UNIT HYDROGRAPH AND EFFECTIVE RAINFALL S INFLUENCE OVER THE STORM RUNOFF HYDROGRAPH

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1 UNIT HYDROGRAPH AND EFFECTIVE RAINFALL S INFLUENCE OVER THE STORM RUNOFF HYDROGRAPH INTRODUCTION Water is a common chemical substance essential for the existence of life and exhibits many notable and unique properties despite being composed by very simple molecules One of such properties is the ability to naturally present itself in solid, liquid or gasous form depending on the surrounding conditions It has a melting temperature of 0 and a boiling temperature of 100 (at sea level), thus appearing as ice in the coldest locations of the planet and as vapour in the atmosphere Despite its abundance, only a relatively small fraction is actually fresh water thus leading to an increasing number of situations in which the demand exceeds supply Therefore, it is increasingly important to correctly manage water resources by building an efficient network of water adductors and reservoirs Hydrology plays a major role in water resource management by supplying data and analysing it in order to provide a basis to the subsequent planning and building of the necessary structures The continuous movement of water on, below and above the surface of the planet is known as Hydrologic Cycle This endless process is driven by the Sun s energy, which heats the water in the oceans, making it evaporate into its gasous form in the atmosphere Water also evaporates from the soil and transpired from plants The rising air then takes the vapour to higher altitudes where lower temperatures cause it to condense into clouds The constant movement of cloud particles cause them to collide, grow and eventually fall out of the sky as precipitation Some of the precipitation falls onto land, where it flows over the ground as surface runoff, creating streamflows and moving towards the oceans to complete the cycle This is, however, a simplification of the actual phenomenae, because in reality not all water flows into rivers Some of it soaks into the ground (infiltration) replenishing aquifers Over time, however, subterranean water seeps back into surface water bodies as groundwater discharge However, this isn t relevant to the present study In the scope of hydrology, precipitation and runoff are perhaps the most important concepts applied to civil engineering Precipitation occurs when the atmosphere becomes saturated with water vapour both because of the lower temperature and the increasing amount of vapour in the air Precipitation forms as the small water particles collide, merge and grow, eventually becoming too heavy to be sustained in the cloud Rain drops range in size Larger drops resemble flattened pancakes while smaller drops are spherical Precipitation can be measured with rain gauges and tipping buckets Rain gauges have wide openings at the top to capture rainfall which is funneled into a narrow tube Since the tube is 1

2 thinner than the top of the funnel, precise measuring to the one tenth of a milimeter This method is used to determine the total amount of precipitation between two readings, but does not provide adequate information about the intensity of the precipitation over time A tipping bucket, however, records precipitation on a rotating drum or electronically thus providing useful data for an analysis Surface runoff is the water flow which occurs when soil is infiltrated to full capacity and excess water (in this case from precipitation) flows over the land A land area which produces runoff draining to a common section is called a watershed or a drainage basin Each drainage basin is separated topographically from adjacent basins by a geographical barrier such as a ridge, hill or mountain These barriers are known as water divides A drainage basin is usually described by a number of properties such as its area, which is the most significant factor in determining the likelihood of floods occurring Catchment is determined by the basin s topography, shape, size, soil type and land use While the former two determine the time taken for rain to reach the river, the latter determine the amount of water that reaches the river The time taken for the rain to reach the common section that defines the watershed is know as time of concentration and it is a critical concept for this study When no definite data is available, a watershed s time of concentration can be estimated using one of several mathematical expressions depending on the basin s size and the available data and establishing relationships between different geomorphological parameters such as the basin s form, average slope, vegetation, the streams length and slope and the condition of the soil at the beginning of the rainfall Hydrographs are charts that display the change of a hydrologic variable over time One of the most frequently created hydrographs is the stream discharge hydrograph which, as the name states, is a time record of the discharge of a stream, river or watershed outlet Rainfall is typically the main input to a watershed and the the streamflow is often considered the output of the watershed In fact and as a way to decrease the number of variables to the problem at hand, these are usually the only relevant forms of input and output, while all other are ignored Such an hydrograph is a representation of how a watershed responds to rainfall A unit hydrograph is used to more easily represent the effect of rainfall on a particular basin It is a hypothetical response of the watershed to a unit input of effective rainfall This allows easy calculation of the response to any arbitrary input (effective rainfall), by simply performing a convolution between the rain input and the unit hydrograph output A convolution is a mathematical operation on two functions (effective rainfall hyetograph and the basin s unit hydrograph) producing a third function, the direct stream discharge hydrograph of a particular rain event over a particular watershed An instantaneous unit hydrograph is a further refinement of the concept; for an IUH, the input rainfall is assumed to all take place at a discrete point in time (obviously, this isn't the case for actual rainstorms) Making this assumption can greatly simplify the analysis involved in constructing a unit hydrograph, and it is necessary for the creation of a geomorphologic instantaneous unit hydrograph 2

