝑡,1

(7-81)

With the coefficients:

𝐶1 =

Δ𝑡 − 2 ⋅ 𝐾 ⋅ 𝑋 2 ⋅ 𝐾 ⋅ (1 − 𝑋) + Δ𝑡

(7-82)

𝐶2 =

Δ𝑡 + 2 ⋅ 𝐾 ⋅ 𝑋 2 ⋅ 𝐾 ⋅ (1 − 𝑋) + Δ𝑡

(7-83)

𝐶3 =

2 ⋅ 𝐾 ⋅ (1 − 𝑋) − Δ𝑡 2 ⋅ 𝐾 ⋅ (1 − 𝑋) + Δ𝑡

(7-84)

To maintain numerical stability and avoid the computation of negative outflows, the following condition must be met:

𝑉𝑠𝑡𝑜𝑟𝑒𝑑,2 − 𝑉𝑠𝑡𝑜𝑟𝑒𝑑,1 = Δ𝑡 ⋅ (

𝑞𝑖𝑛,1 + 𝑞𝑖𝑛,2 𝑞𝑜𝑢𝑡,1 + 𝑞𝑜𝑢𝑡,2 ) − Δ𝑡 ⋅ ( ) 2 2

(7-85)

The weighting factor 𝑋 is given by the user as input. The storage time constant 𝐾 needs to be estimated as:

𝐾 = 𝑐𝑜𝑒𝑓1 ∙ 𝐾𝑏𝑛𝑘𝑓𝑢𝑙𝑙 + 𝑐𝑜𝑒𝑓2 ∙ 𝐾0.1,𝑏𝑛𝑘𝑓𝑢𝑙𝑙

(7-86)

𝑐𝑜𝑒𝑓1 and 𝑐𝑜𝑒𝑓2 are weighting factors defined by the user. 𝐾𝑏𝑛𝑘𝑓𝑢𝑙𝑙 is the storage time constant calculated for the reach segment with bankfull flows. 𝐾0.1,𝑏𝑛𝑘𝑓𝑢𝑙𝑙 is the storage time constant calculated for the reach segment with 1/10 of bankfull flows. To calculate both of them the following equation is used:

𝐾=

1000 ∙ 𝐿𝑐ℎ 𝑐𝑘

(7-87)

𝑐𝑘 is the velocity corresponding to the flow for a specified depth [𝑚/𝑠], which describes the velocity with which a variation in flow rate travels along the channel. It is defined as:

5 𝑅𝑐ℎ 2/3 ∙ 𝑠𝑙𝑝𝑐ℎ 1/2 𝑐𝑘 = ⋅ ( ) 3 𝑛

104

(7-88)


Or substituting with Mannnig´s flow velocity 𝑣𝑐 :

𝑐𝑘 =

5 ∙𝑣 3 𝑐

(7-89)

7.4.5 Transmission Losses, Evaporation and Bank Storage 7.4.5.1 Transmission Losses Streams gain water thorugh seepage from the ground water body. They can be classified as: Ephemeral

Contain water during and immediately after a storm event. The rest of the year they are dry.

Intermittent Those streams are dry part of the year, but contain water when the ground water is high enough as well as during and after a strom event. Perennial

The stream receives continuously ground water and flows throughout the year.

In the case when the stream doesn´t receive water transmission losses 𝑡𝑙𝑜𝑠𝑠 are possible:

𝑡𝑙𝑜𝑠𝑠 = 𝐾𝑐ℎ ⋅ 𝑇𝑇 ⋅ 𝑃𝑐ℎ ⋅ 𝐿𝑐ℎ

(7-90)

𝐾𝑐ℎ accounts for the effective hydraulic conductivity of the channel alluvium [𝑚𝑚⁄ℎ].

7.4.5.2 Evaporation Losses The daily evaporation 𝐸𝑐ℎ from the reach is calculated as:

𝐸𝑐ℎ = 𝑐𝑜𝑒𝑓𝑒𝑣 ⋅ 𝐸0 ⋅ 𝐿𝑐ℎ ⋅ 𝑊 ⋅ 𝑓𝑟∆𝑡

(7-91)

𝑐𝑜𝑒𝑓𝑒𝑣 is a calibration parameter between 0 and 1. 𝐸0 is the potential evaporation [𝑚𝑚⁄𝑑], 𝐿𝑐ℎ is the channel length, 𝑊 is the width and 𝑓𝑟∆𝑡 is the time step fraction in which water is flowing in the channel.

105


7.4.5.3 Bank Storage

Figure 7-9: Bank Storage

Bank storage accounts for water, which is literally stored in the bank of the river, as shown in Figure 7-9. The amount of water that enters the bank 𝑏𝑛𝑘𝑖𝑛 [𝑚³] can be calculated as:

𝑏𝑛𝑘𝑖𝑛 = 𝑡𝑙𝑜𝑠𝑠 ⋅ (1 − 𝑓𝑟𝑡𝑟𝑛𝑠 )

(7-92)

The bank flow 𝑉𝑏𝑛𝑘 [𝑚³ 𝐻2 𝑂] is simulated as with the bank flow recession constant 𝛼𝑏𝑛𝑘 :

𝑉𝑏𝑛𝑘 = 𝑏𝑛𝑘 ⋅ (1 − 𝑒𝑥𝑝[−𝛼𝑏𝑛𝑘 ])

(7-93)

Water can move from bank storage into the unsaturated zone. SWAT models this movement of water into the adjacent unsaturated zone as a function of water demand for evapotranspiration. It is called ‘revap’ to avoid confusion with soil evapotranspiration. The maximum amount of water that can be removed from the storage via revap is:

𝑏𝑛𝑘𝑟𝑒𝑣𝑎𝑝,𝑚𝑥 = 𝛽𝑟𝑒𝑣 ⋅ 𝐸0 ⋅ 𝐿𝑐ℎ ⋅ 𝑊

(7-94)

𝛽𝑟𝑒𝑣 is the revap coefficent. The actual amount of water moving into the unsaturated zone (revap) 𝑏𝑛𝑘𝑟𝑒𝑣𝑎𝑝 that will occur on a day is calculated as:

𝑏𝑛𝑘𝑟𝑒𝑣𝑎𝑝 = 𝑏𝑛𝑘 𝑏𝑛𝑘𝑟𝑒𝑣𝑎𝑝 = 𝑏𝑛𝑘𝑟𝑒𝑣𝑎𝑝,𝑚𝑥

𝑖𝑓 𝑖𝑓

𝑏𝑛𝑘 < 𝑏𝑛𝑘𝑟𝑒𝑣𝑎𝑝,𝑚𝑥 𝑏𝑛𝑘 ≥ 𝑏𝑛𝑘𝑟𝑒𝑣𝑎𝑝,𝑚𝑥

7.4.5.4 Channel Water Balance The final water balance at the end of a time step is:

106

(7-95)

7 Lateral Flows 7.5 Further Reading

𝑣𝑠𝑡𝑜𝑟𝑒𝑑,2 = 𝑣𝑠𝑡𝑜𝑟𝑒𝑑,1 + 𝑉𝑖𝑛 − 𝑉𝑜𝑢𝑡 − 𝑡𝑙𝑜𝑠𝑠 − 𝐸𝑐ℎ + 𝑑𝑖𝑣 + 𝑉𝑏𝑛𝑘

(7-96)

It includes also the volume of water added or removed 𝑑𝑖𝑣 for the time step through water dirversions.

7.5 Further Reading In the following paper SWAT is used to predict the monthly volume inflow to Simly Dam, an important drinking water reservoir north of Islamabad, Pakistan: Ghoraba, S. M.: Hydrological modelling of the Simly Dam watershed (Pakistan) using GIS and SWAT model, Alexandria Engineering Journal (2015) Article in press http://dx.doi.org/10.1016/j.aej.2015.05.018

7.6 Nomenclature 𝐴𝑐ℎ 𝐴𝑟𝑒𝑎 𝑎𝑟

Cross sectional area of the channel [𝑚²] Subbasin area [km²] Unit channel regression intercept [𝑚³]

𝑎𝑞𝑑𝑝,𝑖

Water stored in the deep aquifer on day 𝑖 [𝑚𝑚 𝐻2 𝑂]

𝑎𝑞𝑠ℎ,𝑖

Water stored in the shallow aquifer on day 𝑖 [𝑚𝑚 𝐻2 𝑂]

𝑎𝑞𝑠ℎ𝑡ℎ𝑟,𝑟𝑣𝑝

threshold water level in the shallow aquifer for revap [𝑚𝑚 𝐻2 𝑂]

𝑎𝑥

Regression intercept for channel of length 𝐿 and width 𝑊 [𝑚³]

𝐵𝐹𝐷

Number of base flow days for the watershed

𝑏𝑟

Unit channel regression slope [ℎ𝑟]

𝑏𝑥

Regression slope for channel of length 𝐿 and width 𝑊 [-]

𝑏𝑛𝑘𝑖𝑛

Amount of water that enters the bank [𝑚³ 𝐻2 𝑂]

𝑏𝑛𝑘𝑟𝑒𝑣𝑎𝑝,𝑚𝑥

Maximum amount of water that can be removed from the storage via 𝑟𝑒𝑣𝑎𝑝 [𝑚³ 𝐻2 𝑂]

𝑏𝑛𝑘𝑟𝑒𝑣𝑎𝑝

Actual amount of water moving into the unsaturated zone via 𝑟𝑒𝑣𝑎𝑝 [𝑚³ 𝐻2 𝑂]

𝐶

Runoff coefficient [-]

𝑐𝑘

Celerity corresponding to the flow for a specified depth [𝑚/𝑠]

𝑐𝑜𝑒𝑓1 , 𝑐𝑜𝑒𝑓2 𝑐𝑜𝑒𝑓𝑒𝑣 𝑑𝑖𝑣 𝑑𝑢𝑟𝑓𝑙𝑤 𝐸0

Weighting factors defined by the user [-] Calibration parameter [-] Volume of water added or removed through water diversions [𝑚³ 𝐻2 𝑂] Flow duration [ℎ𝑟] Potential evaporation [𝑚𝑚⁄𝑑] 107

7 Lateral Flows 7.6 Nomenclature

𝐸𝑐ℎ

Daily evaporation [𝑚𝑚⁄𝑑]

𝑓𝑟∆𝑡

Time step fraction in which water is flowing in the channel [-]

𝐻0

Saturated thickness [𝑚𝑚/𝑚𝑚]

ℎ𝑡𝑜𝑡

Total water depth [𝑚]

ℎ𝑤𝑡𝑏𝑙

Water table height [𝑚]

𝑖

Rainfall intensity [𝑚𝑚/ℎ𝑟]

𝐾

Muskingum ratio of storage to discharge [𝑠]

𝐾0.1,𝑏𝑛𝑘𝑓𝑢𝑙𝑙

Storage time constant calculated for the reach segment with 1/10 of bankfull flows [-]

𝐾𝑏𝑛𝑘𝑓𝑢𝑙𝑙

Storage time constant calculated for the reach segment with bankfull flows [-]

𝐾𝑐ℎ

Effective hydraulic conductivity of the natural channel [𝑚𝑚/ℎ𝑟].

𝐾𝑐ℎ

Effective hydraulic conductivity of the channel alluvium [𝑚𝑚⁄ℎ]

𝑘𝑟

Decay factor [1/𝑘 𝑘𝑚]

𝐾𝑠𝑎𝑡,𝑚𝑥 𝐾𝑠𝑎𝑡

Hydraulic conductivity [𝑚𝑚/𝑑𝑎𝑦]

𝐿

Channel length from the most distant point to the subbasin outlet [𝑘𝑚]

𝐿𝑐

Average flow channel length for the subbasin [𝑘𝑚]

𝐿𝑐ℎ

Channel length [𝑚]

𝐿𝑐𝑒𝑛

Distance along the channel to the subbasin centroid [𝑘𝑚]

𝐿𝑔𝑤

Distance from the ridge or subbasin divide for the groundwater system to the main channel [𝑚]

𝐿ℎ𝑖𝑙𝑙

Hillslope length [𝑚]

𝐿𝑠𝑙𝑝

Subbasin slope length [m]

𝑛 𝑃𝑐ℎ

Manning´s roughness coefficient [-] Wetted perimeter [𝑚]

𝑄𝑔𝑤,𝑖

Ground water flow on day 𝑖 [𝑚𝑚 𝐻2 𝑂]

𝑄𝑔𝑤

Ground water flow/ base flow into the main channel [𝑚𝑚 𝐻2 𝑂]

𝑄𝑙𝑎𝑡

Discharge at the hillslope exit [𝑚𝑚 𝐻2 𝑂/𝑑𝑎𝑦]

𝑄𝑙𝑎𝑡 ′

Amount of lateral flow generated in the subbasin on a given day [𝑚𝑚 𝐻2 𝑂]

𝑄𝑙𝑎𝑡𝑠𝑡𝑜𝑟, 𝑖−1 𝑄𝑠𝑡𝑜𝑟,𝑖−1 𝑄𝑠𝑢𝑟𝑓

108

Highest saturated hydraulic conductivity in the soil [𝑚𝑚/𝑑𝑎𝑦]

Lateral flow stored from the previous day [𝑚𝑚 𝐻2 𝑂] Surface runoff stored or lagged from the previous day [𝑚𝑚 𝐻2 𝑂] Surface runoff [𝑚𝑚 𝐻2 𝑂]

𝑞0 ∗

Source area flow rate [𝑚𝑚/ℎ𝑟]

𝑞𝑐ℎ

Average channel flow rate [𝑚³/𝑠]

𝑞𝑐ℎ ∗

Average channel flow rate in [𝑚𝑚/ℎ𝑟]


𝑞𝑖𝑛,1 , 𝑞𝑖𝑛,2 𝑞𝑜𝑢𝑡,1 , 𝑞𝑜𝑢𝑡,2 𝑞𝑜𝑣 𝑞𝑝𝑒𝑎𝑘,𝑓

Inflow rate at the beginning (1) and the end (2) [𝑚³/s] Outflow rate at the beginning and the end of the time step [𝑚³/s] Average overland flow rate [m³/s] Peak rate after transmission losses [𝑚³/𝑠]

𝑞𝑝𝑒𝑎𝑘

Peak runoff rate [m³/s]


peak flow [𝑚𝑚/ℎ𝑟]

𝑅𝑐ℎ

Hydraulic radius for a given depth of flow [𝑚]

𝑅𝑑𝑎𝑦

Rainfall for the day [𝑚𝑚 𝐻2 𝑂]

𝑅𝑡𝑐

Amount of rain falling during the time of concentration [𝑚𝑚 𝐻2 𝑂]

𝑆𝐶

Storage coefficient [-]

𝑆𝑊𝑙𝑦,𝑒𝑥𝑐𝑒𝑠𝑠 𝑠𝑙𝑝 𝑠𝑙𝑝𝑐ℎ 𝑠𝑢𝑟𝑙𝑎𝑔 𝑇𝑇 𝑇𝑇𝑙𝑎𝑔 𝑡𝑐ℎ 𝑡𝑐𝑜𝑛𝑐 𝑡𝑖𝑙𝑒𝑙𝑎𝑔 𝑡𝑜𝑣

Volume of water stored in the saturated zone of the hill slope segment [𝑚𝑚 𝐻2 𝑂] Average slope in the sub basin [m/m] Channel slope [𝑚/𝑚] surface runoff lag coefficient [-] Travel time through the channel [𝑠] Lateral flow travel time [day] Channel flow time of concentration [hr] Time of concentration [hr] Tile drainages Overland flow time of concentration [hr]

𝑡𝑙𝑜𝑠𝑠

Transmission losses

𝑉𝑏𝑛𝑘

Bank flow [𝑚³ 𝐻2 𝑂]

𝑉𝑐ℎ

Volume of water held in the channel [𝑚3 ]

𝑉𝑜𝑙𝑄𝑠𝑢𝑟𝑓,𝑓

Runoff volume [𝑚³]

𝑉𝑜𝑢𝑡,2

Output volume [𝑚3 ]

𝑉𝑠𝑡𝑜𝑟𝑒𝑑, 𝑝𝑟𝑖𝑠𝑚

Volume of prism storage [𝑚3 ]

𝑉𝑠𝑡𝑜𝑟𝑒𝑑, 𝑤𝑒𝑑𝑔𝑒

Volume of the wedge [𝑚3 ]

𝑉𝑠𝑡𝑜𝑟𝑒𝑑 𝑣𝑐 𝑣𝑙𝑎𝑡 𝑣𝑜𝑙𝑄𝑠𝑢𝑟𝑓,𝑖 𝑣𝑜𝑙𝑡ℎ𝑟

Water volume stored in the water body [𝑚3 ] Average channel velocity [𝑚/𝑠] Velocity flow at the outlet [mm/h] Runoff volume prior to the losses [𝑚³] Threshold volume for a channel of length L and width W [𝑚³]

𝑣𝑜𝑣

Overland flow velocity [m/s]

𝑊

Width of the channel at water level [𝑚]

𝑊𝑏𝑡𝑚

Bottom width of the channel [𝑚] 109


𝑤𝑏𝑡𝑚,𝑓𝑙𝑑 𝑤𝑑𝑒𝑒𝑝

Water percolating from the shallow aquifer into the deep aquifer [𝑚𝑚 𝐻2 𝑂]

𝑤𝑝𝑢𝑚𝑝,𝑑𝑝

Water removed from the deep aquifer by pumping [𝑚𝑚 𝐻2 𝑂]

𝑤𝑝𝑢𝑚𝑝,𝑠ℎ

Water removed from the shallow aquifer by pumping [𝑚𝑚 𝐻2 𝑂]

𝑤𝑟𝑐ℎ𝑟𝑔,𝑠ℎ

Amount of water that enters the aquifer via recharge [𝑚𝑚 𝐻2 𝑂]


Water moving into the soil zone in response to water deficiencies [𝑚𝑚 𝐻2 𝑂]

𝑋

Muskingum weighting factor [-]

∆𝑡

Time step

𝑑ℎ𝑤𝑡𝑏𝑙 ⁄𝑑𝑡

110

bottom width of the flood plain [𝑚]

Water table change with time [𝑚𝑚/𝑑𝑎𝑦]

𝛼ℎ𝑖𝑙𝑙

Angle of the hill slope [°]

𝛼𝑏𝑛𝑘

Bank flow recession constant [-]

𝛼𝑔𝑤

Base flow recession constant [ℎ𝑟]

𝛼𝑡𝑐

Fraction of daily rainfall during the time of concentration [-]

𝛽𝑟𝑒𝑣

Revap coefficient [-]

𝜙𝑑

Drainable porosity of the soil [𝑚𝑚/𝑚𝑚]

𝜙𝑓𝑐

Porosity of the soil layer when the layer is at field capacity [𝑚𝑚/𝑚𝑚]

𝜙𝑠𝑜𝑖𝑙

Total porosity of the soil layer [𝑚𝑚/𝑚𝑚]

