This article discusses the impact of water cycle on the environment and its importance for plant, animal, and human life. It also explores the methods for measuring flow rate in natural watercourses and how to separate base flow from the hydrograph of a stream's discharge.
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Engineering Hydrology1 ENGINEERING HYDROLOGY Name Course Professor University City/state Date
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Engineering Hydrology2 Engineering Hydrology Task 1 a)Impact of water cycle on the environment Most or all the water that the earth receives is as a result of water cycle. Water cycle is a virtuous cycle that starts with evaporation of water from any watercourses such as oceans, rivers, streams or seas and transpiration from soils and plants, after being heated by the sun. The evaporated water rises up into the atmosphere as a gas (water vapour) and then starts cooling and condensing after reaching a certain height to become clouds. The clouds continue accumulating and when they become heavy enough, they fall back onto the earth as precipitation(Supriya, (n.d.)).The precipitation, which is in form of rain, is a very essential component for human, plant and animal life. After falling onto the surface of the earth, the water runs off back to watercourses where it evaporates again and the process repeats itself making water to circulate continuously in the earth-atmosphere system(IPCC, 2013).The water cycle processes are presented in Figure 1 below. Therefore water cycle is a complex but very essential constituent of the planetary system that plays a key role in regulating human, plant and animal life(Sohoulande & Singh, 2015).
Engineering Hydrology3 Figure 1: Water cycle processes(Khan Academy, 2016) The main impact of water cycle on the environment is that it determines the amount of water available for plant, animal and human life. The integral part that water plays in the life of all living organisms on earth cannot be overemphasized. Water cycle promotes provision of water needed for plant growth and therefore it directly affects agricultural production. Plants grow well if they get adequate supply of water. This implies that water cycle affects agricultural activities – it enhances crop or food production and farm yields. Animals also depend on water for survival. The water provided by water cycle promotes animal life because these animals cannot survive without water. In relation to human beings, all human life is dependent on water. Humans depend on water for food production, industrial activities, transportation, and domestic uses (heating, cooling, cleaning and cooking, among others). Therefore water cycle enhances the quality of human life. In general, the life of all living organisms depend upon water and the water cycle distributes the fresh water needed by these living organisms. The main reason why water cycle
Engineering Hydrology4 lays a very important role in the life of plants, animals and humans is because it supplies fresh water. About 97.5% of water on earth is salty water. Over 99% of the remaining water is either ice or underground water. Additionally, less than 1% of fresh water available on earth is available in surface forms such as rivers, lakes and streams(Khan Academy, 2016).All living organisms depend on the small amount of surface fresh water available for survival. Therefore water cycle increases provision and supply of purified water needed for plant, animal and human life, and promotes distribution of this water on earth. The water gets purified naturally during evaporation into the atmosphere and infiltration into the soil or underground water(McQuade, 2018).In other words, water cycle promotes the entire ecosystem. It is also important to note that the water cycle has greatly been affected by climate change over the past few decades(Le, et al., 2011).The climate change is affecting the main components of the water cycle: evaporation, condensation and precipitation. This is also causing direct impacts on the environment because the rates and trends of evaporation, transpiration, condensation, precipitation and runoff are constantly changing. The higher temperatures due to global warming have increased evaporation rates. These higher temperatures increase the capacity of air in the atmosphere to hold more water vapour, which can result to more intense rainstorms(Grover, 2015).With the intense rainstorms, the risk of flooding increases. When flooding occurs, the largest percentage of water runs off into oceans, rivers and streams, with very little water infiltrating into the soil. All these increases drought risks(The Climate Reality Project, 2016).This means that the impact of water cycle on the environment is also being affected by climate change. b)Methods for measuring flow rate in natural watercourses
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Engineering Hydrology5 Measurement of flow rate or discharge in natural watercourses is a necessity in water supply, energy production, flood control and agriculture. This is very important in engineering fields because it helps engineers to design water structures that are able to perform intended functions efficiently and effectively(Xu, et al., 2016).Some of the methods used to measure flow rate in watercourses include the following: i)Eyeball method This method involves measuring the cross sectional area and velocity of the watercourse and multiplying the two. The cross-sectional area can be estimated using tape measure or ruler while the velocity can be determined by dividing a set distance by the time taken by a floating piece of paper to travel over the distance. The accuracy of this method is relatively low and therefore it is used where the flow of the watercourse is low and steady, and when the only property needed is the flow’s order of magnitude(Czachorski, 2018). ii)Depth to flow method In this method, the flow rate in natural watercourse is determined by measuring the depth of the watercourse and using the Manning’s equation (Equation 1 below). Q=AR 2 3√S n ……………………………………………….. (1) Where Q = flow rate (m3/s), A = cross sectional area of the channel (m2), R = depth of the channel (m), S = slope of the channel, and n = Manning’s roughness coefficient. The method is used when the flow conditions of the watercourse are uniform (depth of the watercourse remains constant over the length of the channel) and the slope and cross sectional area of the channel are known.