3 The geomorphological unit hydrograph (GUH) arises in the context of attempting to relate the instantaneous unit hydrograph of a catchment to the geometry of the stream network and, ideally, to some expression of flow velocity or stream length, so that the instantaneous unit hydrograph may be synthesized from information contained on a topographical map or, by extension, from the general relations observed in the geometrical configurations of stream networks, relations known as the geomorphological laws The concept is presented as an alternative to the traditional method of seeking empirical relations between measures of instantaneous unit hydrograph scale and shape and appropriate catchment characteristics, by the regression of the former on the latter, and the use of such relations, once established, to predict the instantaneous unit hydrograph s for ungauged catchments Another very important hydrograph for the subject at hand is the precipitation hydrograph, also known as hyetograph Although a precipitation hydrograph usually plots the cumulative record of precipitation for a given rainfall, a visual representation of the intensity is far more useful Determining a hypothetical rainfall for a given duration is accomplished through the application of concepts such as the return time and the probability distribution function By gathering the annual maximum values of precipitation, it is possible to plot them in a chart and adjust to an exponential (or logarithmic) function in the form of In this way, and are parameters that establish a relationship between the duration of rainfall and the total precipitation that occurs These parameters have been determined for all of mainland Portugal and for different durations, making it in fact possible to estimate the probable precipitation for any intense rainfall event anywhere in the country METHODS A number of different unit hydrographs were considered These hydrographs all have the same total duration, equal to and have a triangular shape but differ in the moment in whichh the peak flow occurs The hydrographs are shown in the following chart and table: Figure 1 Non Dimensional Triangular Unit Hydrographs 3

4 1 1,000 0,875 0,750 0,625 0,500 0,375 0,250 0, ,500 1,000 0,857 0,714 0,571 0,429 0,286 0, ,333 0,667 1,000 0,833 0,667 0,500 0,333 0, ,250 0,500 0,750 1,000 0,800 0,600 0,400 0, ,200 0,400 0,600 0,800 1,000 0,750 0,500 0, ,167 0,333 0,500 0,667 0,833 1,000 0,667 0, ,143 0,286 0,429 0,571 0,714 0,857 1,000 0, ,125 0,250 0,375 0,500 0,625 0,750 0,875 1,000 Table 1 - Flow values of the unit hydrographs (divided by the maximum flow) Each of these hydrographs is then combined with the rainfall event that leads to the largest value of peak flow The rainfall is divided into parts of an equal duration of and the precipitation in each of the time intervals is determined using the concept of precipitation probability in the following expression, in which and are descriptive parameters of the rainfall (1) The duration of the rainfall for a storm runoff analysis must be no shorter than the time of concentration of the basin A rainfall with a duration equal to the time of concentration is called a critical rainfall In order to obtain comparable results for basins with different characteristics, the actual precipitation is divided by the value of the critical precipitation ( ) (2) (3) Once the precipitation for each individual time interval has been determined, the blocks must be distributed in a symetrical arrangement relative to the magnitude of the unit hydrograph s values, in this way making sure that the resulting direct runoff hydrograph has the highest possible peak value The following table shows the precipitation of each block for a given value of 4