8 Soil Erosion 8.1 Background

8 Soil Erosion 8.1 Background Soil erosion is defined as the wearing away of the upper soil layer. Topsoil is the top layer of soil and is the most fertile because it contains the most organic, nutrient-rich materials. Therefore, this is the layer that farmers want to protect for growing their crops and ranchers want to protect for growing grasses for their cattle to graze on. One of the main causes of soil erosion is water erosion, which is the loss of topsoil due to water. Raindrops fall directly on topsoil. The impact of the raindrops loosens the material bonding it together, allowing small fragments to detach. If the rainfall continues, water gathers on the ground, causing water flow on the land surface, generating surface water runoff. This runoff carries the detached soil materials away and deposits them elsewhere. There are some conditions that can accentuate surface water runoff and therefore soil erosion. For example, if the land is sloped, there is a greater potential for soil erosion due to the fact that gravity pulls the water and soil materials down the slope. Also, water will have an easier time running across the surface, carrying topsoil with it, if the ground is already saturated due to heavy rains or the soil lacks vegetation to keep the soil in place. Early soil surveys were made to help farmers locate soils responsive to different management practices and to help them decide what crops and management practices were most suitable for the particular kinds of soil on their farms. Many of the early workers were geologists because only geologists were skilled in the necessary field methods and in scientific correlation appropriate to the study of soils. They conceived soils as mainly the weathering products of geologic formations, defined by landform and lithologic composition. Most of the soil surveys published before 1910 were strongly influenced by these concepts. Those published from 1910 to 1920 gradually added greater refinements and recognized more soil features but retained fundamentally geological concepts. Understanding about erosion and its effect on cropland productivity emerged in the 1920‘s. The „dust-bowl“ provided the impetus for provision of larger funding for research on soil erosion. Research lead to knowledge about how the following parameter influence the soil erosion: •

Slope steepness

•

Slope length

•

Crop type and rotations applied

•

Conservation practices

111

8 Soil Erosion 8.2 USLE and MUSLE

8.2 USLE and MUSLE 8.2.1 USLE The Universal Soil Loss Equation (USLE) was developed in the 1950s by the United States Department of Agriculture and predicts the long-term average annual rate of erosion on a field slope based on rainfall pattern, soil type, topography, crop system and management practices. USLE only predicts the amount of soil loss that results from sheet or rill erosion on a single slope and does not account for additional soil losses that might occur from gully, wind or tillage erosion. It also doesn´t account for surface runoff or the area of the HRU. This erosion model was created for use in selected cropping and management systems. USLE can be used to compare soil losses from a particular field with a specific crop and management system to "tolerable soil loss" rates. Alternative management and crop systems may also be evaluated to determine the adequacy of conservation measures in farm planning. NOTE: SWAT does not use the USLE model, though it prints out the sediment loadings calculated with USLE. Nevertheless, it is presented here, as it is the predecessor of the modified USLE (MUSLE) equation. The Universal Soil Loss Equation for the sediment yield 𝑠𝑒𝑑 [𝑡𝑜𝑛𝑠/ℎ𝑎] is:

𝑠𝑒𝑑 = 1.292 ∙ 𝐸𝐼𝑢𝑠𝑙𝑒 ⋅ 𝐾𝑢𝑠𝑙𝑒 ∙ 𝐶𝑢𝑠𝑙𝑒 ∙ 𝑃𝑢𝑠𝑙𝑒 ∙ 𝐿𝑆𝑢𝑠𝑙𝑒 ∙ 𝐶𝐹𝑅𝐺

(8-1)

The factor 1.292 is slightly a conversion factor, as the equation was developed initially for the English dimension system (𝑓𝑜𝑜𝑡, 𝑎𝑐𝑟𝑒, 𝑖𝑛𝑐ℎ) and converts it into the SI system. 𝐸𝐼𝑈𝑆𝐿𝐸 (Sometimes also called 𝑅 factor) is the rainfall erosion index [0.017 𝑡𝑜𝑛𝑠 𝑐𝑚⁄(𝑚2 ℎ) ], it quantifies the erosive effect from raindrop impact and contains information on amount and rate of surface runoff. It is taken as an average annual value (based on data) for a specific region. 𝐾𝑈𝑆𝐿𝐸 is the USLE soil erodibility factor [0.013 𝑡𝑜𝑛𝑠 𝑚2 ℎ⁄(𝑚3 𝑡𝑜𝑛𝑠 𝑐𝑚) ], 𝐶𝑢𝑠𝑙𝑒 is the USLE cover and management factor, 𝑃𝑢𝑠𝑙𝑒 is the USLE support practice factor, 𝐿𝑆𝑢𝑠𝑙𝑒 is the USLE topographic factor and 𝐶𝐹𝑅𝐺 is the coarse fragment factor.

8.2.2 MUSLE While USLE predicts average annual gross erosion as a function of rainfall energy, MULSE incorporates the runoff factor instead of the energy. This improves the sediment yield prediction, and allows the equation to be applied to individual storm events. The equation is:

𝑠𝑒𝑑 = 11.8 ⋅ (𝑄𝑠𝑢𝑟𝑓 ∙ 𝑞𝑝𝑒𝑎𝑘 ∙ 𝑎𝑟𝑒𝑎ℎ𝑟𝑢 )

0.56

∙ 𝐾𝑢𝑠𝑙𝑒 ∙ 𝐶𝑢𝑠𝑙𝑒 ∙ 𝑃𝑢𝑠𝑙𝑒 ∙ 𝐿𝑆𝑢𝑠𝑙𝑒 ∙ 𝐶𝐹𝑅𝐺

(8-2)

𝑠𝑒𝑑 is the sediment yield on a given day. 𝑄𝑠𝑢𝑟𝑓 is the surface runoff [𝑚𝑚𝐻2 𝑂/ℎ𝑎], 𝑞𝑝𝑒𝑎𝑘 is the peak runoff rate [𝑚³/𝑠] and 𝑎𝑟𝑒𝑎ℎ𝑟𝑢 is the area of the HRU [ℎ𝑎]. In the following the different factors are explained.

Soil erodibility factor K 112


Soils differ in their erodibility, this is due to the nature of the soils themselves. As direct measurement of this erodibility Wischmeier et al. (1971) developed a general equation to calculate the soil erodibility factor when the silt and very fine sand conent makes up less than 70 % of the soil particle size distribution. Assumptions for a unit plot are a continuously falling slope of a length of 22.1 𝑚 and an inclination of 9 %. Furthermore, it is vegetation free and tilled for more than 2 years. The equation is:

𝐾𝑢𝑠𝑙𝑒

0.00021 ∙ 𝑀1.14 ∙ (12 − 𝑂𝑀) + 3.25 ∙ (𝑐𝑠𝑜𝑖𝑙𝑠𝑡𝑟 − 2) + 2.5 ⋅ (𝑐𝑝𝑒𝑟𝑚 − 3) = 100

(8-3)

The particle size parameter 𝑀 is:

𝑀 = (𝑚𝑠𝑖𝑙𝑡 + 𝑚𝑣𝑓𝑠 ) ∙ (100 − 𝑚𝑐 )

(8-4)

𝑚𝑠𝑖𝑙𝑡 is the percentage of silt paricles (0.002-0.05 𝑚𝑚 diameter), 𝑚𝑣𝑓𝑠 is the percentage of very fine sand (0.05 – 0.1 𝑚𝑚 diameter), and 𝑚𝑐 is the percentage of clay (< 0.002 𝑚𝑚 diameter). 𝑂𝑀 is the percentage of organic matter, 𝑜𝑟𝑔𝐶 accounts for the percentage of organic carbon: 𝑂𝑀 = 1.72 ∙ 𝑜𝑟𝑐𝐶

(8-5)

The soil structure 𝑐𝑠𝑜𝑖𝑙𝑠𝑡𝑟 refers to aggregation of primary soil particles into compound particles. The soil structures are divided into 1. 2. 3. 4.

very fine granular fine granular medium or coarse granular blocky, platy, prism-like or massive

Shape of structure Size classes

Platy

Prismatic and columnar

Blocky

Granular

Very fine

< 1 𝑚𝑚

< 10 𝑚𝑚

< 5 𝑚𝑚

< 1 𝑚𝑚

Fine

1- 2 𝑚𝑚

10-20 𝑚𝑚

5-10 𝑚𝑚

1-2 𝑚𝑚

Medium

2-5 𝑚𝑚

20-50 𝑚𝑚

10-20 𝑚𝑚

2-5 𝑚𝑚

Coarse

5-10 𝑚𝑚

50-100 𝑚𝑚

20-50 𝑚𝑚

5-10 𝑚𝑚

Very coarse

> 10 𝑚𝑚

> 100 𝑚𝑚

> 50 𝑚𝑚

> 10 𝑚𝑚

Cover and Management Factor C 113


The plant canopy affects erosion by reducing the effective rainfall energy of intercepted raindrops. Water drops falling from the canopy may regain appreciable velocity but it will be less than the terminal velocity of free-falling raindrops. The USLE cover and management factor, 𝐶𝑈𝑆𝐿𝐸 , is defined as the ratio of soil loss from land cropped under specified conditions to the corresponding loss from clean-tilled, continuous fallow. The average fall height of drops from the canopy and the density of the canopy will determine the reduction in rainfall energy expended at the soil surface. A given percentage of residue on the soil surface is more effective that the same percentage of canopy cover. Residue intercepts falling raindrops so near the surface that drops regain no fall velocity. Residue also obstructs runoff flow, reducing its velocity and transport capacity. Because plant cover varies during the growth cycle of the plant, SWAT updates 𝐶𝑈𝑆𝐿𝐸 daily using the equation: 𝐶𝑈𝑆𝐿𝐸 = exp([𝑙𝑛(0.8) − 𝑙𝑛(𝐶𝑈𝑆𝐿𝐸,𝑚𝑛 )] ⋅ 𝑒𝑥𝑝[−0.00115 ⋅ 𝑟𝑠𝑑𝑠𝑢𝑟𝑓 ] + 𝑙𝑛[𝐶𝑈𝑆𝐿𝐸,𝑚𝑛 ])

(8-6)

where 𝐶𝑈𝑆𝐿𝐸,𝑚𝑛 is the minimum value for the cover and management factor for the land cover, and 𝑟𝑠𝑑𝑠𝑢𝑟𝑓 is the amount of residue on the soil surface [𝑘𝑔/ℎ𝑎]. The minimum 𝐶 factor can be estimated from a known average annual C factor using the following equation: 𝐶𝑈𝑆𝐿𝐸,𝑚𝑛 = 1.463 ln[𝐶𝑈𝑆𝐿𝐸,𝑎𝑎 ] + 0.1034

(8-7)

where 𝐶𝑈𝑆𝐿𝐸,𝑚𝑛 is the minimum C factor for the land cover and 𝐶𝑈𝑆𝐿𝐸,𝑎𝑎 is the average annual 𝐶 factor for the land cover.

Support Practice Factor P The support practice factor, 𝑃𝑈𝑆𝐿𝐸 , is defined as the ratio of soil loss with a specific support practice to the corresponding loss with up-and-down slope culture. Support practices include contour tillage, stripcropping on the contour, and terrace systems. Stabilized waterways for the disposal of excess rainfall are a necessary part of each of these practices. Contour tillage and planting provides almost complete protection against erosion from storms of low to moderate intensity, but little or no protection against occasional severe storms that cause extensive break overs of contoured rows. Contouring is most effective on slopes of 3 to 8 percent. Values for 𝑃𝑈𝑆𝐿𝐸 and slope-length limits for contour support practices are given in the following Table 8-1:

114


Table 8-1: 𝑃𝑈𝑆𝐿𝐸 factor values and slope-length limits for contouring

Land Slope (%)

𝑷𝑼𝑺𝑳𝑬

Max. Length [𝒎]

1-2

0.6

122

3-5

0.5

91

6-8

0.5

61

9-12

0.6

37

13-16

0.7

24

17-20

0.8

18

21-25

0.9

15

Stripcropping is a practice in which contoured strips of sod are alternated with equal-width strips of row crop or small grain. Recommended values for contour stripcropping are given in Table 8-2: Table 8-2: 𝑃𝑈𝑆𝐿𝐸 factor values, maximum strip width and slope-length limits for contour stripcropping

𝑷𝑼𝑺𝑳𝑬

Land Slope (%)

C

Strip width [m]

Max. Length [𝒎]

A

B

1-2

0.3

0.45

0.6

40

244

3-5

0.25

0.38

0.5

30

183

6-8

0.25

0.38

0.5

30

122

9-12

0.3

0.45

0.6

24

73

13-16

0.35

0.52

0.7

24

49

17-20

0.4

0.6

0.8

18

37

21-25

0.45

0.68

0.9

15

30

Terraces are a series of horizontal ridges made in a hillside. There areseveral types of terraces. Broadbase terraces are constructed on gently sloping land and the channel and ridge are cropped the same as the interterrace area. The steep backslope terrace, where the backslope is in sod, is most common on steeper land. Impoundment terraces are terraces with underground outlets. Terraces divide the slope of the hill into segments equal to the horizontal terrace interval. With terracing, the slope length is the terrace interval. For broadbase terraces, the horizontal terrace interval is the distance from the center of the ridge to the center of the channel for the terrace below. The horizontal terrace interval for steep backslope terraces is the distance from the point where cultivation begins at the base of the ridge to the base of the frontslope of the terrace below. Values for 𝑷𝑼𝑺𝑳𝑬 for contour farming terraced fields are listed in Table 8-3: 115


Table 8-3: 𝑃𝑈𝑆𝐿𝐸 factor values for contour-farmed terraced fields1

Land Slope (%)

Computing sediment yield2

Farm planning Contour 𝑃𝑈𝑆𝐿𝐸 factor3

Stripcrop 𝑃𝑈𝑆𝐿𝐸 factor

Graded channels sod outlets

Steep backslope underground outlets

1-2

0.6

0.3

0.12

0.05

3-8

0.5

0.25

0.10

0.05

9-12

0.6

0.3

0.12

0.05

13-16

0.7

0.35

0.14

0.05

17-20

0.8

0.4

0.16

0.06

21-25

0.8

0.45

.018

0.06

Topographic Factor The topographic factor, 𝐿𝑆𝑈𝑆𝐿𝐸 , is the expected ratio of soil loss per unit area from a field slope to that from a 22.1-m length of uniform 9 percent slope under otherwise identical conditions. The topographic factor is calculated: 𝐿ℎ𝑖𝑙𝑙 𝑚 𝐿𝑆𝑈𝑆𝐿𝐸 = ( ) ⋅ (65.41 ⋅ sin2(𝛼ℎ𝑖𝑙𝑙 ) + 4.56 ⋅ sin(𝛼ℎ𝑖𝑙𝑙 ) + 0.065) 22.1

(8-8)

where 𝐿ℎ𝑖𝑙𝑙 is the slope length[𝑚], m is the exponential term, and 𝛼ℎ𝑖𝑙𝑙 is the angle of the slope. The exponential term, 𝑚, is calculated: 𝑚 = 0.6 ⋅ (1 − exp[−35.835 ⋅ 𝑠𝑙𝑝])

(8-9)

where 𝑠𝑙𝑝 is the slope of the HRU expressed as rise over run [𝑚/𝑚]. The relationship between 𝛼ℎ𝑖𝑙𝑙 and slp is: 𝑠𝑙𝑝 = tan 𝛼ℎ𝑖𝑙𝑙

(8-10)

1

Slope length is the horizontal terrace interval. The listed values are for contour farming. No additional contouring

factor is used in the computation 2 These values include entrapment efficiency and are used for control of offsite sediment within limits and forestimating the field’s contribution to watershed sediment yield. 3 Use these values for control of interterrace erosion within specified soil loss tolerances.

116


Coarse Fragment Factor The coarse fragment factor 𝐶𝐹𝑅𝐺 is calculated: 𝐶𝐹𝑅𝐺 = exp(−0.053 ⋅ 𝑟𝑜𝑐𝑘)

(8-11)

where 𝑟𝑜𝑐𝑘 is the percent rock in the first soil layer (%).

117

8 Soil Erosion 8.3 Excursion: SWAT Erosion Modelling in Ethiopia

8.3 Excursion: SWAT Erosion Modelling in Ethiopia4 8.3.1 Background

Figure 8-1: The Nile Basin (left) and the Blue Nile Basin (right) (MSc Theses H. Huber)

The Blue Nile catchment area comprises 311,548 𝑘𝑚2 , has a mean annual discharge of 48.5 𝑘𝑚3 and it contributes around 60% of the discharge reaching Aswan (Figure 8-1). The

4

Huber, H.: SWAT erosion modelling and HRU resolution effects on the output, applied to a small catchment near Lake Tana, Ethiopia; M.Sc. Thesis, TUM 2015 118

8 Soil Erosion 8.3 Excursion: SWAT Erosion Modelling in Ethiopia

river originates in Gish Abay in West Gojam, which flows as Gilgel Abay into Lake Tana. This lake is the biggest one in Ethiopia and is approximately 73 km long and 68 km wide. The Blue Nile enters a clay plain just before entering Sudan, where it stays for 630 km until it reaches Khartoum and joins the White Nile. Erosion is a major problem throughout the Blue Nile Basin. Great parts of people depend on subsistence agriculture and therefore are directly affected by soil degradation there. Reported soil loss rates vary between averagely 20 𝑡⁄ℎ𝑎 𝑦 and more than 150 𝑡⁄ℎ𝑎 𝑦. The transformation of great parts of the Ethiopian Highlands from forestland to cropland caused a remarkable reduction of vegetative cover protecting the soil. This results in an enhanced exposure to the different forces giving rise to soil erosion. The process of erosion can be divided into three steps:   

detachment of individual soil grains transportation of detached grains over the land surface deposition of grains as sediment

8.3.2 SWAT Model In a Master´s Thesis at our Chair a SWAT model was set up to study the soil erosion in the Ribb River catchment. A sub catchment of the Blue Nile, which drains into the Lake Tana. Ribb River encompasses 1495 𝑘𝑚2 and is located on the east side of Lake Tana. Countermeasure tests where primarily applied at field studies. Implementation of stone bunds where observed in northern Ethiopia at a variety of field studies over different times. All in all the effect on soil erosion was very positive, reducing it on average by 68%. Furthermore, favorable increased infiltration rates and improved crop yields were observed. For a successful implementation of technical long-term solutions, it is unavoidable to take care of social acceptance, economic and ecological viability. Testing of exclosures in the Tigray highlands of Ethiopia revealed a high sediment trapping capacity. Exclosures are areas closed for grazing and agriculture. SWAT was used to investigate different scenarios for sediment yield management. It was applied to the entire Blue Nile Basin. Three different scenarios were simulated: (i) filter strips, (ii) stone bunds and (iii) reforestation. The best result was obtained for the filter strips case, where filter strips reduced sediment yield by 44 %. Simulation of stone bunds and reforestation predicted a reduction of 41 and 11%, respectively. SWAT has proven as a powerful tool to assess different counter measures, but mere technical solutions are not the key method. Among others it is necessary to include land managers, farmers, and hydrologists along with local stakeholders. It was recognized that it is most important to create awareness among farmers to conserve the soil rather than sole construction of mechanical measures. Ethiopian agricultural policy has recognized the necessity of soil conservation for agricultural development and food supply.