Engineering Hydrology6 iii)Immersed current meter This is a conventional method for measuring flow rate in natural watercourses. In this method, a current meter is immersed at different points across the watercourse to measure and record the mean flow velocity of the section(National Environmental Monitoring Standards (NEMS) Group, 2013).The mean flow velocity is the used to calculate the flow rate using computational methods (i.e. by multiplying it with the cross-sectional area of the flow at each of the sections). The flow rate is determined by finding the average of the products in the selected sections of the watercourse. This is a simple and reliable method of measuring flow rate in natural water courses and is mainly used for measuring low flow rate. However, it also becomes difficult to measure flow rate when the water depth is not enough for immersion of the current meter or when the velocity of the flow is below the minimum required by the meter's setting. Use of immersed current meters is also hampered by turbulence. iv)Surface velocity area method This method is used to measure flow rate in natural watercourses by using surface devices to determine the velocity if the flow, which is then multiplied by the cross sectional area of the open channel(Alvisi, et al., 2014). The velocity of the flow can be measured using electromagnetic, mechanical or acoustic open channel flow meters(Almedia & Souza, 2017). The velocity of the watercourse can be measured from the surface sing close range photogrammetry, total stations or even remotely piloted aircraft systems (also known as drones) (Tauro, et al., 2015);(Bolognesi, et al., 2017).This method is mainly used in areas where it is difficult to access the watercourse in order to measure the flow velocity. It is also suitable for use in watercourses with uniform flow conditions and cross sectional area.
Engineering Hydrology7 v)Artificial tracing method This method involves use of tracers such as chemical tracers, radioactive tracers and fluorescent tracers. Chemical tracers can be used in small and wide watercourses but are most suitable for use in small watercourses because they are easy to handle, low cost, provide satisfactory results and have low impact on the watercourse. In this method, a tracer of known concentration is released instantaneously at the release section of the watercourse then its concentration is determined in a downstream measuring section(Christiansen, 2009).The flow rate is then calculated using computation method where the inputs are the concentration of the tracer at the release and measuring sections, the volume of the water between the two sections, and the time taken by the tracer to travel between the two sections. The choice of most suitable tracer is determined by: chemical water characteristics, distance between release and measuring section, type of suspended sediment in the watercourse, type of flow, watercourse’s hydrological characteristics(Rowinski & Chrzanowski, 2011),and trace measurement equipment’s sensitivity (Tazioli, 2011).This method is suitable for use when measuring very high flow rate in a watercourse, such as during flood events. However, the method is not very effective when the water is turbid because in such a case, the tracers may be easily absorbed by the suspended sediments. vi)Primary device method This method measures the flow rate in natural watercourses using hydraulic structures such as a weir, flume or dam, which enables flow rate measurement by measuring depth(Open Channel Flow, (n.d.)).The measured depth can then be converted to a flow rate using a rating curve equation or computational formula. Primary devices are designed to force the flow of the watercourse to pass through critical depth(Goel, et al., 2015), which corresponds to a single flow
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Engineering Hydrology8 rate(Samani, 2017). These devices are easy to use and maintain.The method is very accurate and mainly used to measure flow rate in watercourses that have these primary devices. However, primary devices can cause backwater and head loss in the system thus reducing the accuracy of the flow rate measurement. The main factors determining the choice of suitable method for measuring flow rate in natural watercourses are type of flow and conditions of the watercourse. c)How to separate base flow from the hydrograph of a stream’s discharge There are various methods that can be used to separate base flow from the hydrograph of a stream’s discharge. In this case, it is assumed that the hydrograph (a curve showing the relationship between discharge of the stream and time) has already been drawn. The hydrograph comprises of two limbs: rising limb and falling limb. Rising limb is the curve between the start of runoff and peak discharge whereas the falling limb is the curve between the peak point and end of storm flow. The method used to separate base flow from the hydrograph in this scenario is straight line method. This is the simplest method for separating the base flow from a hydrograph and is only suitable for individual storm events. The first step is to identify the time when direct runoff started on the hydrograph. This point is easy to locate because it is basically the point where the hydrograph starts rising sharply and is determined by simply inspecting the hydrograph visually. The rising limb is also referred to as the concentration curve and it is the ascending part of the hydrograph.(Riya, (n.d.))The point corresponding to this time is marked on the hydrograph (point A in Figure 2 below).
Engineering Hydrology9 Figure 2: Components of a hydrograph(Shiksha, 2014) The second step is to identify the time when direct runoff ends and is also determined by inspecting the stream flow hydrograph visually. It is quite difficult to locate the exact point because it is not definite hence it is assumed to be a point near the end of the falling limb (Science Education Resource Center at Carleton College , 2017). Alternatively, this point can also be estimated using equation 2 below. The point on the graph corresponding to this time is also marked on the hydrograph (point D in Figure 2 above). The third step is to draw a straight line between the two points. After drawing this straight line, the area below the straight line or the straightline itself represents the base flow while the area above it represents the surface or direct runoff(Bosch, et al., 2017). N = 0.83A0.2…………………………..…………………….. (2) Where N = is the number of days after the peak point of hydrograph (peak discharge) and A = watershed area (km2)(Riya, (n.d.)).