5 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 0,812 0,660 0,536 0,435 0,354 0,287 0,233 0,189 0,154 0,058 0,098 0,124 0,139 0,146 0,148 0,146 0,140 0,133 0,036 0,064 0,085 0,101 0,112 0,120 0,124 0,126 0,126 0,026 0,049 0,067 0,082 0,095 0,105 0,112 0,118 0,122 0,021 0,040 0,056 0,071 0,083 0,095 0,104 0,112 0,119 0,018 0,034 0,049 0,063 0,075 0,087 0,098 0,108 0,117 0,015 0,030 0,043 0,057 0,069 0,082 0,093 0,104 0,115 0,013 0,026 0,039 0,052 0,065 0,077 0,089 0,101 0,113 Table 2 - Precipitation for different values of n, in a non-uniform rainfall (divided by the critical precipitation) Obtaining the amount of precipitation in each block is a lot simpler if the rainfall is considered to be of uniform intensity since it s only necessary to determine the precipitation for the rainfall s duration and then divide that value by the number of blocks However, this procedure leads to lower values of precipitation in some of the blocks and is usually not suitable for practical application STORM RUNOFF HYDROGRAPH The storm direct runoff hydrograph is obtained by means of the method of discrete convolution once both a unit and a precipitation hydrograph have been determined This method attempts to summarize the Unit Hydrograph Theory in the following mathematical equations: (4) where is the precipitation (in mm), is the flow in the unit hydrograph and is the flow in the storm direct runoff hydrograph The resulting data can be interpreted as a set of discrete values and can be statistically studied as such The standard deviation (), assymetry coefficient ( ) and kurtosis ( ) of each resulting storm runoff hydrograph was determined, since the focus of the study is its shape The results for the storm runoff hydrographs resulting from non-uniform intensity rainfalls are displayed in the following table, in which the base time equals 1 5

6 ,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 0,1528 0,1395 0,1304 0,1251 0,1251 0,1298 0,1395 0,1528-0,2928-0,1632-0,0766-0,0108 0,0108 0,0667 0,1632 0,2928 3,6669 3,3879 3,0819 2,8666 2,8666 3,0935 3,3879 3,6669 0,1699 0,1533 0,1419 0,1355 0,1355 0,1415 0,1533 0,1699-0,3663-0,2709-0,1679-0,0464 0,0464 0,1723 0,2709 0,3663 3,3001 3,2277 3,0684 2,9540 2,9540 3,1053 3,2277 3,3001 0,1802 0,1627 0,1507 0,1443 0,1443 0,1507 0,1627 0,1802-0,3112-0,2569-0,1694-0,0505 0,0505 0,1851 0,2569 0,3112 2,9666 2,9970 2,9460 2,9160 2,9160 2,9847 2,9970 2,9666 0,1864 0,1694 0,1578 0,1518 0,1518 0,1581 0,1694 0,1864-0,2303-0,2079-0,1407-0,0431 0,0431 0,1630 0,2079 0,2303 2,7435 2,8135 2,8244 2,8416 2,8416 2,8538 2,8135 2,7435 0,1898 0,1742 0,1636 0,1584 0,1584 0,1641 0,1742 0,1898-0,1483-0,1492-0,1021-0,0316 0,0316 0,1269 0,1492 0,1483 2,6044 2,6810 2,7223 2,7599 2,7599 2,7398 2,6810 2,6044 0,1915 0,1778 0,1685 0,1641 0,1641 0,1692 0,1778 0,1915-0,0721-0,0892-0,0616-0,0191 0,0191 0,0855 0,0892 0,0721 2,5227 2,5886 2,6398 2,6808 2,6808 2,6467 2,5886 2,5227 0,1919 0,1805 0,1728 0,1693 0,1693 0,1735 0,1805 0,1919-0,0034-0,0312-0,0222-0,0067 0,0067 0,0429 0,0312 0,0034 2,4795 2,5264 2,5734 2,6077 2,6077 2,5727 2,5264 2,4795 0,1914 0,1826 0,1766 0,1739 0,1739 0,1772 0,1826 0,1914 0,0577 0,0239 0,0146 0,0052-0,0052 0,0007-0,0239-0,0577 2,4623 2,4864 2,5193 2,5416 2,5416 2,5149 2,4864 2,4623 0,1904 0,1842 0,1800 0,1781 0,1781 0,1804 0,1842 0,1904 0,1118 0,0759 0,0485 0,0165-0,0165-0,0402-0,0759-0,1118 2,4623 2,4633 2,4747 2,4821 2,4821 2,4706 2,4633 2,4623 Table 3 - Statistical analysis of dimensionless storm runoff hydrographs These results show that the storm runoff hydrographs are symmetrical to each other around a central moment Under the present conditions, for instance, the storm runoff hydrograph calculated from the unit hydrograph with the peak moment is symmetrical to the one that is obtained from the unit hydrograph with the peak moment It is observed that for each unit hydrograph, the corresponding storm runoff hydrograph s standard deviation increases, the kurtosis decreases and the asymmetry tends to change signal as the value of increases 6