119

8 Soil Erosion 8.4 Further Reading

8.4 Further Reading Read about the soil erosion and sedimentation simulation, and reduction strategies at the Blue Nile Basin in: Betrie G.et al., Sediment management modelling in the Blue Nile Basin using SWAT model, Hydrol. Earth Syst. Sci., 15, 807–818, 2011 http://dx.doi.org/10.5194/hess-15-807-2011 Read about the integration of remotely sensed C Factor into SWAT for modelling sediment yield: Song X. et al., Integration of remotely sensed C factor into SWAT for modelling sediment yield, Hydrological Processes, 25, 3387-3398 http://dx.doi.org/ 10.1002/hyp.8066

8.5 Nomenclature 𝑎𝑟𝑒𝑎ℎ𝑟𝑢 𝐶𝑢𝑠𝑙𝑒

USLE cover and management factor

𝐶𝑢𝑠𝑙𝑒,𝑎𝑎

Average annual 𝐶 factor for the land cover

𝐶𝑢𝑠𝑙𝑒,𝑚𝑛

Minimum 𝐶 factor for the land cover

𝐶𝐹𝑅𝐺

coarse fragment factor

𝐸𝐼𝑈𝑆𝐿𝐸

Rainfall erosion index [0.017 𝑡𝑜𝑛𝑠 𝑐𝑚⁄(𝑚2 ℎ) ]

𝐾𝑈𝑆𝐿𝐸

USLE soil erodibility factor [0.013 𝑡𝑜𝑛𝑠 𝑚2 ℎ⁄(𝑚3 𝑡𝑜𝑛𝑠 𝑐𝑚) ]

𝐿ℎ𝑖𝑙𝑙 𝐿𝑆𝑢𝑠𝑙𝑒

120

Area of the HRU [ℎ𝑎]

Slope length [𝑚] USLE topographic factor

𝑚

Exponential Term

𝑚𝑐

Percentage of clay [%]

𝑚𝑠𝑖𝑙𝑡

Percentage of silt particles [%]

𝑚𝑣𝑓𝑠

Percentage of very fine sand [%]

𝑂𝑀

Percentage of organic matter [%]

𝑜𝑟𝑔𝐶

Percentage of organic carbon [%]

𝑃𝑢𝑠𝑙𝑒

USLE support practice factor


Surface runoff [𝑚𝑚𝐻2 𝑂/ℎ𝑎]


Peak runoff rate [𝑚³/𝑠]

𝑟𝑜𝑐𝑘

Percent rock in the first soil layer [%]

𝑠𝑒𝑑

Sediment yield [𝑡𝑜𝑛𝑠/ℎ𝑎]

𝛼ℎ𝑖𝑙𝑙

Angle of the slope [°]

9 Crop Growth and Crop Yield 9.1 Potential Heat Units

9 Crop Growth and Crop Yield 9.1 Potential Heat Units The crop growth is controlled by the growth cycle of the specific plant, which is determined by the temperature. It is the most important factor governing the growth and bio mass production of plants. Each plant has its own temperature range for minimum, maximum and optimum growth. Before growth can take place the temperature needs to exceed the minimum base temperature, as shown in Figure 9-1. In general, the higher the temperature gets the faster the plant will grow. When the optimum temperature is reached the growth rate will slow down and finally stop when a maximum temperature is reached.

Figure 9-1: Daily temperature profile and base temperature for a specific crop; growth will occur above the base temperature

As a plant will not grow when the mean temperature falls below its base temperature, the only portion of the mean daily temperature that contributes towards the plant´s development is the amount that exceeds the base temperature. This information is important to determine the plants maturity. Therefore, to quantify the total heat requirements of a plant, the accumulation of daily mean air temperature above the plant´s base temperature is recorded over the period of the plant´s growth and expressed in terms of heat units. Example:

Assume apples are growing with a base temperature of 7 °C. If the mean temperature on a given day is 22 °C, the heat units accumulated on that day are 22 – 7 = 15 heat units. This information helps to determine the maturity date of the apples.

SWAT uses a direct simulated heat index. Each degree of the daily mean temperature above the base temperature accounts for one unit. This method assumes that the growth rate is di121

9 Crop Growth and Crop Yield 9.2 Dormancy

rectly proportional to the increase in temperature. SWAT assumes all heat above the base temperature to accelerate the crop growth, so it does not account for harmful high temperatures. The Heat Units can be calculated as:

𝐻𝑈 = ̅̅̅̅ 𝑇𝑎𝑣 − 𝑇𝑏𝑎𝑠𝑒 𝑤ℎ𝑒𝑛 ̅̅̅̅ 𝑇𝑎𝑣 > 𝑇𝑏𝑎𝑠𝑒

(9-1)

With 𝐻𝑈 as the number of heat units, 𝑇𝑎𝑣 [°𝐶] the mean air temperature for a day, 𝑇𝑏𝑎𝑠𝑒 the plant´s base temperature [°𝐶]. The number of heat units required for a plant to reach maturity after 𝑚 days is calculated with the potential heat units 𝑃𝐻𝑈: 𝑚

𝑃𝐻𝑈 = ∑ 𝐻𝑈

(9-2)

𝑑=1

When calculating the potential heat units for a specific plant, the number of days to reach maturity must be known. For most crops, these numbers are quantified and easily accessible.

9.2 Dormancy SWAT assumes dormancy periods during which plants do not grow. This might be because of a lack of sun light or temperature. The beginning and end of dormancy are defined by a threshold daytime length. The threshold day length to start dormancy 𝑇𝐷𝐿,𝑡ℎ𝑟 [ℎ] is calculated as:

𝑇𝐷𝐿,𝑡ℎ𝑟 = 𝑇𝐷𝐿,𝑚𝑛 + 𝑡𝑑𝑜𝑟𝑚

(9-3)

𝑇𝐷𝐿,𝑚𝑛 is the minimum day length [ℎ] for the entire watershed during the year, and 𝑡𝑑𝑜𝑟𝑚 is the dormancy threshold [ℎ]. When the days become shorter than 𝑇𝐷𝐿,𝑚𝑛 in the fall, the plants enter dormancy; they stop it again in springe once the day length exceeds𝑇𝐷𝐿,𝑚𝑛 . The dormancy threshold 𝑡𝑑𝑜𝑟𝑚 itself is dependent on the latitude 𝜙 [°]:

𝑡𝑑𝑜𝑟𝑚 = 1 𝜙 − 20 𝑡𝑑𝑜𝑟𝑚 = 20 𝑡𝑑𝑜𝑟𝑚 = 0

122

𝑖𝑓 𝜙 > 40° N or S 𝑖𝑓 20° N or S < 𝜙 < 40° N or S 𝑖𝑓 𝜙 < 20° N or S

(9-4)

9 Crop Growth and Crop Yield 9.3 Biomass production and Crop Yield

9.3 Biomass production and Crop Yield 9.3.1 Biomass Production In SWAT the plant growth is modelled by simulating lead area development, interception of light and assuming a plant species-specific radiation-use efficiency. The amount of daily solar radiation intercepted by the leaf area of the plant calculated with Beer‘s law:

𝐻𝑝ℎ𝑜𝑠𝑦𝑛 = 0.5 ∙ 𝐻𝑑𝑎𝑦 ∙ (1 − exp(−𝑘𝑙 ∙ 𝐿𝐴𝐼))

(9-5)

𝐻𝑝ℎ𝑜𝑠𝑦𝑛 [𝑀𝐽⁄𝑚²] is the intercepted photosynthetically active radiation on a given day, 𝐻𝑑𝑎𝑦 [𝑀𝐽⁄𝑚² 𝑑] is the solar radiation reaching the ground on the current day of the simulation, 𝑘𝑙 is the light extinction coefficient [-] and 𝐿𝐴𝐼 is the leaf area index of the canopy. The photosynthetically active light has a wavelength between 400 and 700 nm. Direct solar radiation contains about 45 % of photosynthetically active radiation, whereas diffuse radiation has 60 %. Radiation use efficiency is the amount of dry biomass produced per unit intercepted solar radiation. The radiation-use efficiency is defined in the plant growth database and is assumed to be independent of the plant´s growth stage. The maximum increase in biomass on a given day that will result from the intercepted photosynthetically active radiation is approximated as:

∆𝑏𝑖𝑜 = 𝑅𝑈𝐸 ∙ 𝐻𝑝ℎ𝑜𝑠𝑦𝑛

(9-6)

∆𝑏𝑖𝑜 is the potential increase in total plant biomass on a given day [𝑘𝑔⁄ℎ𝑎], 𝑅𝑈𝐸 is the radiation use efficency of the earth [10−1 𝑔⁄𝑀𝐽]. The total biomass 𝑏𝑖𝑜 [𝑘𝑔⁄ℎ𝑎] on a given day is: 𝑑

𝑏𝑖𝑜 = ∑ ∆𝑏𝑖𝑜𝑖

(9-7)

𝑖=1

In dependence of the different CO2 levels [𝑝𝑝𝑚𝑣] in the atmosphere the radiation use efficiency of the plant 𝑅𝑈𝐸 has to be adjusted

𝑅𝑈𝐸 =

100 ⋅ 𝐶𝑂2 𝐶𝑂2 + exp (𝑟1 − 𝑟2 ⋅ 𝐶𝑂2 )

(9-8)

The shape coefficients for the radiation use efficiency curves 𝑟1, 𝑟2 are determined as:

𝑟1 = 𝑙𝑛 [

𝐶𝑂2 , 𝑎𝑚𝑏 − 𝐶𝑂2,𝑎𝑚𝑏 ] + 𝑟2 ⋅ 𝐶𝑂2,𝑎𝑚𝑏 (0.01 ⋅ 𝑅𝑈𝐸𝑎𝑚𝑏 )

(9-9) 123


(𝑙𝑛 [ 𝑟2 =

𝐶𝑂2,𝑎𝑚𝑏 𝐶𝑂2,ℎ𝑖 − 𝐶𝑂2,𝑎𝑚𝑏 ] − 𝑙𝑛 [ − 𝐶𝑂2,ℎ𝑖 ]) (0.01 ⋅ 𝑅𝑈𝐸𝑎𝑚𝑏 ) (0.01 ⋅ 𝑅𝑈𝐸ℎ𝑖 ) 𝐶𝑂2,ℎ𝑖 − 𝐶𝑂2,𝑎𝑚𝑏

(9-10)

𝐶𝑂2,𝑎𝑚𝑏 is the ambient atmospheric CO2 concentration of carbon dioxide in the atmosphere [𝑝𝑝𝑚𝑣] (default in SWAT is set to 330 𝑝𝑝𝑚𝑣, even though the ambient atmospheric CO2 concentration is now higher), 𝑅𝑈𝐸𝑎𝑚𝑏 is the radiation use efficency of the plant at ambient atmospheric CO2 concentration [10−1 𝑔⁄𝑀𝐽], 𝐶𝑂2,ℎ𝑖 is an elevated atmospheric concentration [𝑝𝑝𝑚𝑣], and 𝑅𝑈𝐸ℎ𝑖 is the radiaition use efficiency of the plant at elevated atmospheric CO2 concentration [10−1 𝑔⁄𝑀𝐽].

9.3.2 Nitrogen Uptake Plant nitrogen uptake is controlled by the plant nitrogen equation. The plant nitrogen equation calculates the fraction of nitrogen in the plant biomass as a function of growth stage given optimal growing conditions:

𝑓𝑟𝑁 = (𝑓𝑟𝑁,1 − 𝑓𝑟𝑁,3 ) ∙ ⌊1 −

𝑓𝑟𝑃𝐻𝑈 ⌋ + 𝑓𝑟𝑁,3 𝑓𝑟𝑃𝐻𝑈 + 𝑒𝑥𝑝(𝑛1 − 𝑛2 ∙ 𝑓𝑟𝑃𝐻𝑈 )

(9-11)

𝑓𝑟𝑁 is the optimal fraction of nitrogen in the plant biomass for current growth stage. 𝑓𝑟𝑁,1 the normal fraction of nitrogen in the plant biomass at emergence, 𝑓𝑟𝑁,3 the normal fraction of nitrogen in the plant biomass at maturity, 𝑓𝑟𝑃𝐻𝑈 the fraction of potential heat units accumulated for the plant on a given day in the growing season, and 𝑛1 , 𝑛2 are shape coefficients. To determine the mass of nitrogen that should be stored in the plant biomass on a given day, the nitrogen fraction is multiplied by the total plant biomass:

𝑏𝑖𝑜𝑁,𝑜𝑝𝑡 = 𝑓𝑟𝑁 ∙ 𝑏𝑖𝑜

(9-12)

𝑏𝑖𝑜 is the total biomass on a given day [𝑘𝑔/ℎ𝑎]. Originally, SWAT calculated the plant nitrogen demand for a given day by taking the difference between the nitrogen content of the plant bio mass expected for the plant’s growth stage and the actual nitrogen content 𝑁𝑢𝑝 = 𝑏𝑖𝑜𝑁,𝑜𝑝𝑡 − 𝑏𝑖𝑜𝑁 . This method was found to calculate an excessive nitrogen demand immediately after a cutting (i.e. harvest operation). The depth distribution of nitrogen uptake is calculated with the function:

𝑁𝑢𝑝,𝑧 =

𝑁𝑢𝑝 𝑧 ⋅ [1 − exp (−𝛽𝑛 ⋅ )] [1 − exp(−𝛽_𝑛) ] 𝑧𝑟𝑜𝑜𝑡

(9-13)

where 𝑁𝑢𝑝,𝑧 is the potential nitrogen uptake from the soil surface to depth z [𝑘𝑔 𝑁⁄ℎ𝑎], 𝑁𝑢𝑝 is the potential nitrogen uptake [𝑘𝑔 𝑁/ℎ𝑎], 𝛽𝑛 is the nitrogen uptake distribution parameter, 𝑧 is 124


the depth from the soil surface [𝑚𝑚], and 𝑧𝑟𝑜𝑜𝑡 is the depth of root development in the soil [𝑚𝑚].

9.3.3 Crop Yield The harvest operation in SWAT removes a portion of the plant biomass from the HRU as yield. The nutrients and plant material contained in the yield are lost and no longer available in the system (watershed). The fraction of the above-ground plant biomass removed as dry economic yield is called the harvest index. For the majority of crops it is between 0.0 and 1.0. However, crops like potatoes, whose crop roots are also harvested may have a harvest index greater than 1.0. The economic yield of cash crops is the reproductive portion of the plant. Often, the harvest index is relatively stable across a range of environmental conditions. SWAT calculates the potential harvest index 𝐻𝐼 for each day of the plant´s growing season using the relationship:

𝐻𝐼 = 𝐻𝐼𝑜𝑝𝑡 ⋅

100 ⋅ 𝑓𝑟𝑃𝐻𝑈 100 ⋅ 𝑓𝑟𝑃𝐻𝑈 + exp(11.1 − 10 ⋅ 𝑓𝑟𝑃𝐻𝑈 )

(9-14)

𝐻𝐼𝑜𝑝𝑡 is the potential harvest index for the plant at maturity given ideal growing conditions, and 𝑓𝑟𝑃𝐻𝑈 is the fraction of potential heat units accumulated for the plant on a given day in the growing season. Figure 9-2 shows the variation of the optimal harvest index (𝐻𝐼/𝐻𝐼𝑜𝑝𝑡 ) during the fraction of growth season (which starts with 0 and ends with 1).

125


Figure 9-2: Variation in optimal harvest index ( 𝐻𝐼𝑖 /𝐻𝐼𝑜𝑝𝑡 ) with fraction of growing season (𝑓𝑟𝑃𝐻𝑈 )

The final crop yield 𝑦𝑙𝑑 [𝑘𝑔/ℎ𝑎] calculated as:

𝑦𝑙𝑑 = 𝑏𝑖𝑜𝑎𝑔 ⋅ 𝐻𝐼 𝑦𝑙𝑑 = 𝑏𝑖𝑜 ⋅ (1 −

1 ) (1 + 𝐻𝐼)

𝑤ℎ𝑒𝑛 𝐻𝐼 ≤ 1.00

(9-15)

𝑤ℎ𝑒𝑛 𝐻𝐼 > 1.00

(9-16)

𝑏𝑖𝑜 is the total plant biomass on the day of the harvest [𝑘𝑔/ℎ𝑎], and 𝑏𝑖𝑜𝑎𝑔 is the above ground biomass on the day of the harvest [𝑘𝑔/ℎ𝑎], it is calculated as:

𝑏𝑖𝑜𝑎𝑔 = (1 − 𝑓𝑟𝑟𝑜𝑜𝑡 ) ⋅ 𝑏𝑖𝑜

(9-17)

With 𝑓𝑟𝑟𝑜𝑜𝑡 as the fraction of total biomass in the roots on a given day in the growing season.

126

9 Crop Growth and Crop Yield 9.4 Stress factors

9.4 Stress factors Plant growth can be delayed or limited because of extreme temperatures, and a lack of water, nitrogen or phosphorus. The amount of stress for each of these four parameters is calculated on a daily basis using the equations summarized in the following sections.

9.4.1 Water Stress Under perfect water conditions water stress is 0.0 and approaches 1.0 as the soil water conditions vary from the optimum. By comparison of actual and potential plant transpiration water stress can be simulated as:

𝑤𝑠𝑡𝑟𝑠 = 1 −

𝑤𝑎𝑐𝑡𝑢𝑎𝑙𝑢𝑝 𝐸𝑡,𝑎𝑐𝑡 =1− 𝐸𝑡 𝐸𝑡

(9-18)

Where 𝑤𝑠𝑡𝑟𝑠 is the water stress for the given day [𝑚𝑚𝐻2 𝑂], 𝐸𝑡 is the maximum plant transpiration on a given day [𝑚𝑚𝐻2 𝑂], 𝐸𝑡,𝑎𝑐𝑡 is the actual amount of transpiration on a given day [𝑚𝑚𝐻2 𝑂], and 𝑤𝑎𝑐𝑡𝑢𝑎𝑙𝑢𝑝 is the total water uptake for the day [𝑚𝑚𝐻2 𝑂] (see also Section 4.4).

9.4.2 Temperature Stress Temperature stress can be expressed as a function of the daily average air temperature and the optimal temperature for plant growth. Near the optimal temperature there will be no stress for the plant growth. The equations to calculate temperature stress 𝑡𝑠𝑡𝑟𝑠 as a fraction of optimal plant growth are:

𝑤ℎ𝑒𝑛 𝑇̅𝑎𝑣 ≤ 𝑇𝑏𝑎𝑠𝑒

𝑡𝑠𝑡𝑟𝑠 = 1

(9-19)

2

𝑡𝑠𝑡𝑟𝑠 = 1 − 𝑒𝑥𝑝 [

−0.1054 ⋅ (𝑇𝑜𝑝𝑡 − 𝑇̅𝑎𝑣 ) ] (𝑇̅𝑎𝑣 − 𝑇𝑏𝑎𝑠𝑒 )2

𝑡𝑠𝑡𝑟𝑠 = 1 − 𝑒𝑥𝑝 [ 𝑡𝑠𝑡𝑟𝑠 = 1

−0.1054 ⋅ (𝑇𝑜𝑝𝑡 − 𝑇̅𝑎𝑣 )

𝑤ℎ𝑒𝑛 𝑇𝑏𝑎𝑠𝑒 < 𝑇̅𝑎𝑣 ≤ 𝑇𝑜𝑝𝑡

2

2 (2 ⋅ 𝑇𝑜𝑝𝑡 − 𝑇̅𝑎𝑣 − 𝑇𝑏𝑎𝑠𝑒 )

] 𝑤ℎ𝑒𝑛 𝑇𝑜𝑝𝑡 < 𝑇̅𝑎𝑣 ≤ 2 ⋅ 𝑇𝑜𝑝𝑡 − 𝑇𝑏𝑎𝑠𝑒 𝑤ℎ𝑒𝑛 𝑇̅𝑎𝑣 > 2 ⋅ 𝑇𝑜𝑝𝑡 − 𝑇𝑏𝑎𝑠𝑒

̅̅̅̅ 𝑇 𝑎𝑣 is the mean air temperature [°𝐶], 𝑇𝑏𝑎𝑠𝑒 is the plant´s base or minimum temperature for growth [°𝐶], and 𝑇𝑜𝑝𝑡 is the plant´s optimal temperature for growth [°𝐶]. Figure 9-3 illustrates the impact of mean daily air temperature on the plant growth for a plant with a base temperature of 0°C and an optimal temperature of 15°C. (NOTE: It does not show 𝑡𝑠𝑡𝑟𝑠 itself, it shows the fraction of optimal growth, in dependence of 𝑡𝑠𝑡𝑟𝑠.)