7 Plotting the storm runoff hydrographs in charts shows that although the peak runoff value is unaffected by the unit hydrograph s shape, the storm runoff hydrograph s shape is very quite dependent on it This is even more noticeable in extreme values of, closer to 0,1 and 0,9 The storm runoff hydrographs are a lot more similar to each other if is closer to 0,4 or 0,6 It can be said that if a basin s unit hydrograph is unknown, it s real shape is of no importance when determining the peak runoff value either the rainfall s intensity is constant or not DAM DISCHARGE HYDROGRAPH Knowledge of the area of a particular basin and the width of the spillway is necessary in order to obtain values of a dam s discharge hydrograph Combinations of three different basins of 25, 50 and 100 with spillways 15, 30 and 45 in width were analyzed with different values of precipitation The critical rainfall of each scenario is a function of the time of concentration and the value of The following table shows the precipitation (in ) used for each scenario as well as the the time of concentration of each basin:,,,, , , , , , , , , Table 4 - Precipitation (in ) for different basins and different values of The temporal distribution of the rainfall was determined as described before in order to obtain the highest peak flow value for a storm runoff hydrograph ( The peak runoff value ( and the height of the water above the discharger was determined through a process of iteration for each basin area and spillway width and for each value of The value of is an indicator of the capacity of the dam to act as a buffer The higher this value is, the smaller the buffer A smaller discharger will naturally result in a lower value of thus creating a more noticeable buffer effect The following 7

8 chart shows how this value changes for a given value of as the moment in which the unit hydrograph s peak value occurs changes, in a 50 basin and a 30 discharger Figure 2 Dam runoff results in a non-uniform rainfall The chart leads to the conclusion that the buffer capability of this basin/discharger combination is fairly consistent for any value of, except for 0,1 This pattern is also found in the remaining possible combinations The analysis of the results also leads to an interesting conclusion regarding the buffering in terms of the basin s area and the discharger s width and the effect of the unit hydrograph s shape over the dam s runoff As a matter of fact, the unit hydrograph s shape is less important the larger the basin s area and the smaller the discharger s width However, for a given combination, the unit hydrograph s shape has a more noticeable effect when the value of is approximately 0,5, which is close to the values usually found in Mainland Portugal when precipitation is measured in and time in hours or days On the other hand, if the rainfall is considered to be of uniform intensity, the same procedure shows that the influence of the unit hydrograph s shape is all but negligible However, these effects are merely details and may even be ignored in the larger scope of all other variables that condition the design of a structure It can be said that the unit hydrograph s shape has no significant influence over a dam s runoff These conclusions are valid in the design stages of a project, where the peak value is particularly important However, in a flood simulation a much higher precision is required and the conclusions no longer apply 8

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