127

9 Crop Growth and Crop Yield 9.4 Stress factors

Figure 9-3: Impact of mean air temperature on plant growth for a plant with Tbase=0°C and Topt=15 °C

9.4.3 Nitrogen Stress Nitrogen stress is calculated in SWAT only for non-legumes (e.g. oat, wheat, or maize), this is per definition not allowed. Nitrogen stress is quantified by comparing actual and optimal plant nitrogen levels. Nitrogen stress varies non-linearly between 0.0 at the best conditions and 1.0 when the nitrogen content of the plant is 50% or less of the optimal value. The stress 𝑛𝑠𝑡𝑟𝑠 is computed as:

𝑛𝑠𝑡𝑟𝑠 = 1 −

𝜑𝑛 𝜑𝑛 + exp[3.535 − 0.02597 ⋅ 𝜑𝑛 ]

(9-20)

𝜑𝑛 is a scaling factor for nitrogen stress, which is calculated as:

𝑏𝑖𝑜𝑁 𝜑𝑛 = 200 ⋅ ( − 0.5) 𝑏𝑖𝑜𝑁,𝑜𝑝𝑡

(9-21)

𝑏𝑖𝑜𝑁,𝑜𝑝𝑡 is the optimal mass of nitrogen stored in the plant at the current stage of growth [𝑘𝑔 𝑁/ℎ𝑎]. 𝑏𝑖𝑜𝑁 is the actual mass of nitrogen stored in the plant [𝑘𝑔 𝑃/ℎ𝑎].

128

9 Crop Growth and Crop Yield 9.5 Actual Growth

9.4.4 Phosphorus Stress Phosphorous stress is quantified by comparing actual and optimal plant nitrogen levels. Phosphorous stress varies non-linearly between 0.0 at the best conditions and 1.0 when the phosphorous content of the plant is 50% or less of the optimal value. The stress 𝑝𝑠𝑡𝑟𝑠 is computed as:

𝑝𝑠𝑡𝑟𝑠 = 1 −

𝜑𝑝 𝜑𝑝 + exp[3.535 − 0.02597 ⋅ 𝜑𝑝 ]

(9-22)

𝜑𝑝 is a scaling factor for phosphorous stress, which is calculated as:

𝑏𝑖𝑜𝑝 𝜑𝑝 = 200 ⋅ ( − 0.5) 𝑏𝑖𝑜𝑝,𝑜𝑝𝑡

(9-23)

𝑏𝑖𝑜𝑝,𝑜𝑝𝑡 is the optimal mass of phosphorous stored in the plant at the current stage of growth [𝑘𝑔 𝑁/ℎ𝑎]. 𝑏𝑖𝑜𝑝 is the actual mass of phosphorous stored in the plant [𝑘𝑔 𝑁/ℎ𝑎].

9.5 Actual Growth The plant growth factor quantifies the fraction of potential growth achieved on a given day and is calculated as:

𝛾𝑟𝑒𝑔 = 1 − max(𝑤𝑠𝑡𝑟𝑠, 𝑡𝑠𝑡𝑟𝑠, 𝑛𝑠𝑡𝑟𝑠, 𝑝𝑠𝑡𝑟𝑠)

(9-24)

Where 𝛾𝑟𝑒𝑔 is the plant growth factor (0.0-1.0). The potential biomass production (see Section 9.3.1) can then be adjusted if one of the four stress factors is greater than 0.0 as:

∆𝑏𝑖𝑜𝑎𝑐𝑡 = ∆𝑏𝑖𝑜 ⋅ 𝛾𝑟𝑒𝑔

(9-25)

∆𝑏𝑖𝑜𝑎𝑐𝑡 accounts for the actual increase in total plant biomass on a given day [𝑘𝑔/ℎ𝑎]. The potential leaf area added on a given day is also adjusted daily for plant stress:

∆𝐿𝐴𝐼𝑎𝑐𝑡,𝑖 = ∆𝐿𝐴𝐼𝑖 ⋅ √𝛾𝑟𝑒𝑔

(9-26)

∆𝐿𝐴𝐼𝑎𝑐𝑡,𝑖 is the actual leaf area added on day 𝑖, ∆𝐿𝐴𝐼𝑖 is the potential leaf area added on day 𝑖.

9.6 Further Reading Read about how SWAT is used to model vegetation and crop growth in a tropical central Brazilian watershed: 129

9 Crop Growth and Crop Yield 9.7 Nomenclature

Strauch, M. & Volk, M.: SWAT plant growth modification for improved modelling of perennial vegetation in the tropics, Ecological Modelling 269 (2013) 98–112 http://dx.doi.org/10.1016/j.ecolmodel.2013.08.013

Read about crop yield estimation using remote sensing techniques based on evapotranspiration (evaporative fraction, EF): Bastiaanssen and Ali: A new crop yield forecasting model based on satellite measurements applied across the Indus Basin, Pakistan, Agriculture, ecosystems and environment 94 (2003) 321-340 http://dx.doi.org/ 10.1016/S0167-8809(02)00034-8

9.7 Nomenclature 𝑏𝑖𝑜 𝑏𝑖𝑜𝑎𝑔

Above ground biomass on the day of the harvest [𝑘𝑔/ℎ𝑎]

𝑏𝑖𝑜𝑁

Actual mass of nitrogen stored in the plant [𝑘𝑔 𝑁/ℎ𝑎]

𝑏𝑖𝑜𝑁,𝑜𝑝𝑡 𝑏𝑖𝑜𝑝

Optimal mass of nitrogen stored in the plant at the current stage of growth [𝑘𝑔 𝑁/ℎ𝑎] Actual mass of phosphorous stored in the plant [𝑘𝑔 𝑁/ℎ𝑎]

𝑏𝑖𝑜𝑝,𝑜𝑝𝑡

Optimal mass of phosphorous stored in the plant at the current stage of growth [𝑘𝑔 𝑁/ℎ𝑎]

𝐶𝑂2,𝑎𝑚𝑏

ambient atmospheric CO2 concentration of carbon dioxide in the atmosphere [𝑝𝑝𝑚𝑣]

𝐶𝑂2,ℎ𝑖 𝐸𝑡

Elevated atmospheric concentration [𝑝𝑝𝑚𝑣] Maximum plant transpiration on a given day [𝑚𝑚𝐻2 𝑂]

𝐸𝑡,𝑎𝑐𝑡

Actual amount of transpiration on a given day [𝑚𝑚𝐻2 𝑂]

𝑓𝑟𝑃𝐻𝑈

Fraction of potential heat units accumulated for the plant

𝑓𝑟𝑟𝑜𝑜𝑡

Fraction of total biomass in the roots

𝐻𝑑𝑎𝑦

Solar radiation reaching the ground [𝑀𝐽⁄𝑚² 𝑑]

𝐻𝐼𝑜𝑝𝑡

Potential harvest index

𝐻𝑝ℎ𝑜𝑠𝑦𝑛

130

Total biomass [𝑘𝑔⁄ℎ𝑎]

Intercepted photosynthetically active radiation [𝑀𝐽⁄𝑚²]

𝐻𝑈

Number of heat units

𝑘𝑙

Light extinction coefficient [-]

𝐿𝐴𝐼

Leaf area index of the canopy

𝑛𝑠𝑡𝑟𝑠

Nitrogen stress [%]

𝑃𝐻𝑈

Potential heat units

𝑝𝑠𝑡𝑟𝑠

Phosphorous stress [%]

𝑅𝑈𝐸

Radiation use efficiency of the earth [10−1 𝑔⁄𝑀𝐽]

9 Crop Growth and Crop Yield 9.7 Nomenclature

𝑅𝑈𝐸𝑎𝑚𝑏

Radiation use efficiency of the plant at ambient atmospheric CO 2 concentration [10−1 𝑔⁄𝑀𝐽]

𝑅𝑈𝐸ℎ𝑖

Radiation use efficiency of the plant at elevated atmospheric CO2 concentration [10−1 𝑔⁄𝑀𝐽]

𝑟1, 𝑟2

Shape coefficients for the radiation use efficiency [-]

𝑇𝑎𝑣

Mean air temperature for a day [°𝐶]

𝑇𝑏𝑎𝑠𝑒

Plant´s base temperature [°𝐶]

𝑇𝐷𝐿,𝑚𝑛

Minimum day length [ℎ]

𝑇𝑜𝑝𝑡 𝑡𝑑𝑜𝑟𝑚

Plant´s optimal temperature for growth [°𝐶] Dormancy threshold [ℎ]

𝑤𝑎𝑐𝑡𝑢𝑎𝑙𝑢𝑝

Total water uptake for the day [𝑚𝑚𝐻2 𝑂]

𝑤𝑠𝑡𝑟𝑠

Water stress for the given day [𝑚𝑚𝐻2 𝑂]

∆𝐿𝐴𝐼𝑎𝑐𝑡,𝑖 ∆𝑏𝑖𝑜 ∆𝑏𝑖𝑜𝑎𝑐𝑡

Actual leaf area added on day 𝑖 Potential increase in total plant biomass on a given day [𝑘𝑔⁄ℎ𝑎] Actual increase in total plant biomass on a given day [𝑘𝑔/ℎ𝑎]

𝛾𝑟𝑒𝑔

plant growth factor [-]

𝜑𝑛

Scaling factor for nitrogen stress [-]

𝜑𝑝

Scaling factor for phosphorous stress [-]

131

10 Water Quality Modelling 10.1 Problems with Nitrification and Eutorphication

10 Water Quality Modelling 10.1 Problems with Nitrification and Eutorphication Nutrients, such as nitrogen and phosphorous, are a serious problem threatening water quality. Excessive nutrient loads cause water quality deterioration, including toxic algal blooms, oxygen deficiency, fish death and eutrophication of the river networks and lakes. Arable land are a major source of nitrogen (𝑁) and phosphorus (𝑃) due to intensive agricultural activities like fertilization. From these areas considerable nutrients are discharged into natural water bodies. Thus, 𝑁 and 𝑃 transport processes from agricultural land into water bodies is a significant issue for environment managers and policy makers worldwide. Unlike point-source pollution, nutrient fluxes from agricultural land are difficult to measure and control, because they are heterogeneously distributed and derive from a variety of diffuse sources and may occur randomly and intermittently. Furthermore, transport processes are complex, since they are controlled by a variety of natural and anthropogenic driving forces such as hydrology, climate, geology, soil characteristic, and land use. Despite the uncertainty related to model predictions, a process-based modeling approach is necessary to simulate nutrient fate and transport at the catchment scale and to support water managers and decision makers. Process based models consist of a rather complete representation of the environmental system, which combines hydrological, soil and nutrient processes. These models are able to calculate long time series of relevant physical quantities (e.g., nutrient fluxes) with variable spatial distributions.

10.2 Nitrogen and Phosphorous Cycles 10.2.1

Nitrogen Cycle

Apart of carbon, oxygen and hydrogen, plants require nitrogen more than any other chemical compound for their growth. The nitrogen cycle is a dynamics system which occurs in water, air and soil. In SWAT it is modeled in the shallow aquifer and in the soil profile. There are three major forms of organic nitrogen in mineral soils: nitrogen in humus, nitrogen held by colloids, and nitrogen in solution. Nitrogen can be added to the soil by fertilization, the application of residue, fixation through bacteria, and rain. Nitrogen is removed from the soil by plant uptake, leaching, denitrification, volatilization, and erosion. Nitrogen is also considered to be a very reactive element, as it has the ability to exist in a different number of valence states, what makes it a highly mobile element. To predict the movement between the different pools is the soil is difficult, but important to successful management of this element. SWAT considers five different pools of nitrogen in the soil, as shown in Figure 10-1. Two pools are of inorganic form, 𝑁𝐻4+ and 𝑁𝑂3− (Nitrate). The other three pools are organic forms of nitrogen. Fresh organic 𝑁 is associated with crop residue and microbial biomass while the active and stable organic 𝑁 pools are associated with soil humus. The organic nitrogen associated

132

10 Water Quality Modelling 10.2 Nitrogen and Phosphorous Cycles

with humus is partitioned into two pools to account for the variation in availability of active humic substances, which can mineralize to 𝑁𝑂3− .

Figure 10-1: SWAT nitrogen pools and processes that move nitrogen between the pools

10.2.2

Phosphorous Cycle

Phosphorus may be added to the soil by fertilizer, manure or residue application. Phosphorus is removed from the soil by plant uptake and erosion. The three major forms of phosphorus in mineral soils are organic phosphorus associated with humus, insoluble forms of mineral phosphorus, and plant-available phosphorus in soil solution. Unlike nitrogen which is highly mobile, phosphorus solubility is limited in most environments. SWAT monitors six different pools of phosphorus in the soil (Figure 10-2). Three pools are inorganic forms of phosphorus while the other three pools are organic forms of phosphorus. Fresh organic P is associated with crop residue and microbial biomass while the active and stable organic P pools are associated with the soil humus. The organic phosphorus associated with humus is partitioned into two pools to account for the variation in availability of humic substances to mineralization. Soil inorganic P is divided into solution, active, and stable pools. The solution pool is in rapid equilibrium (several days or weeks) with the active pool. The active pool is in slow equilibrium with the stable pool.

133

10 Water Quality Modelling 10.3 Processes in Soils

Figure 10-2: SWAT soil phosphorus pools and processes that move P between the pools

10.3 Processes in Soils 10.3.1

Nitrogen Fixation

Legumes are able to obtain a portion of their nitrogen demand through fixation of atmospheric 𝑁2 performed by rhizobia living in association with the plant. In exchange for nitrogen, the plant supplies the bacteria with carbohydrates. SWAT simulates nitrogen fixation by legumes when the soil does not supply the plant with the amount of nitrogen needed for growth. The nitrogen obtained by fixation is incorporated directly into the plant biomass and never enters the soil. If nitrate levels in the root zone are insufficient to meet the demand of a legume, SWAT allows the plant to obtain additional nitrogen through nitrogen fixation. Nitrogen fixation is calculated as a function of soil water, soil nitrate content and growth stage of the plant:

𝑁𝑓𝑖𝑥 = 𝑁𝑑𝑒𝑚𝑎𝑛𝑑 ⋅ 𝑓𝑔𝑟 ⋅ min(𝑓𝑠𝑤 , 𝑓𝑁𝑂3 , 1)

(10-1)

where 𝑁𝑓𝑖𝑥 is the amount of nitrogen added to the plant biomass by fixation [𝑘𝑔 𝑁/ℎ𝑎], 𝑁𝑑𝑒𝑚𝑎𝑛𝑑 is the plant nitrogen demand not met by uptake from the soil [𝑘𝑔 𝑁/ℎ𝑎], 𝑓𝑔𝑟 is the growth stage factor (0.0-1.0), 𝑓𝑠𝑤 is the soil water factor (0.0-1.0), and 𝑓𝑛𝑜3 is the soil nitrate factor (0.0-1.0). The maximum amount of nitrogen that can be fixed by the plant on a given day is 𝑁𝑑𝑒𝑚𝑎𝑛𝑑 .

10.3.2

Degradation of Nitrogen

The user can define the amount of nitrate and organic nitrogen contained in humic substances for all soil layers at the beginning of the simulation. Initial nitrate levels in the soil are varied by depth using the equation:

134


𝑁𝑂3𝑐𝑜𝑛𝑐,𝑧 = 7 ⋅ exp(−

𝑧 ) 1000

(10-2)

𝑁𝑂3𝑐𝑜𝑛𝑐,𝑧 is the concentration [𝑚𝑔/𝑘𝑔] of nitrate in the soil depth 𝑧 [𝑚𝑚] from the soil surface. As surface concentration a value of 7 𝑚𝑔/𝑘𝑔 was assumed. Figure 10-3 shows the plot nitrate concentration with depth.

Figure 10-3: Decrease of nitrate concentration with depth

10.3.2.1 Humus Mineralization Nitrogen is allowed to move between the different types of organic pools. The amount of nitrogen which is transferred between the pools can be calculated as:

1 𝑁𝑡𝑟𝑛𝑠,𝑙𝑦 = 𝛽𝑡𝑟𝑛𝑠 ⋅ 𝑜𝑟𝑔𝑁𝑎𝑐𝑡,𝑙𝑦 ⋅ ( − 1) − 𝑜𝑟𝑔𝑁𝑠𝑡𝑎,𝑙𝑦 𝑓𝑟𝑎𝑐𝑡𝑁

(10-3)

𝑁𝑡𝑟𝑛𝑠,𝑙𝑦 is the amount of nitrogen transferred between the active and stable organic pools [𝑘𝑔 𝑁/ℎ𝑎], 𝛽𝑡𝑟𝑛𝑠 is the rate constant [10−5], 𝑜𝑟𝑔𝑁𝑎𝑐𝑡,𝑙𝑦 is the amount of nitrogen in the active organic pool [𝑘𝑔 𝑁/ℎ𝑎], 𝑓𝑎𝑎𝑐𝑡𝑁 is the amount of nitrogen in the stable organic pool [𝑘𝑔 𝑁/ℎ𝑎]. Mineralization from the humus active organic nitrogen pool is then calculated as: 1 2

𝑁𝑚𝑖𝑛𝑎,𝑙𝑦 = 𝛽𝑀𝑖𝑛 ⋅ (𝛾𝑡𝑚𝑝,𝑙𝑦 ⋅ 𝛾𝑠𝑤,𝑙𝑦 ) ⋅ 𝑜𝑟𝑔𝑁𝑎𝑐𝑡,𝑙𝑦

(10-4)

𝑁𝑚𝑖𝑛𝑎,𝑙𝑦 is the nitrogen mineralized from the humus active organic nitrogen pool [𝑘𝑔 𝑁/ℎ𝑎], 𝛽𝑀𝑖𝑛 is the rate coefficient for mineralization of the humus active organic nutrients, 𝛾𝑡𝑚𝑝,𝑙𝑦 is the nu135


trient cycling temperature factor for layer ly, 𝛾𝑠𝑤,𝑙𝑦 is the nutrient cycling water factor for layer ly, 𝑜𝑟𝑔𝑁𝑎𝑐𝑡,𝑙𝑦 is the amount of nitrogen in the active organic pool [𝑘𝑔 𝑁/ℎ𝑎]. Nitrogen mineralized from the humus active organic pool is added to the nitrate pool in the layer.

10.3.2.2 Residue Decomposition The decomposition of the fresh organic nitrogen pool is allowed only in the first soil layer. Decomposition and mineralization are controlled by a decay rate constant that is updated daily. The decay rate constant is calculated as a function of the ratio between C:N and C:P of the residue, temperature and soil water content. The C:N ratio of the residue is calculated as:

𝜖𝐶:𝑁 =

0.58 ⋅ 𝑟𝑠𝑑𝑙𝑦 𝑜𝑟𝑔𝑁𝑓𝑟𝑠ℎ,𝑙𝑦 + 𝑁𝑂3𝑙𝑦

(10-5)

𝜖𝐶:𝑁 is the C:N ratio of the residue in the soil layer, 𝑟𝑠𝑑𝑙𝑦 is the residue in layer 𝑙𝑦 [𝑘𝑔/ℎ𝑎], 0.58 is the fraction of residue that is carbon, 𝑜𝑟𝑔𝑁𝑓𝑟𝑠ℎ,𝑙𝑦 is the nitrogen in the fresh organic pool in layer ly [𝑘𝑔 𝑁/ℎ𝑎], and 𝑁𝑂3𝑙𝑦 is the amount of nitrate in layer 𝑙𝑦 [𝑘𝑔 𝑁/ℎ𝑎].

𝜖𝐶:𝑃 =

0.58 ⋅ 𝑟𝑠𝑑𝑙𝑦 𝑜𝑟𝑔𝑃𝑓𝑟𝑠ℎ,𝑙𝑦 + 𝑃𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛,𝑙𝑦

(10-6)

𝜖𝐶:𝑃 is the C:P ratio of the residue in the soil layer, 𝑟𝑠𝑑𝑙𝑦 is the residue in layer 𝑙𝑦 [𝑘𝑔/ℎ𝑎], 0.58 is the fraction of residue that is carbon, 𝑜𝑟𝑔𝑃𝑓𝑟𝑠ℎ,𝑙𝑦 is the phosphorus in the fresh organic pool in layer 𝑙𝑦 [𝑘𝑔 𝑃/ℎ𝑎], and 𝑃𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛,𝑙𝑦 is the amount of nitrate in layer 𝑙𝑦 [𝑘𝑔 𝑃/ℎ𝑎]. The decay rate constant 𝛿𝑛𝑡𝑟,𝑙𝑦 defines the fraction of residue that is decomposed. The decay rate constant is calculated as: 1

𝛿𝑛𝑡𝑟,𝑙𝑦 = 𝛽𝑟𝑠𝑑 ⋅ 𝛾𝑛𝑡𝑟,𝑙𝑦 ⋅ (𝛾𝑡𝑒𝑚𝑝,𝑙𝑦 ⋅ 𝛾𝑠𝑤,𝑙𝑦 )2

(10-7)

𝛽𝑟𝑠𝑑 is the rate coefficient for mineralization of the residue fresh organic nutrients, 𝛾𝑛𝑡𝑟,𝑙𝑦 is the nutrient cycling residue composition factor for layer 𝑙𝑦 , 𝛾𝑡𝑚𝑝,𝑙𝑦 is the nutrient cycling temperature factor for layer 𝑙𝑦 . 𝛾𝑛𝑡𝑟,𝑙𝑦 is calculated as:

(𝜖𝐶:𝑁 − 25) ] 25 = min (𝜖𝐶:𝑃 − 200) exp [−0.693 ⋅ ] 200 { 1.0 exp [−0.693 ⋅

𝛾𝑛𝑡𝑟,𝑙𝑦

136

(10-8)


Finally, mineralization from the residue fresh organic nitrogen pool 𝑁𝑚𝑖𝑛𝑓,𝑙𝑦 [𝑘𝑔 𝑁/ℎ𝑎 ] can be calculated as:

𝑁𝑚𝑖𝑛𝑓,𝑙𝑦 = 0.8 ⋅ 𝛿𝑛𝑡𝑟,𝑙𝑦 ⋅ 𝑜𝑟𝑔𝑁𝑓𝑟𝑠ℎ,𝑙𝑦

(10-9)

Decomposition from the residue fresh organic nitrogen pool 𝑁𝑑𝑒𝑐,𝑙𝑦 is calculated as: 𝑁𝑑𝑒𝑐,𝑙𝑦 = 0.2 ⋅ 𝛿𝑛𝑡𝑟,𝑙𝑦 ⋅ 𝑜𝑟𝑔𝑁𝑓𝑟𝑠ℎ,𝑙𝑦

(10-10)

Nitrogen decomposed from the fresh organic pool is added to the humus active organic pool in the layer.

10.3.3

Degradation of Phosphorus

10.3.3.1 Humus Mineralization Phosphorus in the humus fraction is partitioned between the active and stable organic pools using the ratio of humus active organic 𝑁 to stable organic 𝑁. The amount of phosphorus in the active and stable organic pools is calculated:

𝑜𝑟𝑔𝑃𝑎𝑐𝑡,𝑙𝑦 = 𝑜𝑟𝑔𝑃ℎ𝑢𝑚,𝑙𝑦 ⋅

𝑜𝑟𝑔𝑁𝑎𝑐𝑡,𝑙𝑦 𝑜𝑟𝑔𝑁𝑎𝑐𝑡,𝑙𝑦 + 𝑜𝑟𝑔𝑁𝑠𝑡𝑎,𝑙𝑦

(10-11)

𝑜𝑟𝑔𝑃𝑠𝑡𝑎,𝑙𝑦 = 𝑜𝑟𝑔𝑃ℎ𝑢𝑚,𝑙𝑦 ⋅

𝑜𝑟𝑔𝑁𝑠𝑡𝑎,𝑙𝑦 𝑜𝑟𝑔𝑁𝑠𝑡𝑎,𝑙𝑦 + 𝑜𝑟𝑔𝑁𝑠𝑡𝑎,𝑙𝑦

(10-12)

where 𝑜𝑟𝑔𝑃𝑎𝑐𝑡,𝑙𝑦 is the amount of phosphorus in the active organic pool [𝑘𝑔 𝑃/ℎ𝑎], 𝑜𝑟𝑔𝑃𝑠𝑡𝑎,𝑙𝑦 is the amount of phosphorus in the stable organic pool [𝑘𝑔 𝑃/ℎ𝑎 ], 𝑜𝑟𝑔𝑃𝑎𝑐𝑡,𝑙𝑦 is the concentration of humic organic phosphorus in the layer [𝑘𝑔 𝑃/ℎ𝑎 ], 𝑜𝑟𝑔𝑁𝑎𝑐𝑡,𝑙𝑦 is the amount of nitrogen in the active organic pool [𝑘𝑔 𝑁/ℎ𝑎], and 𝑜𝑟𝑔𝑁𝑠𝑡𝑎,𝑙𝑦 is the amount of nitrogen in the stable organic pool [𝑘𝑔 𝑁/ℎ𝑎]. Mineralization from the humus active organic P pool is calculated: 1

𝑃𝑚𝑖𝑛𝑎,𝑙𝑦 = 1.4 ⋅ 𝛽𝑚𝑖𝑛 ⋅ (𝛾𝑡𝑚𝑝,𝑙𝑦 ⋅ 𝛾𝑠𝑤,𝑙𝑦 )2 ⋅ 𝑜𝑟𝑔𝑃𝑎𝑐𝑡,𝑙𝑦

(10-13)

where 𝑃𝑚𝑖𝑛𝑎,𝑙𝑦 is the phosphorus mineralized from the humus active organic 𝑃 pool [𝑘𝑔 𝑃/ℎ𝑎 ], 𝛽𝑚𝑖𝑛 is the rate coefficient for mineralization of the humus active organic nutrients, 𝛾𝑡𝑚𝑝,𝑙𝑦 is the nutrient cycling temperature factor for layer ly, 𝛾𝑠𝑤,𝑙𝑦 is the nutrient cycling water factor for layer ly, and 𝑜𝑟𝑔𝑃𝑎𝑐𝑡,𝑙𝑦 is the amount of phosphorus in the active organic pool [𝑘𝑔 𝑃/ℎ𝑎 ]. Phosphorus mineralized from the humus active organic pool is added to the solution 𝑃 pool in the layer. 137


10.3.3.2 Residue Decomposition The decomposition of the fresh organic phosphorus pool is allowed only in the first soil layer. Decomposition and mineralization are controlled by a decay rate constant that is updated daily in SWAT. The decay rate constant is calculated as a function of the ratio between C:N and C:P of the residue, temperature and soil water content. The C:N ratio of the residue is calculated as:

𝜖𝐶:𝑁 =

0.58 ⋅ 𝑟𝑠𝑑𝑙𝑦 𝑜𝑟𝑔𝑁𝑓𝑟𝑠ℎ,𝑙𝑦 + 𝑁𝑂3𝑙𝑦

(10-14)

𝜖𝐶:𝑁 is the C:N ratio of the residue in the soil layer, 𝑟𝑠𝑑𝑙𝑦 is the residue in layer 𝑙𝑦 [𝑘𝑔/ℎ𝑎], 0.58 is the fraction of residue that is carbon, 𝑜𝑟𝑔𝑁𝑓𝑟𝑠ℎ,𝑙𝑦 is the nitrogen in the fresh organic pool in layer ly [𝑘𝑔 𝑁/ℎ𝑎], and 𝑁𝑂3𝑙𝑦 is the amount of nitrate in layer 𝑙𝑦 [𝑘𝑔 𝑁/ℎ𝑎].

𝜖𝐶:𝑃 =

0.58 ⋅ 𝑟𝑠𝑑𝑙𝑦 𝑜𝑟𝑔𝑃𝑓𝑟𝑠ℎ,𝑙𝑦 + 𝑃𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛,𝑙𝑦

(10-15)

𝜖𝐶:𝑃 is the C:P ratio of the residue in the soil layer, 𝑟𝑠𝑑𝑙𝑦 is the residue in layer 𝑙𝑦 [𝑘𝑔/ℎ𝑎], 0.58 is the fraction of residue that is carbon, 𝑜𝑟𝑔𝑃𝑓𝑟𝑠ℎ,𝑙𝑦 is the phosphorus in the fresh organic pool in layer 𝑙𝑦 [𝑘𝑔 𝑃/ℎ𝑎], and 𝑃𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛,𝑙𝑦 is the amount of (mineralized) phosphorus in layer 𝑙𝑦 [𝑘𝑔 𝑃/ℎ𝑎]. The decay rate constant 𝛿𝑛𝑡𝑟,𝑙𝑦 defines the fraction of residue that is decomposed. The decay rate constant is calculated as: 1

𝛿𝑛𝑡𝑟,𝑙𝑦 = 𝛽𝑟𝑠𝑑 ⋅ 𝛾𝑛𝑡𝑟,𝑙𝑦 ⋅ (𝛾𝑡𝑒𝑚𝑝,𝑙𝑦 ⋅ 𝛾𝑠𝑤,𝑙𝑦 )2

(10-16)

𝛽𝑟𝑠𝑑 is the rate coefficient for mineralization of the residue fresh organic nutrients, 𝛾𝑛𝑡𝑟,𝑙𝑦 is the nutrient cycling residue composition factor for layer 𝑙𝑦, 𝛾𝑡𝑚𝑝,𝑙𝑦 is the nutrient cycling temperature factor for layer 𝑙𝑦. 𝛾𝑛𝑡𝑟,𝑙𝑦 is calculated as:

(𝜖𝐶:𝑁 − 25) ] 25 = min (𝜖𝐶:𝑃 − 200) exp [−0.693 ⋅ ] 200 { 1.0 exp [−0.693 ⋅


(10-17)

Finally, mineralization from the residue fresh organic phosphorus pool 𝑃𝑚𝑖𝑛𝑓,𝑙𝑦 [𝑘𝑔 𝑁/ℎ𝑎] can be calculated as:

138

10 Water Quality Modelling 10.4 Processes on the Land Surface

𝑃𝑚𝑖𝑛𝑓,𝑙𝑦 = 0.8 ⋅ 𝛿𝑛𝑡𝑟,𝑙𝑦 ⋅ 𝑜𝑟𝑔𝑃𝑓𝑟𝑠ℎ,𝑙𝑦

(10-18)

Decomposition from the residue fresh organic phosphorus pool 𝑃𝑑𝑒𝑐,𝑙𝑦 is calculated as: 𝑝𝑑𝑒𝑐,𝑙𝑦 = 0.2 ⋅ 𝛿𝑛𝑡𝑟,𝑙𝑦 ⋅ 𝑜𝑟𝑔𝑃𝑓𝑟𝑠ℎ,𝑙𝑦

(10-19)

Phosphorus decomposed from the fresh organic pool is added to the humus active organic pool in the layer 𝑙𝑦.

10.4 Processes on the Land Surface 10.4.1

Nitrate Pathways

The transport of nitrate can be divided into surface runoff, lateral flow, and percolation. To calculate the amount of nitrate moved with the water, the concentration of nitrate in the mobile water has to be calculated. Then, with the mass flux of water moving in each pathway and the concentration the mass of nitrate which is lost from the soil layer can be calculated. The concentration 𝑐𝑜𝑛𝑐𝑁𝑂3,𝑚𝑜𝑏𝑖𝑙𝑒 [𝑘𝑔 𝑁⁄𝑚𝑚 𝐻2 𝑂] is calculated as:

𝑁𝑂3𝑙𝑦 ⋅ (1 − 𝑒𝑥𝑝 [ 𝑐𝑜𝑛𝑐𝑁𝑂3,𝑚𝑜𝑏𝑖𝑙𝑒 =

−𝑤𝑚𝑜𝑏𝑖𝑙𝑒 ]) (1 − 𝜃𝑒 ) ⋅ 𝑆𝐴𝑇𝑙𝑦

(10-20)

𝑤𝑚𝑜𝑏𝑖𝑙𝑒

𝑁𝑂3𝑙𝑦 is the amount of nitrate in the layer [𝑘𝑔 𝑁/ℎ𝑎], 𝑤𝑚𝑜𝑏𝑖𝑙𝑒 is the amount of mobile water in the layer [𝑚𝑚 𝐻2 𝑂], 𝜃𝑒 is the fraction of porosity, 𝑆𝐴𝑇𝑙𝑦 is the saturated water content of the soil layer [𝑚𝑚 𝐻2 𝑂]. The amount of mobile water in the soil layer is the amount of water lost by surface runoff, lateral flow or percolation:

𝑤𝑚𝑜𝑏𝑖𝑙𝑒 = 𝑄𝑠𝑢𝑟𝑓 + 𝑄𝑙𝑎𝑡,𝑙𝑦 + 𝑤𝑝𝑒𝑟𝑐,𝑙𝑦 𝑓𝑜𝑟 𝑡𝑜𝑝 10 𝑚𝑚

(10-21)

𝑤𝑚𝑜𝑏𝑖𝑙𝑒 = 𝑄𝑙𝑎𝑡,𝑙𝑦 + 𝑤𝑝𝑒𝑟𝑐,𝑙𝑦

(10-22)

𝑓𝑜𝑟 𝑙𝑜𝑤𝑒𝑟 𝑠𝑜𝑖𝑙 𝑙𝑎𝑦𝑒𝑟𝑠

𝑤𝑚𝑜𝑏𝑖𝑙𝑒 is the amount of mobile water in the layer [𝑚𝑚 𝐻2 𝑂], 𝑄𝑠𝑢𝑟𝑓 is the surface runoff generated at a given day [𝑚𝑚 𝐻2 𝑂], 𝑄𝑙𝑎𝑡,𝑙𝑦 is the water discharged from the layer by lateral flow [𝑚𝑚 𝐻2 𝑂], and 𝑤𝑝𝑒𝑟𝑐,𝑙𝑦 is the amount of water percolating to the underlying soil layer on a given day [𝑚𝑚 𝐻2 𝑂]. Only surface runoff is in the upper 10 𝑚𝑚 of the soil layer is allowed to interact with and transport nutrients. Nitrate removed in surface runoff 𝑁𝑂3𝑠𝑢𝑟𝑓 [𝑘𝑔 𝑁/ℎ𝑎] is calculated as: 139


𝑁𝑂3𝑠𝑢𝑟𝑓 = 𝛽𝑁𝑂3 ⋅ 𝑐𝑜𝑛𝑐𝑁𝑂3,𝑚𝑜𝑏𝑖𝑙𝑒 ⋅ 𝑄𝑠𝑢𝑟𝑓

(10-23)

𝛽𝑁𝑂3 is the nitrate percolation coefficient, 𝑐𝑜𝑛𝑐𝑁𝑂3,𝑚𝑜𝑏𝑖𝑙𝑒 is the concentration of nitrate in the mobile water [𝑘𝑔 𝑁⁄𝑚𝑚 𝐻2 𝑂], and 𝑄𝑠𝑢𝑟𝑓 is the surface runoff generated on a given day [𝑚𝑚 𝐻2 𝑂]. Nitrate removed in lateral flow 𝑁𝑂3𝑙𝑎𝑡,𝑙𝑦 [𝑘𝑔 𝑁/ℎ𝑎] is calculated as:

𝑁𝑂3𝑙𝑎𝑡,𝑙𝑦 = 𝛽𝑁𝑂3 ⋅ 𝑐𝑜𝑛𝑐𝑁𝑂3,𝑚𝑜𝑏𝑖𝑙𝑒 ⋅ 𝑄𝑙𝑎𝑡,𝑙𝑦 𝑓𝑜𝑟 𝑡𝑜𝑝 10 𝑚𝑚 𝑁𝑂3𝑙𝑎𝑡,𝑙𝑦 = 𝑐𝑜𝑛𝑐𝑁𝑂3,𝑚𝑜𝑏𝑖𝑙𝑒 ⋅ 𝑄𝑙𝑎𝑡,𝑙𝑦

(10-24)

𝑓𝑜𝑟 𝑙𝑜𝑤𝑒𝑟 𝑙𝑎𝑦𝑒𝑟𝑠

Nitrate moved to the underlying layer by percolation 𝑁𝑂3𝑝𝑒𝑟𝑐,𝑙𝑦 [𝑘𝑔 𝑁/ℎ𝑎] is calculated as: 𝑁𝑂3𝑝𝑒𝑟𝑐,𝑙𝑦 = 𝑐𝑜𝑛𝑐𝑁𝑂3,𝑚𝑜𝑏𝑖𝑙𝑒 ⋅ 𝑤𝑝𝑒𝑟𝑐,𝑙𝑦

(10-25)

𝑤𝑝𝑒𝑟𝑐,𝑙𝑦 is the amount of water percolating to the underlying soil layer on a given day [𝑚𝑚 𝐻2 𝑂].

10.4.1.1 Organic Nitrate in Surface Runoff Organic 𝑁 attached to soil particles may be transported by surface runoff to the main channel. This form of nitrogen is associated with the sediment loading from the HRU and changes in sediment loading will be reflected in the organic nitrogen loading. The amount of organic nitrogen transported with sediment to the stream is calculated with a loading function:

𝑜𝑟𝑔𝑁𝑠𝑢𝑟𝑓 = 0.001 ⋅ 𝑐𝑜𝑛𝑐𝑜𝑟𝑔𝑁 ⋅

𝑠𝑒𝑑 ⋅𝜖 𝑎𝑟𝑒𝑎ℎ𝑟𝑢 𝑁,𝑠𝑒𝑑

(10-26)

where 𝑜𝑟𝑔𝑁𝑠𝑢𝑟𝑓 is the amount of organic nitrogen transported to the main channel in surface runoff [𝑘𝑔 𝑁/ℎ𝑎], 𝑐𝑜𝑛𝑐𝑜𝑟𝑔𝑁 is the concentration of organic nitrogen in the top 10 𝑚𝑚 layer [𝑔 𝑁/𝑡𝑜𝑛], 𝑠𝑒𝑑 is the sediment yield on a given day [𝑡𝑜𝑛], 𝑎𝑟𝑒𝑎ℎ𝑟𝑢 is the HRU area [ℎ𝑎], and 𝜖𝑁,𝑠𝑒𝑑 is the nitrogen enrichment ratio.

𝑐𝑜𝑛𝑐𝑜𝑟𝑔𝑁 = 100 ⋅

(𝑜𝑟𝑔𝑁𝑓𝑟𝑠ℎ,𝑠𝑢𝑟𝑓 + 𝑜𝑟𝑔𝑁𝑠𝑡𝑎,𝑠𝑢𝑟𝑓 + 𝑜𝑟𝑔𝑁𝑎𝑐𝑡,𝑠𝑢𝑟𝑓 ) 𝜌𝑏 ⋅ 𝑑𝑒𝑝𝑡ℎ𝑠𝑢𝑟𝑓

(10-27)

where 𝑜𝑟𝑔𝑁𝑓𝑟𝑠ℎ,𝑠𝑢𝑟𝑓 is nitrogen in the fresh organic pool in the top 10 𝑚𝑚 [𝑘𝑔 𝑁/ℎ𝑎], 𝑜𝑟𝑔𝑁𝑠𝑡𝑎,𝑠𝑢𝑟𝑓 is nitrogen in the stable organic pool [𝑘𝑔 𝑁/ℎ𝑎], 𝑜𝑟𝑔𝑁𝑎𝑐𝑡,𝑠𝑢𝑟𝑓 is nitrogen in the active organic pool in the top 10 𝑚𝑚 of the layer [𝑘𝑔 𝑁/ℎ𝑎], 𝜌𝑏 is the bulk density of the first soil layer [𝑡𝑜𝑛/𝑚3 ], and 𝑑𝑒𝑝𝑡ℎ𝑠𝑢𝑟𝑓 is the depth of the soil surface layer (here: 10 𝑚𝑚).

140


10.4.1.2 Enrichment Ratio Part of the water’s energy is used to pick up and transport soil particles, as surface runoff flows over the soil surface. The smaller particles have less weight, thus, they are more easily transported than coarser particles. When the particle size distribution of the transported sediment is compared to that of the soil surface layer, the sediment load to the main channel has a greater proportion of clay sized particles. That means, the sediment load is enriched in clay particles. Organic nitrogen in the soil is attached primarily to colloidal (clay) particles, so the sediment load will also contain a greater proportion or concentration of organic N than that found in the soil surface layer. The enrichment ratio is defined as the ratio of the concentration of organic nitrogen transported with the sediment to the concentration in the soil surface layer. SWAT will calculate an enrichment ratio for each storm event, or allow the user to define a particular enrichment ratio for organic nitrogen that is used for all storms during the simulation. To calculate the enrichment ratio, SWAT uses a logarithmic relationship. The equation used to calculate the nitrogen enrichment ratio, 𝜖𝑁:𝑆𝑒𝑑 , for each storm event is:

𝜖𝑁:𝑆𝑒𝑑 = 0.78 ⋅ (𝑐𝑜𝑛𝑐𝑠𝑒𝑑,𝑠𝑢𝑟𝑞 )

−0.2468

(10-28)

where 𝑐𝑜𝑛𝑐𝑠𝑒𝑑,𝑠𝑢𝑟𝑞 is the concentration of sediment in surface runoff [𝑡𝑜𝑛 𝑠𝑒𝑑⁄𝑚3 𝐻2 𝑂]. The concentration of sediment in surface runoff is calculated:

𝑐𝑜𝑛𝑐𝑠𝑒𝑑,𝑠𝑢𝑟𝑞 =

𝑠𝑒𝑑 10 ⋅ 𝑎𝑟𝑒𝑎ℎ𝑟𝑢 𝑄𝑠𝑢𝑟𝑓

(10-29)

𝑄𝑠𝑢𝑟𝑓 is the amount of surface runoff for a given day [𝑚𝑚 𝐻2 𝑂], 𝑠𝑒𝑑 is the sediment yield for a given day [𝑡𝑜𝑛] and 𝑎𝑟𝑒𝑎ℎ𝑟𝑢 is the area of the HRU [ℎ𝑎].

10.4.2

Soluble Phosphorus Movement

The primary mechanism of phosphorus movement in the soil is by diffusion. Diffusion is the migration of ions over small distances (1-2 𝑚𝑚) in the soil solution in response to a concentration gradient. Due to the low mobility of solution phosphorus, surface runoff will only partially interact with the solution 𝑃 stored in the top 10 𝑚𝑚 of soil. The amount of solution 𝑃 transported in surface runoff is:

𝑃𝑠𝑢𝑟𝑓 =

𝑃𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛,𝑠𝑢𝑟𝑓 ⋅ 𝑄𝑠𝑢𝑟𝑓 𝜌𝑏 ⋅ 𝑑𝑒𝑝𝑡ℎ𝑠𝑢𝑟𝑓 ⋅ 𝑘𝑑,𝑠𝑢𝑟𝑓

(10-30)

141


10.4.2.1 Organic and Mineral Phosphorus in surface Runoff Organic and mineral 𝑃 attached to soil particles may be transported by surface runoff to the main channel. This form of phosphorus is associated with the sediment loading from the HRU and changes in sediment loading will be reflected in the loading of these forms of phosphorus. The amount of phosphorus transported with sediment to the stream is calculated as:

𝑠𝑒𝑑𝑃𝑠𝑢𝑟𝑓 = 0.001 ⋅ 𝑐𝑜𝑛𝑐𝑠𝑒𝑑𝑃 ⋅

𝑠𝑒𝑑 ⋅ 𝜖 𝑎𝑟𝑒𝑎𝐻𝑅𝑈 𝑃:𝑠𝑒𝑑

(10-31)

𝑠𝑒𝑑𝑃𝑠𝑢𝑟𝑓 is the amount of phosphorus transported with sediment to the main channel in surface runoff [𝑘𝑔 𝑃/ℎ𝑎], 𝑐𝑜𝑛𝑐𝑠𝑒𝑑𝑃 is the concentration of phosphorus attached to sediment in the top 10 𝑚𝑚 [𝑔 𝑃/𝑡𝑜𝑛], 𝑠𝑒𝑑 is the sediment yield on a given day [𝑡𝑜𝑛], 𝑎𝑟𝑒𝑎𝐻𝑅𝑈 is the HRU area [ℎ𝑎], and 𝜖𝑃:𝑠𝑒𝑑 is the phosphorus enrichment ratio. The concentration of phosphorus attached to sediment in the soil surface layer, 𝑐𝑜𝑛𝑐𝑠𝑒𝑑𝑃 , is calculated:

𝑐𝑜𝑛𝑐𝑠𝑒𝑑𝑃 = 100 ⋅

(𝑚𝑖𝑛𝑃𝑎𝑐𝑡,𝑠𝑢𝑟𝑓 + 𝑚𝑖𝑛𝑃𝑠𝑡𝑎,𝑠𝑢𝑟𝑓 + 𝑜𝑟𝑔𝑃ℎ𝑢𝑚,𝑠𝑢𝑟𝑓 + 𝑜𝑟𝑔𝑃𝑓𝑟𝑠ℎ,𝑠𝑢𝑟𝑓 ) 𝜌𝑏 ⋅ 𝑑𝑒𝑝𝑡ℎ𝑠𝑢𝑟𝑓

(10-32)

where 𝑜𝑟𝑔𝑃𝑓𝑟𝑠ℎ,𝑠𝑢𝑟𝑓 is phosphorus in the fresh organic pool in the top 10 𝑚𝑚 [𝑘𝑔 𝑃/ℎ𝑎], 𝑜𝑟𝑔𝑃𝑠𝑡𝑎,𝑠𝑢𝑟𝑓 is phosphorus in the stable organic pool [𝑘𝑔 𝑃/ℎ𝑎], 𝑜𝑟𝑔𝑃𝑎𝑐𝑡,𝑠𝑢𝑟𝑓 is phosphorus in the active organic pool in the top 10 𝑚𝑚 of the layer [𝑘𝑔 𝑃/ℎ𝑎], 𝜌𝑏 is the bulk density of the first soil layer [𝑡𝑜𝑛/𝑚3 ], and 𝑑𝑒𝑝𝑡ℎ𝑠𝑢𝑟𝑓 is the depth of the soil surface layer (here: 10 𝑚𝑚).

10.4.2.2 Enrichment Ratio The enrichment ratio is defined as the ratio of the concentration of phosphorus transported with the sediment to the concentration of phosphorus in the soil surface layer. The enrichment ratio is defined as the ratio of the concentration of phosphorus transported with the sediment to the concentration of phosphorus in the soil surface layer. The equation used to calculate the phosphorus enrichment ratio, 𝜖𝑃:𝑆𝑒𝑑 , for each storm event is:

𝜖𝑃:𝑆𝑒𝑑 = 0.78 ⋅ (𝑐𝑜𝑛𝑐𝑠𝑒𝑑,𝑠𝑢𝑟𝑞 )

−0.2468

(10-33)

The concentration of sediment 𝑐𝑜𝑛𝑐𝑠𝑒𝑑,𝑠𝑢𝑟𝑞 in surface runoff is calculated:

𝑐𝑜𝑛𝑐𝑠𝑒𝑑,𝑠𝑢𝑟𝑞 =

𝑠𝑒𝑑 10 ⋅ 𝑎𝑟𝑒𝑎ℎ𝑟𝑢 𝑄𝑠𝑢𝑟𝑓

(10-34)

𝑄𝑠𝑢𝑟𝑓 is the amount of surface runoff for a given day [𝑚𝑚 𝐻2 𝑂], 𝑠𝑒𝑑 is the sediment yield for a given day [𝑡𝑜𝑛] and 𝑎𝑟𝑒𝑎ℎ𝑟𝑢 is the area of the HRU [ℎ𝑎]. 142


10.4.3

Nutrient Lag in Surface Runoff and Lateral Flow

In very large subbasins with a time of concentration greater than 1 day, only a portion of the surface runoff and lateral flow will reach the main channel on the day it is generated. SWAT incorporates a storage feature to lag a portion of the surface runoff and lateral flow release to the main channel. Nutrients in the surface runoff and lateral flow are lagged as well. Once the nutrient load in surface runoff and lateral flow is determined, the amount of nutrients released to the main channel is calculated as:

𝑁𝑂3𝑠𝑢𝑟𝑓 = (𝑁𝑂3′𝑠𝑢𝑟𝑓 + 𝑁𝑂3𝑠𝑢𝑟𝑠𝑡𝑜𝑟,𝑖−1 ) ⋅ (1 − 𝑒𝑥𝑝 [

−𝑠𝑢𝑟𝑙𝑎𝑔 ]) 𝑡𝑐𝑜𝑛𝑐

−𝑠𝑢𝑟𝑙𝑎𝑔 𝑁𝑂3𝑙𝑎𝑡 = (𝑁𝑂3′𝑙𝑎𝑡 + 𝑁𝑂3𝑙𝑎𝑡𝑠𝑡𝑜𝑟,𝑖−1 ) ⋅ (1 − 𝑒𝑥𝑝 [ ]) 𝑇𝑇𝑙𝑎𝑔 ′ 𝑜𝑟𝑔𝑁𝑠𝑢𝑟𝑓 = (𝑜𝑟𝑔𝑁𝑠𝑢𝑟𝑓 + 𝑜𝑟𝑔𝑁𝑠𝑡𝑜𝑟,𝑖−1 ) ⋅ (1 − 𝑒𝑥𝑝 [

′ 𝑃𝑠𝑢𝑟𝑓 = (𝑃𝑠𝑢𝑟𝑓 + 𝑃𝑠𝑡𝑜𝑟,𝑖−1 ) ⋅ (1 − 𝑒𝑥𝑝 [



′ 𝑠𝑒𝑑𝑃𝑠𝑢𝑟𝑓 = (𝑠𝑒𝑑𝑃𝑠𝑢𝑟𝑓 + 𝑠𝑒𝑑𝑃𝑠𝑡𝑜𝑟,𝑖−1 ) ⋅ (1 − 𝑒𝑥𝑝 [

(10-35)

(10-36)

(10-37)

(10-38)


(10-39)

𝑁𝑂3𝑠𝑢𝑟𝑓 is the amount of nitrate discharged to the main channel in surface runoff on a given day [𝑘𝑔 𝑁/ℎ𝑎], 𝑁𝑂3′𝑠𝑢𝑟𝑓 is the amount of surface runoff nitrate generated in the HRU on a given day [𝑘𝑔 𝑁/ℎ𝑎], 𝑁𝑂3𝑠𝑢𝑟𝑠𝑡𝑜𝑟,𝑖−1 is the surface runoff nitrate stored or lagged from the previous day [𝑘𝑔 𝑁/ℎ𝑎], 𝑁𝑂3𝑙𝑎𝑡 is the amount of nitrate discharged to the main channel in lateral flow on a given day [𝑘𝑔 𝑁/ℎ𝑎], 𝑁𝑂3′𝑙𝑎𝑡 is the amount of lateral flow nitrate generated in the HRU on a given day [𝑘𝑔 𝑁/ℎ𝑎], 𝑁𝑂3𝑙𝑎𝑡𝑠𝑡𝑜𝑟,𝑖−1 is the lateral flow nitrate stored or lagged from the previous day [𝑘𝑔 𝑁/ℎ𝑎], 𝑜𝑟𝑔𝑁𝑠𝑢𝑟𝑓 is the amount of organic 𝑁 discharged to the main channel in surface ′ runoff on a given day [𝑘𝑔 𝑁/ℎ𝑎], 𝑜𝑟𝑔𝑁𝑠𝑢𝑟𝑓 is the organic 𝑁 loading generated in the HRU on a given day [𝑘𝑔 𝑁/ℎ𝑎], 𝑜𝑟𝑔𝑁𝑠𝑡𝑜𝑟,𝑖−1 is the organic 𝑁 stored or lagged from the previous day [𝑘𝑔 𝑁/ℎ𝑎], 𝑃𝑠𝑢𝑟𝑓 is the amount of solution 𝑃 discharged to the main channel in surface runoff on ′ a given day [𝑘𝑔 𝑃/ℎ𝑎], 𝑃𝑠𝑢𝑟𝑓 is the amount of solution 𝑃 loading generated in the HRU on a given day [𝑘𝑔 𝑃/ℎ𝑎], 𝑃𝑠𝑡𝑜𝑟,𝑖−1 is the solution 𝑃 loading stored or lagged from the previous day [𝑘𝑔 𝑃/ℎ𝑎], 𝑠𝑒𝑑𝑃𝑠𝑢𝑟𝑓 is the amount of sediment-attached 𝑃 discharged to the main channel in ′ surface runoff on a given day [𝑘𝑔 𝑃/ℎ𝑎], 𝑠𝑒𝑑𝑃𝑠𝑢𝑟𝑓 is the amount of sediment-attached 𝑃 loading generated in the HRU on a given day [𝑘𝑔 𝑃/ℎ𝑎], 𝑠𝑒𝑑𝑃𝑠𝑡𝑜𝑟,𝑖−1 is the sediment-attached 𝑃 stored or lagged from the previous day [𝑘𝑔 𝑃/ℎ𝑎], 𝑠𝑢𝑟𝑙𝑎𝑔 is the surface runoff lag coefficient, 𝑡𝑐𝑜𝑛𝑐 is the time of concentration for the HRU [ℎ] and 𝑇𝑇𝑙𝑎𝑔 is the lateral flow travel time [𝑑𝑎𝑦].

143

10 Water Quality Modelling 10.5 Instream Processes

10.5 Instream Processes In addition to sediment, nutrients and pesticides, SWAT can calculate (optional module) the amount of algae, dissolved oxygen and carbonaceous biological oxygen demand (CBOD) entering the main channel with surface runoff. Loadings of these three parameters impact the quality of stream water. Because field data for the water quality algorithms is not readily available the user is allowed to switch on or off the module when necessary. SWAT uses the QUAL2E steady state model by EPA, which integrates inputs from point and non-point sources, in order to determine impacts on water quality in the water bodies, best management practices and to predict the time until a system has completely recovered from any alteration of its original state.

10.5.1

Algae

Algae is increasing the stream’s dissolved oxygen concentration via photosynthesis. At night, algae reduce the concentration via respiration. As algae grow and die, they form part of the instream nutrient cycle. Growth and decay of algae is calculated as a function of the growth rate, the respiration rate, the settling rate and the amount of algae present in the stream. The change in algal biomass for a given day is:

𝜎1 Δ𝑎𝑙𝑔𝑎𝑒 = ((𝜇𝑎 ⋅ 𝑎𝑙𝑔𝑎𝑒) − (𝜌𝑎 ⋅ 𝑎𝑙𝑔𝑎𝑒) − ( ⋅ 𝑎𝑙𝑔𝑎𝑒)) ⋅ 𝑇𝑇 𝑑𝑒𝑝𝑡ℎ

(10-40)

Δ𝑎𝑙𝑔𝑎𝑒 is the change in algal biomass concentration [𝑚𝑔 𝑎𝑙𝑔/𝐿], 𝜇𝑎 is the local specific growth rate of algae [𝑑𝑎𝑦], 𝜌𝑎 is the local respiration or death rate of algae [𝑑𝑎𝑦], 𝜎1 is the local settling rate for algae [𝑚/𝑑𝑎𝑦], depth is the depth of water in the channel [𝑚], 𝑎𝑙𝑔𝑎𝑒 is the algal biomass concentration at the beginning of the day [𝑚𝑔 𝑎𝑙𝑔/𝐿], and 𝑇𝑇 is the flow travel time in the reach segment [𝑑𝑎𝑦].

10.5.2

Organic Nitrogen

The amount of organic nitrogen in the stream may be increased by the conversion of algal biomass nitrogen to organic nitrogen. Organic nitrogen concentration in the stream may be decreased by the conversion of organic nitrogen to 𝑁𝐻4+ or the settling of organic nitrogen with sediment. The change in organic nitrogen for a given day is:

Δ𝑜𝑟𝑔𝑁𝑠𝑡𝑟 = (𝛼1 ⋅ 𝜌𝑎 ⋅ 𝑎𝑙𝑔𝑎𝑒 − 𝛽𝑁,3 ⋅ 𝑜𝑟𝑔𝑁𝑠𝑡𝑟 − 𝜎4 ⋅ 𝑜𝑟𝑔𝑁𝑠𝑡𝑟 ) ⋅ 𝑇𝑇

(10-41)

Δ𝑜𝑟𝑔𝑁𝑠𝑡𝑟 is the change in organic nitrogen concentration [𝑚𝑔 𝑁⁄𝐿], 𝛼1 is the fraction of algal biomass that is nitrogen [𝑚𝑔 𝑁⁄𝑚𝑔 𝑎𝑙𝑔 𝑏𝑖𝑜𝑚𝑎𝑠𝑠], 𝜌𝑎 is the local respiration or death rate of algae [𝑑𝑎𝑦], algae is the algal biomass concentration at the beginning of the day [𝑚𝑔 𝑎𝑙𝑔⁄𝐿], 𝛽𝑁,3 is the rate constant for hydrolysis of organic nitrogen to 𝑁𝐻4+ [𝑑𝑎𝑦], 𝑜𝑟𝑔𝑁𝑠𝑡𝑟 is the organic nitrogen concentration at the beginning of the day [𝑚𝑔 𝑁⁄𝐿], 𝜎4 is the rate coefficient for organic nitrogen settling [𝑑𝑎𝑦], and 𝑇𝑇 is the flow travel time in the reach segment [𝑑𝑎𝑦]. The local 144


rate constant βN,3,20 for hydrolysis of organic nitrogen to 𝑁𝐻4+ is user defined at 20°𝐶. To adjust the hydrolysis rate to the local water temperature 𝑇𝑤𝑎𝑡𝑒𝑟 [°𝐶] following relationship is used:

βN,3 = βN,3,20 ⋅ 1.047(𝑇𝑤𝑎𝑡𝑒𝑟 −20)

(10-42)

The user defines the rate coefficient 𝜎4,20 for organic nitrogen settling at 20°𝐶. The organic nitrogen settling rate is adjusted to the local water temperature using the relationship:

𝜎4 = 𝜎4,20 ⋅ 1.024(𝑇𝑤𝑎𝑡𝑒𝑟 −20)

(10-43)

Figure 10-4 shows exemplarily how the rate coefficients σ and β change with temperature, for initial values of 1 at 20°C.

Figure 10-4: Temperature dependent rate coefficients 𝜎 and 𝛽

10.5.3

Ammonium

Through the mineralization of organic nitrogen (Figure 10-1) and diffusion of ammonium from the streambed sediments the amount of ammonium (𝑁𝐻4+ ) in the stream may be increased. Likewise, the ammonium concentration in the stream may be decreased by the conversion of 𝑁𝐻4+ to 𝑁𝑂2− (Nitrite) or the uptake of 𝑁𝐻4+ by algae. The change in ammonium for a given day is: 145


𝛥𝑁𝐻4𝑠𝑡𝑟 = (𝛽𝑁,3 ⋅ 𝑜𝑟𝑔𝑁𝑠𝑡𝑟 − 𝛽𝑁,1 ⋅ 𝑁𝐻4𝑠𝑡𝑟 +

𝜎3 − 𝑓𝑟𝑁𝐻4 ⋅ 𝛼1 ⋅ 𝜇𝑎 ⋅ 𝑎𝑙𝑔𝑎𝑒) ⋅ 𝑇𝑇 (1000 ⋅ 𝑑𝑒𝑝𝑡ℎ)

(10-44)

𝛥𝑁𝐻4𝑠𝑡𝑟 is the change in ammonium concentration [𝑚𝑔 𝑁⁄𝐿], 𝑜𝑟𝑔𝑁𝑠𝑡𝑟 is the organic nitrogen concentration at the beginning of the day [𝑚𝑔 𝑁⁄𝐿], 𝛽𝑁,1 is the rate constant for biological oxidation of ammonia nitrogen [𝑑𝑎𝑦], 𝑁𝐻4𝑠𝑡𝑟 is the ammonium concentration at the beginning of the day [𝑚𝑔 𝑁⁄𝐿 ], 𝜎3 is the benthos (sediment) source rate for ammonium [𝑚𝑔 𝑁⁄𝑚2 𝑑𝑎𝑦], 𝑑𝑒𝑝𝑡ℎ is the depth of water in the channel [𝑚], 𝑓𝑟𝑁𝐻4 is the fraction of algal nitrogen uptake from ammonium pool, 𝛼1 is the fraction of algal biomass that is nitrogen [𝑚𝑔 𝑁⁄𝑚𝑔 𝑎𝑙𝑔 𝑏𝑖𝑜𝑚𝑎𝑠𝑠], and 𝜇𝑎 is the local growth rate of algae [𝑑𝑎𝑦]. The rate constant for biological oxidation 𝛽𝑁,1 of ammonia nitrogen will vary as a function of instream oxygen concentration 𝑂𝑥𝑠𝑡𝑟 [𝑚𝑔 𝑂2 /𝐿], temperature 𝑇𝑤𝑎𝑡𝑒𝑟 , and rate constant for biological oxidation of ammonia nitrogen at 20°C, 𝛽𝑁,1,20. The rate constant is calculated:

(1 − exp[−0.6 ⋅ 𝑂𝑥𝑠𝑡𝑟 ]) ⋅ 1.083(𝑇𝑤𝑎𝑡𝑒𝑟 −20) 𝛽𝑁,1 = 𝛽𝑁,1,20 ⋅ ⏟

(10-45)

𝑖𝑛ℎ𝑖𝑏𝑖𝑡𝑖𝑜𝑛 𝑓𝑎𝑐𝑡𝑜𝑟

The inhibition factor takes into account that little till no nitrification will take place at low dissolved oxygen concentrations, as plotted in Figure 10-5. (According to the EU Water Framework Directive water bodies with less than 2 𝑚𝑔 𝑂2 /𝐿 are considered highly polluted)

Figure 10-5: Nitrification inhibition factor at low oxygen concentrations

146


The sediment source rate 𝜎3 is adjusted to the water temperature. The user defines the source rate 𝜎3,20 at 20°𝐶.

𝜎3 = 𝜎3,20 ⋅ 1.074(𝑇𝑤𝑎𝑡𝑒𝑟 −20)

(10-46)

The fraction 𝑓𝑟𝑁𝐻4 of algal nitrogen uptake from ammonium pool is calculated as:

𝑓𝑟𝑁𝐻4 =

𝑓𝑁𝐻4 ⋅ 𝑁𝐻4𝑠𝑡𝑟 (𝑓𝑁𝐻4 ⋅ 𝑁𝐻4𝑠𝑡𝑟 + (1 − 𝑓𝑁𝐻4 ) ⋅ 𝑁𝑂3𝑠𝑡𝑟 )

(10-47)

𝑓𝑁𝐻4 is the preference factor for ammonia nitrogen, 𝑁𝐻4𝑠𝑡𝑟 is the ammonium concentration in the stream [𝑚𝑔 𝑁⁄𝐿], and 𝑁𝑂3𝑠𝑡𝑟 is the nitrate concentration in the stream [𝑚𝑔 𝑁/𝐿 ].

10.5.4

Nitrite

The ammount of nitrte (𝑁𝑂2− ) in the stream will be increased by the conversion of 𝑁𝐻4+ to 𝑁𝑂2− and decreased by the conversion of 𝑁𝑂2− to 𝑁𝑂3− . The conversion of 𝑁𝑂2− to 𝑁𝑂3− occurs more rapidly than the conversion of 𝑁𝐻4+ to 𝑁𝑂2− , so the amount of nitrite present in the stream is usually very small. The change in nitrite Δ𝑁𝑂2𝑠𝑡𝑟 for a given day is:

Δ𝑁𝑂2𝑠𝑡𝑟 = (𝛽𝑁,1 ⋅ 𝑁𝐻4𝑠𝑡𝑟 − 𝛽𝑁,2 ⋅ 𝑁𝑂2𝑠𝑡𝑟 ) ⋅ 𝑇𝑇

(10-48)

𝛽𝑁,1 is the rate constant for biological oxidation of ammonia nitrogen [1/𝑑𝑎𝑦], 𝑁𝐻4𝑠𝑡𝑟 is the ammonium concentration at the beginning of the day, 𝑁𝑂2𝑠𝑡𝑟 is the nitrite concentration at the beginning of the day [𝑚𝑔 𝑁/𝐿], and 𝛽𝑁,2 is the rate constant for biological oxidation of nitrite to nitrate [1/𝑑𝑎𝑦] is (according to section 10.5.3):

𝛽𝑁,2 = 𝛽𝑁,2,20 ⋅ (1 − exp[−0.6 ⋅ 𝑂𝑥𝑠𝑡𝑟 ]) ⋅ 1.047(𝑇𝑤𝑎𝑡𝑒𝑟 −20)

10.5.5

(10-49)

Nitrate

The ammount of nitrate (𝑁𝑂3− ) in the stream may be increased by the oxidation of 𝑁𝑂2− . The nitrate concentration in the stream may be decreased by the uptake of 𝑁𝑂3− by algae. The change in nitrate Δ𝑁𝑂3𝑠𝑡𝑟 for a given day is [𝑚𝑔 𝑁/𝐿]:

Δ𝑁𝑂3𝑠𝑡𝑟 = (𝛽𝑁,2 ⋅ 𝑁𝑂2𝑠𝑡𝑟 − (1 − 𝑓𝑟𝑁𝐻4 ) ⋅ 𝛼1 ⋅ 𝜇𝑎 ⋅ 𝑎𝑙𝑔𝑎𝑒) ⋅ 𝑇𝑇

(10-50)

For the definition of the parameters see again Section 10.5.3 and 10.5.4.

147


10.5.6

Organic Phosphorus

The amount of organic phosphorus in the stream may be increased by the conversion of algal biomass phosphorus to organic phosphorus. Organic phosphorus concentration in the stream may be decreased by the conversion of organic phosphorus to soluble inorganic phosphorus or the settling of organic phosphorus with sediment. The change in organic phosphorus for a given day is:

Δ𝑜𝑟𝑔𝑃𝑠𝑡𝑟 = (𝛼2 ⋅ 𝜌𝑎 ⋅ 𝑎𝑙𝑔𝑎𝑒 − 𝛽𝑃,4 ⋅ 𝑜𝑟𝑔𝑃𝑠𝑡𝑟 − 𝜎5 ⋅ 𝑜𝑟𝑔𝑃𝑠𝑡𝑟 ) ⋅ 𝑇𝑇

(10-51)

Δ𝑜𝑟𝑔𝑃𝑠𝑡𝑟 is the change in organic phosphorus concentration [𝑚𝑔 𝑃⁄𝐿], 𝛼2 is the fraction of algal biomass that is phosphorus [𝑚𝑔 𝑃⁄𝑚𝑔 𝑎𝑙𝑔 𝑏𝑖𝑜𝑚𝑎𝑠𝑠], 𝜌𝑎 is the local respiration or death rate of algae [𝑑𝑎𝑦], algae is the algal biomass concentration at the beginning of the day [𝑚𝑔 𝑎𝑙𝑔⁄𝐿], 𝛽𝑃,4 is the rate constant for mineralization of phosphorus, 𝑜𝑟𝑔𝑁𝑠𝑡𝑟 is the organic nitrogen concentration at the beginning of the day [𝑚𝑔 𝑁⁄𝐿], 𝜎5 is the rate coefficient for organic nitrogen settling [𝑑𝑎𝑦], and 𝑇𝑇 is the flow travel time in the reach segment [𝑑𝑎𝑦]. The local rate constant β𝑃,4,20 for mineralization of phosphorus is user defined at 20°𝐶. To adjust the mineralization rate to the local water temperature 𝑇𝑤𝑎𝑡𝑒𝑟 [°𝐶] following relationship is used:

𝛽𝑃,4 = 𝛽𝑃,4,20 ⋅ 1.047(𝑇𝑤𝑎𝑡𝑒𝑟 −20)

(10-52)

The user defines the rate coefficient 𝜎5,20 for organic phosphorus settling at 20°𝐶. The organic phosphorus settling rate is adjusted to the local water temperature using the relationship:

𝜎5 = 𝜎5,20 ⋅ 1.024(𝑇𝑤𝑎𝑡𝑒𝑟 −20)

10.5.7

(10-53)

Soluble Phosphorus

The amount of soluble, inorganic phosphorus in the stream may be increased by the mineralization of organic phosphorus and diffusion of inorganic phosphorus from the streambed sediments. The soluble phosphorus concentration in the stream may be decreased by the uptake of inorganic 𝑃 by algae. The change in soluble phosphorus for a given day is:

𝛥𝑠𝑜𝑙𝑃𝑠𝑡𝑟 = (𝛽𝑃,4 ⋅ 𝑜𝑟𝑔𝑃𝑠𝑡𝑟 +

𝜎2 − 𝛼2 ⋅ 𝜇𝑎 ⋅ 𝑎𝑙𝑔𝑎𝑒) ⋅ 𝑇𝑇 (1000 ⋅ 𝑑𝑒𝑝𝑡ℎ)

(10-54)

𝛥𝑠𝑜𝑙𝑃𝑠𝑡𝑟 is the change in phosphorus concentration [𝑚𝑔 𝑃⁄𝐿], 𝑜𝑟𝑔𝑃𝑠𝑡𝑟 is the organic phosphorus concentration at the beginning of the day [𝑚𝑔 𝑃⁄𝐿], 𝛽𝑃,4 is the rate constant for mineralization of organic phosphorus [𝑑𝑎𝑦], 𝜎2 is the benthos (sediment) source rate for phosphorus [𝑚𝑔 𝑃⁄𝑚2 𝑑𝑎𝑦], 𝑑𝑒𝑝𝑡ℎ is the depth of water in the channel [𝑚], 𝛼2 is the fraction of algal bio148

10 Water Quality Modelling 10.6 Streeter – Phelps Equation

mass that is phosphorus [𝑚𝑔 𝑃⁄𝑚𝑔 𝑎𝑙𝑔 𝑏𝑖𝑜𝑚𝑎𝑠𝑠], 𝜇𝑎 is the local growth rate of algae [𝑑𝑎𝑦], and 𝑇𝑇 is the flow travel time in the reach segment [𝑑𝑎𝑦]. The sediment source rate 𝜎2 is adjusted to the water temperature. The user defines the source rate 𝜎2,20 at 20°𝐶:

𝜎2 = 𝜎2,20 ⋅ 1.074(𝑇𝑤𝑎𝑡𝑒𝑟 −20)

(10-55)

10.6 Streeter – Phelps Equation In order to study the water quality in a river or stream along a certain distance the analytical Streeter-Phelps (developed by Streeter and Phelps in 1925 to study the pollution of the Ohio River) model can be applied. It describes how the dissolved oxygen (DO) decreases over time by degradation of biochemical oxygen demand (BOD). Nowadays, with computational means, numerical models are more efficient and accurate than the analytical solution. Nevertheless, it provides easy results under the given assumptions:        

The BOD source (the pollutant which consumes DO) is a point source The BOD source is evenly distributed over the cross section It moves as plug flow along the river without longitudinal mixing Only one DO sink and one DO source is considered No BOD removal by sedimentation No alteration of oxygen balance by photosynthesis Oxidation rate is proportional to remaining concentration of non-oxidated matter Ventilation rate is proportional to oxygen deficit

Figure 10-6: Streeter – Phelps Oxygen sag curve

After adding a pollutant (or in general an oxygen consumer) to a stream Figure 10-6 shows how the dissolved oxygen concentration is initially decreasing downstream until a critical point, 149


where the concentration is the smallest and subsequently increasing again until a maximum value, defined by the critical saturation 𝐶𝑠 . The difference between the maximum available DO in the water, expressed through 𝐶𝑠 the temperature 𝑇 dependent saturation coefficient for oxygen [𝑔/𝑚3]

𝐶𝑠 = 14.61996 − 0.40420 ⋅ 𝑇 + 0.00842 ⋅ 𝑇 2 − 0.00009 ⋅ 𝑇 3

(10-56)

The Streeter–Phelps equation determines the relation between the dissolved oxygen concentration and the biological oxygen demand over time and can be expressed through the differential equation:

𝜕𝐷 = 𝑘1 ⋅ 𝐶 − 𝑘2 ⋅ 𝐷 𝜕𝑡

(10-57)

𝐷 accounts for the saturation deficit [𝑔/𝑚3 ] between 𝐶𝑠 and available DO. 𝐶 is the oxygen demand and defined by available organic matter (expressed as BSB5) [𝑔/𝑚3] at time 𝑡. 𝑘1 is the decay rate coefficient [𝑑𝑎𝑦 −1 ]. For natural streams this value is between 0.1 and 3 at 20°𝐶. The temperature correction is:

𝑘1 = 𝑘1,20 ⋅ 1.05(𝑇−20)

(10-58)

And 𝑘2 is the ventilation or aeration rate of oxygen [𝑑𝑎𝑦 −1 ]. For natural streams this value is between 0.2 and 1.2 at 20°𝐶. The temperature correction is:

𝑘2 = 𝑘2,20 ⋅ exp(0.024 ⋅ (𝑇 − 20))

(10-59)

So, there are two processes going on to change the DO saturation deficit:  

DEGRADATION of dissolved oxygen through biodegradation AEARTION through dissolution of atmospheric 𝑂2

The degradation term on the right side of Eq. (10-57) describes the change of oxygen demand over time 𝜕𝐶 ⁄𝜕𝑡:

𝜕𝐶 = −𝑘1 ⋅ 𝐶 𝜕𝑡 Eq. (10-60) can be solved for 𝐶 through integration:

150

(10-60)


𝐶1 = 𝐶0 ⋅ exp(−𝑘1 ⋅ 𝑡)

(10-61)

𝐶0 is the initial oxygen demand (of organic matter) for 𝑡 = 0 and 𝑥 = 0 (where the pollutant / BSB source is entering the stream). Taking Eq. (10-61), Eq. (10-57) can be solved for 𝐷:

𝐷=

𝑘1 ⋅ 𝐶0 ⋅ (exp(−𝑘1 ⋅ 𝑡) − exp(−𝑘2 ⋅ 𝑡)) + 𝐷0 ⋅ exp(−𝑘2 ⋅ 𝑡) 𝑘2 − 𝑘1

(10-62)

𝐷0 is the initial deficit of oxygen for 𝑡 = 0 and 𝑥 = 0 [𝑔/𝑚3 ]. Figure 10-7 illustrates as an example how the oxygen demand 𝐶, oxygen deficit 𝐷, and dissolved oxygen DO evolve over time. The assumptions are 𝑇 = 20°𝐶, 𝑘1 = 0.9, 𝑘2 = 0.5, 𝐶0 = 9, and 𝐷0 = 1. It can be easily seen that DO can be obtained as the difference between 𝐶𝑠 and 𝐷, which both can be calculated.

Figure 10-7: Temporal evolution of oxygen deficit (D), oxygen demand (C), and dissolved oxygen (DO)

Sometimes it is more important to know where a pollutant is oxidized and not when. If the rivers stream velocity is known 𝑣𝑥 [𝑚/𝑠] following relation helps:

151

10 Water Quality Modelling 10.7 Further Reading

𝑥 = 𝑣𝑥 ⋅ 𝑡

(10-63)

On the DO curve in the given example a minimum concentration occurs after approx. 1.4 days (Figure 10-7). To find the exact minimum the Streeter–Phelps equation (10-62) is differentiated with respect to time, and set equal to zero, the time at which the minimum DO occurs is expressed by:

𝑡𝑐𝑟𝑖𝑡 =

𝑘2 𝐷0 (𝑘2 − 𝑘1 ) ⋅ ln [ ⋅ (1 − )] 𝑘2 − 𝑘1 𝑘1 𝐶0 𝑘1 1

(10-64)

10.7 Further Reading The following paper shows how a SWAT model is set up to model Europe´s hydrology and water quality: Abbaspour, K. C. et al.: A continental-scale hydrology and water quality model for Europe: Calibration and uncertainty of a high-resolution large-scale SWAT model, Journal of Hydrology 524 (2015) 733–752

http://dx.doi.org/10.1016/j.jhydrol.2015.03.027

There is a SWAT Literature Database for Peer- Reviewed Journal Articles online. It has a good interface to find relevant papers.

10.8 Nomenclature 𝑎𝑙𝑔𝑎𝑒 𝑎𝑟𝑒𝑎ℎ𝑟𝑢

HRU area [ℎ𝑎]

𝐶

Oxygen demand and defined by available organic matter [𝑔/𝑚3]

𝐶0

Initial oxygen demand [𝑔/𝑚3 ]

𝑐𝑜𝑛𝑐𝑁𝑂3,𝑚𝑜𝑏𝑖𝑙𝑒 𝑐𝑜𝑛𝑐𝑜𝑟𝑔𝑁 𝑐𝑜𝑛𝑐𝑠𝑒𝑑,𝑠𝑢𝑟𝑞 𝑐𝑜𝑛𝑐𝑠𝑒𝑑𝑃

Concentration of nitrate in the mobile water [𝑘𝑔 𝑁⁄𝑚𝑚 𝐻2 𝑂] Concentration of organic nitrogen in the top 10 𝑚𝑚 layer [𝑔 𝑁/𝑡𝑜𝑛] Concentration of sediment in surface runoff [𝑡𝑜𝑛 𝑠𝑒𝑑⁄𝑚3 𝐻2 𝑂] Concentration of phosphorus attached to sediment in the top 10 𝑚𝑚 [𝑔 𝑃/𝑡𝑜𝑛]

𝐷

Saturation deficit [𝑔/𝑚3 ] between 𝐶𝑠 and available DO

𝐷0

Initial deficit of oxygen [𝑔/𝑚3 ]

𝑑𝑒𝑝𝑡ℎ𝑠𝑢𝑟𝑓 152

Algal biomass concentration at the beginning of the day [𝑚𝑔 𝑎𝑙𝑔/𝐿]

Depth of the soil surface layer [𝑚𝑚]

10 Water Quality Modelling 10.8 Nomenclature

𝑑𝑒𝑝𝑡ℎ 𝑓𝑔𝑟

Depth of water in the channel [𝑚] Growth stage factor (0.0-1.0)

𝑓𝑁𝐻4

Preference factor for ammonia nitrogen

𝑓𝑛𝑜3

Soil nitrate factor (0.0-1.0)

𝑓𝑠𝑤

Soil water factor (0.0-1.0)

𝑓𝑎𝑎𝑐𝑡𝑁

Amount of nitrogen in the stable organic pool [𝑘𝑔 𝑁/ℎ𝑎]

𝑓𝑟𝑁𝐻4

Fraction of algal nitrogen uptake from ammonium pool

𝑘1

Decay rate coefficient [𝑑𝑎𝑦 −1 ]

𝑘2

Ventilation or aeration rate of oxygen [𝑑𝑎𝑦 −1 ]

𝑁𝐻4𝑠𝑡𝑟

Ammonium concentration at the beginning of the day [𝑚𝑔 𝑁/𝐿]

𝑁𝑂2𝑠𝑡𝑟

Nitrite concentration at the beginning of the day [𝑚𝑔 𝑁/𝐿]

𝑁𝑂3𝑐𝑜𝑛𝑐,𝑧 𝑁𝑂3𝑙𝑎𝑡 𝑁𝑂3𝑙𝑎𝑡,𝑙𝑦 𝑁𝑂3′𝑙𝑎𝑡 𝑁𝑂3𝑙𝑎𝑡𝑠𝑡𝑜𝑟,𝑖−1 𝑁𝑂3𝑙𝑦 𝑁𝑂3𝑝𝑒𝑟𝑐,𝑙𝑦

Concentration of nitrate in the soil depth 𝑧 [𝑚𝑔/𝑘𝑔] Amount of nitrate discharged to the main channel in lateral flow on a given day [𝑘𝑔 𝑁/ℎ𝑎] Nitrate removed in lateral flow [𝑘𝑔 𝑁/ℎ𝑎] Amount of lateral flow nitrate generated in the HRU on a given day [𝑘𝑔 𝑁/ℎ𝑎] Lateral flow nitrate stored or lagged from the previous day [𝑘𝑔 𝑁/ℎ𝑎] Amount of nitrate in layer 𝑙𝑦 [𝑘𝑔 𝑁/ℎ𝑎] Nitrate moved to the underlying layer by percolation [𝑘𝑔 𝑁/ℎ𝑎]

𝑁𝑂3𝑠𝑡𝑟

Nitrate concentration in the stream [𝑚𝑔 𝑁/𝐿]

𝑁𝑂3𝑠𝑢𝑟𝑓

Nitrate removed in surface runoff [𝑘𝑔 𝑁/ℎ𝑎]

𝑁𝑂3′𝑠𝑢𝑟𝑓

Amount of surface runoff nitrate generated in the HRU on a given day [𝑘𝑔 𝑁/ℎ𝑎]

𝑁𝑂3𝑠𝑢𝑟𝑠𝑡𝑜𝑟,𝑖−1

Surface runoff nitrate stored or lagged from the previous day [𝑘𝑔 𝑁/ ℎ𝑎]

𝑁𝑑𝑒𝑐,𝑙𝑦

Decomposition from the residue fresh organic nitrogen pool [𝑘𝑔 𝑁/ℎ𝑎]

𝑁𝑑𝑒𝑚𝑎𝑛𝑑 𝑁𝑓𝑖𝑥

Plant nitrogen demand not met by uptake from the soil [𝑘𝑔 𝑁/ℎ𝑎] Amount of nitrogen added to the plant biomass by fixation [𝑘𝑔 𝑁/ℎ𝑎]

𝑁𝑚𝑖𝑛𝑎,𝑙𝑦

Nitrogen mineralized from the humus active organic nitrogen pool [𝑘𝑔 𝑁/ℎ𝑎]

𝑁𝑚𝑖𝑛𝑓,𝑙𝑦

Mineralization from the residue fresh organic nitrogen pool [𝑘𝑔 𝑁/ℎ𝑎]

𝑁𝑡𝑟𝑛𝑠,𝑙𝑦

Amount of nitrogen transferred between the active and stable organic pools [𝑘𝑔 𝑁/ℎ𝑎]

𝑂𝑥𝑠𝑡𝑟

Stream oxygen concentration [𝑚𝑔 𝑂2 /𝐿] 153


𝑜𝑟𝑔𝑁𝑎𝑐𝑡,𝑙𝑦

Amount of nitrogen in the active organic pool [𝑘𝑔 𝑁/ℎ𝑎]

𝑜𝑟𝑔𝑁𝑎𝑐𝑡,𝑠𝑢𝑟𝑓

Nitrogen in the active organic pool in the top 10 𝑚𝑚 of the layer [𝑘𝑔 𝑁/ℎ𝑎]

𝑜𝑟𝑔𝑁𝑓𝑟𝑠ℎ,𝑙𝑦

Nitrogen in the fresh organic pool in layer ly [𝑘𝑔 𝑁/ℎ𝑎]

𝑜𝑟𝑔𝑁𝑓𝑟𝑠ℎ,𝑠𝑢𝑟𝑓 𝑜𝑟𝑔𝑁𝑠𝑡𝑎,𝑙𝑦

Nitrogen in the fresh organic pool in the top 10 𝑚𝑚 [𝑘𝑔 𝑁/ℎ𝑎] Amount of nitrogen in the stable organic pool [𝑘𝑔 𝑁/ℎ𝑎]

𝑜𝑟𝑔𝑁𝑠𝑡𝑎,𝑠𝑢𝑟𝑓

Nitrogen in the stable organic pool [𝑘𝑔 𝑁/ℎ𝑎]

𝑜𝑟𝑔𝑁𝑠𝑡𝑜𝑟,𝑖−1

Organic 𝑁 stored or lagged from the previous day [𝑘𝑔 𝑁/ℎ𝑎]

𝑜𝑟𝑔𝑁𝑠𝑡𝑟

Organic nitrogen concentration at the beginning of the day [𝑚𝑔 𝑁⁄𝐿]

𝑜𝑟𝑔𝑁𝑠𝑢𝑟𝑓

Amount of organic nitrogen transported to the main channel in surface runoff [𝑘𝑔 𝑁/ℎ𝑎]

′ 𝑜𝑟𝑔𝑁𝑠𝑢𝑟𝑓

Organic 𝑁 loading generated in the HRU on a given day [𝑘𝑔 𝑁/ℎ𝑎]

𝑜𝑟𝑔𝑃𝑎𝑐𝑡,𝑙𝑦

Amount of phosphorus in the active organic pool [𝑘𝑔 𝑃/ℎ𝑎]

𝑜𝑟𝑔𝑃𝑎𝑐𝑡,𝑠𝑢𝑟𝑓

Phosphorus in the active organic pool in the top 10 𝑚𝑚 of the layer [𝑘𝑔 𝑃/ℎ𝑎]

𝑜𝑟𝑔𝑃𝑓𝑟𝑠ℎ,𝑙𝑦

Phosphorus in the fresh organic pool in layer 𝑙𝑦 [𝑘𝑔 𝑁/ℎ𝑎]

𝑜𝑟𝑔𝑃𝑓𝑟𝑠ℎ,𝑠𝑢𝑟𝑓 𝑜𝑟𝑔𝑃𝑠𝑡𝑎,𝑙𝑦 𝑜𝑟𝑔𝑃𝑠𝑡𝑎,𝑠𝑢𝑟𝑓

Phosphorus in the fresh organic pool in the top 10 𝑚𝑚 [𝑘𝑔 𝑃/ℎ𝑎] Amount of phosphorus in the stable organic pool [𝑘𝑔 𝑃/ℎ𝑎] Phosphorus in the stable organic pool [𝑘𝑔 𝑃/ℎ𝑎]

𝑃𝑑𝑒𝑐,𝑙𝑦

Decomposition from the residue fresh organic phosphorus pool [𝑘𝑔 𝑃/ℎ𝑎]

𝑃𝑚𝑖𝑛𝑎,𝑙𝑦

Phosphorus mineralized from the humus active organic 𝑃 pool [𝑘𝑔 𝑃/ℎ𝑎]

𝑃𝑚𝑖𝑛𝑓,𝑙𝑦

Mineralization from the residue fresh organic phosphorus pool [𝑘𝑔 𝑃/ℎ𝑎]

𝑃𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛,𝑙𝑦

Amount of nitrate in layer 𝑙𝑦 [𝑘𝑔 𝑁/ℎ𝑎]

𝑃𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛,𝑙𝑦

Amount of phosphorus in layer 𝑙𝑦 [𝑘𝑔 𝑃/ℎ𝑎]

𝑃𝑠𝑡𝑜𝑟,𝑖−1

154

Solution 𝑃 loading stored or lagged from the previous day [𝑘𝑔 𝑁/ℎ𝑎]

𝑃𝑠𝑢𝑟𝑓

Amount of solution phosphorus transported in surface runoff [𝑘𝑔 𝑃/ℎ𝑎]

′ 𝑃𝑠𝑢𝑟𝑓

Amount of solution 𝑃 loading generated in the HRU on a given day [𝑘𝑔 𝑁/ℎ𝑎]

𝑄𝑙𝑎𝑡,𝑙𝑦

Water discharged from the layer by lateral flow [𝑚𝑚 𝐻2 𝑂]


Surface runoff generated at a given day [𝑚𝑚 𝐻2 𝑂]

𝑟𝑠𝑑𝑙𝑦

Residue in layer 𝑙𝑦 [𝑘𝑔/ℎ𝑎]

𝑆𝐴𝑇𝑙𝑦

Saturated water content of the soil layer [𝑚𝑚 𝐻2 𝑂]


𝑠𝑒𝑑

Sediment yield on a given day [𝑡𝑜𝑛]

𝑠𝑒𝑑𝑃𝑠𝑡𝑜𝑟,𝑖−1

Sediment-attached 𝑃 stored or lagged from the previous day [𝑘𝑔 𝑁/ ℎ𝑎]

𝑠𝑒𝑑𝑃𝑠𝑢𝑟𝑓

Amount of sediment-attached 𝑃 discharged to the main channel in surface runoff on a given day [𝑘𝑔 𝑁/ℎ𝑎]

′ 𝑠𝑒𝑑𝑃𝑠𝑢𝑟𝑓

Amount of sediment attached 𝑃 loading generated in the HRU on a given day [𝑘𝑔 𝑁/ℎ𝑎]

𝑠𝑢𝑟𝑙𝑎𝑔

Surface runoff lag coefficient

𝑇𝑤𝑎𝑡𝑒𝑟

Local water temperature [°𝐶]

𝑇𝑇𝑙𝑎𝑔

Lateral flow travel time [𝑑𝑎𝑦]

𝑇𝑇 𝑡𝑐𝑜𝑛𝑐

Flow travel time in the reach segment [𝑑𝑎𝑦] Time of concentration for the HRU [ℎ]

𝑤𝑚𝑜𝑏𝑖𝑙𝑒

Amount of mobile water in the layer [𝑚𝑚 𝐻2 𝑂]

𝑤𝑝𝑒𝑟𝑐,𝑙𝑦

Amount of water percolating to the underlying soil layer on a given day [𝑚𝑚 𝐻2 𝑂]

𝑧 Δ𝑎𝑙𝑔𝑎𝑒

Soil depth from the soil surface [𝑚𝑚] Change in algal biomass concentration [𝑚𝑔 𝑎𝑙𝑔/𝐿]

𝛥𝑁𝐻4𝑠𝑡𝑟

Change in ammonium concentration [𝑚𝑔 𝑁⁄𝐿 ]

Δ𝑜𝑟𝑔𝑁𝑠𝑡𝑟

Change in organic nitrogen concentration [𝑚𝑔 𝑁⁄𝐿 ]

Δ𝑜𝑟𝑔𝑃𝑠𝑡𝑟

Change in organic phosphorus concentration [𝑚𝑔 𝑃⁄𝐿]

𝛥𝑠𝑜𝑙𝑃𝑠𝑡𝑟

Change in phosphorus concentration [𝑚𝑔 𝑃⁄𝐿]

𝛼1

Fraction of algal biomass that is nitrogen [𝑚𝑔 𝑁⁄𝑚𝑔 𝑎𝑙𝑔 𝑏𝑖𝑜𝑚𝑎𝑠𝑠]

𝛼2

Fraction of algal biomass that is phosphorus [𝑚𝑔 𝑃⁄𝑚𝑔 𝑎𝑙𝑔 𝑏𝑖𝑜𝑚𝑎𝑠𝑠]

𝛽𝑀𝑖𝑛 𝛽𝑁,1,20

Rate coefficient for mineralization of the humus active organic nutrients Rate constant for biological oxidation of ammonia nitrogen at 20°C

𝛽𝑁,1

Rate constant for biological oxidation of ammonia nitrogen [𝑑𝑎𝑦]

𝛽𝑁,1

Rate constant for biological oxidation of ammonia nitrogen [1/𝑑𝑎𝑦]

𝛽𝑁,2

Rate constant for biological oxidation of nitrite to nitrate [1/𝑑𝑎𝑦]

𝛽𝑁,3

Rate constant for hydrolysis of organic nitrogen to 𝑁𝐻4+ [𝑑𝑎𝑦]

βN,3,20

Local rate constant for hydrolysis of organic nitrogen to 𝑁𝐻4+

𝛽𝑁𝑂3

Nitrate percolation coefficient

𝛽𝑃,4

rate constant for mineralization of, 𝑜𝑟𝑔𝑁𝑠𝑡𝑟

β𝑃,4,20

Local rate constant for mineralization of phosphorus

155


𝛽𝑟𝑠𝑑

Rate coefficient for mineralization of the residue fresh organic nutrients

𝛽𝑡𝑟𝑛𝑠

Rate constant [10−5]


Nutrient cycling residue composition factor for layer 𝑙𝑦

𝛾𝑠𝑤,𝑙𝑦

Nutrient cycling water factor for layer ly

𝛾𝑡𝑚𝑝,𝑙𝑦

Nutrient cycling temperature factor for layer ly

𝛾𝑡𝑚𝑝,𝑙𝑦

Nutrient cycling temperature factor for layer 𝑙𝑦

𝛿𝑛𝑡𝑟,𝑙𝑦

Fraction of residue that is decomposed

𝛿𝑛𝑡𝑟,𝑙𝑦

Fraction of residue that is decomposed

𝜃𝑒

Fraction of porosity

𝜇𝑎

Local specific growth rate of algae [𝑑𝑎𝑦]

𝜇𝑎

Local growth rate of algae [𝑑𝑎𝑦]

𝜌𝑎

Local respiration or death rate of algae [𝑑𝑎𝑦]

𝜌𝑎

Local respiration or death rate of algae [𝑑𝑎𝑦]

𝜌𝑏

Bulk density of the first soil layer [𝑡𝑜𝑛/𝑚3 ]

𝜎1

Local settling rate for algae [𝑚/𝑑𝑎𝑦]

𝜎3

Benthos (sediment) source rate for ammonium [𝑚𝑔 𝑁⁄𝑚2 𝑑𝑎𝑦]

𝜎4,20

Rate coefficient 𝜎4,20 for organic nitrogen settling

𝜎4

Rate coefficient for organic nitrogen settling [𝑑𝑎𝑦]

𝜎5

Rate coefficient for organic nitrogen settling [𝑑𝑎𝑦]

𝜕𝐶 ⁄𝜕𝑡

156

Change of oxygen demand over time

𝜖𝐶:𝑁

C:N ratio of the residue in the soil layer

𝜖𝐶:𝑃

C:P ratio of the residue in the soil layer

𝜖𝑁,𝑠𝑒𝑑

Nitrogen enrichment ratio

𝜖𝑃:𝑠𝑒𝑑

Phosphorus enrichment ratio


157

List of Figures

11 Literature Allen, Pereira, Raes, Smith. (1998): Crop evapotranspiration-Guidelines for computing crop water requirements-FAO Irrigation and drainage paper 56: Rome, FAO Abtew, Wossenu; Melesse, Assefa M. (2012): Evaporation and evapotranspiration. Measurements and estimations. Dordrecht, London: Springer. Beven, K. J. (2012): Rainfall-runoff modelling. The primer. 2nd ed. Chichester, West Sussex, Hoboken, NJ: Wiley-Blackwell. Flügel, Wolfgang Albert (1996): Hydrological Response Units (HRUs) as modeling entities for hydrological river basin simulation and their methodological potential for modeling complex environmental process systems. In: Die Erde 127, S. 43–62. Gash, J. H. C; Shuttleworth, W. James (2007): Evaporation. Wallingford, Oxfordshire, UK: IAHS Press (Benchmark papers in hydrology, 2). Maidment, David R. (1993): Handbook of hydrology. New York: McGraw-Hill. Masih, Ilyas; Maskey, Shreedhar; Uhlenbrook, Stefan; Smakhtin, Vladimir (2011): Assessing the Impact of Areal Precipitation Input on Streamflow Simulations Using the SWAT Model1. In: JAWRA Journal of the American Water Resources Association 47 (1), S. 179–195. DOI: 10.1111/j.1752-1688.2010.00502.x. Neitsch, S. L.; Arnold, J.; Kiniry, J. R.; Williams, J. R. (2011): Soil and Water Assessment Tool. Theoretical Documentation. Online verfügbar unter http://swat.tamu.edu/documentation/. UCAR/COMET: MetEd » Resource Description: Precipitation Estimates, Part 1: Measurement. Online verfügbar unter https://www.meted.ucar.edu/training_module.php?id=526&tab=01, zuletzt geprüft am 19.03.2015. Wagener, Thorsten; Sivapalan, Murugesu; Troch, Peter; Woods, Ross (2007): Catchment Classification and Hydrologic Similarity. In: Geography Compass 1 (4), S. 901–931. DOI: 10.1111/j.1749-8198.2007.00039.x. Wainwright, John; Mulligan, Mark (2004): Environmental modelling. Finding simplicity in complexity. Chichester, West Sussex, England, Hoboken, NJ: Wiley.

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