Engineering Hydrology10 Task 2 The first step is to draw a graph of flow against –ln(-ln(1 – P)) In this case, P = 1/50 = 0.02 The formula for predicting maximum annual discharge for a river The maximum flow is arranged in descending order The value of p is calculated as follows:p=m N+1; where m = rank and N = total number of sample = 30. The value of T is calculated as follows:T=1 p The various values are provided in Appendix 1 The graph of flow against reduced variate is as shown in Figure 3 below -2.00-1.000.001.002.003.004.00 0 500 1000 1500 2000 2500 3000 3500 Maximum flow (m³/s) Reduced variate, y Maximum flow, m³/s) Figure 3: Flow vs. y
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Engineering Hydrology11 The graph of flow against return period is plotted on a logarithmic scale and provided in Figure 4 below 1.00010.000100.000 0 500 1000 1500 2000 2500 3000 3500 Maximum flow (m³/s) vs. return period Return period, T Maximum flow, m³/s Figure 4: Flow vs. return period The graph of maximum flow versus log T is provided in Figure 5 below 0.00000.20000.40000.60000.80001.00001.20001.40001.60001.8000 0 500 1000 1500 2000 2500 3000 3500 Maximum flow (m³/s) vs. Log T Log T Maximum flow, m³/s) Figure 5: Maximum flow vs. Log T
Engineering Hydrology12 From Figure 5 above, the estimated maximum annual discharge for the river in the 50thyear is 3400 m3/s. The probability for a 50-year floor over the given 30-year period is determined as follows: Exceedence probability = 1 – (1 – p)n In this case, p = 1/50 = 0.02 n = 30 Hence exceedence probability = 1 – (1 – 0.02)30 = 1 – (0.98)30= 1 – 0.5455 = 0.4545 = 45.45% Hence there is a 45.5% chance that a 50-year flood will occur in the river. Task 3 a)Surface runoff and groundwater components The first step in separating the surface runoff and groundwater components of the hydrograph is to draw the hydrograph first. This is drawn by plotting the values of discharge against time. The hydrograph is drawn using the values in Table 1 and is as presented in Figure 3 below. The starting point of the rising limb is at 6 hours and the corresponding discharge is 1.9 m3/s. The point marking the end of direct runoff (falling limb) is determined using equation 2 as follows: N = 0.83A0.2= 0.83 (28.6)0.2= 1.62 days = 38.88 hours from the peak
Engineering Hydrology13 The time at the peak is 24 hours hence the estimated time at the end of direct runoff is: N = 24hrs + 38.95hrs = 62.88hrs.From the data given, the base flow at 62.88hrs or 48 hours is 4.4 m3/s. Table 1: Values of time, discharge and base flow TimeCumulativ e time (hr.) Q (m3/s)Base flow (m3/s) 180001.64.4 210031.54.4 000061.94.4 030095.44.4 06001211.34.4 09001514.14.4 12001816.44.4 15002117.74.4 18002424.54.4 21002713.34.4 0000309.54.4 0300337.34.4 0600365.94.4 0900395.24.4 1200424.74.4 1500454.54.4 1800484.44.4 The values in Table 1 above are used to plot the hydrograph shown in Figure 6 below. The red straight line represents the groundwater component while the blue curve represents the surface runoff component. Additionally, the area below the red linear line represents the groundwater component and the area of the curve under the blue curve but above the red line represents the surface runoff component.
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Engineering Hydrology14 0102030405060 0 5 10 15 20 25 30 Discharge (m³/s) vs. time (hrs) Time (hr) Discharge (m³/s) Figure 6: Separated surface runoff and groundwater components of the hydrograph b)Total volume of surface runoff Direct runoff hydrograph (DRH) is obtained by subtracting base flow from the discharge. The values obtained are given in Table 2 below Table 2: Values of DRH TimeCumulativ e time (hr.) Q (m3/s)Base flow (m3/s)Direct runoff hydrograph (DRH) (m3/s) 180001.64.4-2.8 210031.54.4-2.9 000061.94.4-2.5 030095.44.41 06001211.34.46.9 09001514.14.49.7 12001816.44.412 15002117.74.413.3 18002424.54.420.1 21002713.34.48.9 0000309.54.45.1 0300337.34.42.9 0600365.94.41.5
Engineering Hydrology15 0900395.24.40.8 1200424.74.40.3 1500454.54.40.1 1800484.44.40 The volume of surface runoff is computed using equation 3 below ∑ i n QDRHxΔt………………………….………………. (3) Where QDRHis the DRH in m3/s and Δt = time interval in seconds (1 hour = 3,600 seconds). In this case, the volume of surface runoff is calculated as shown in Table 3 below Table 3: Calculation of volume of surface runoff TimeCumulativ e time (hr.) Q (m3/s)Base flow (m3/s) Direct runoff hydrograph (DRH) (m3/s) Volume of surface runoff DRH x Δt (m3) 180001.64.4-2.8- 210031.54.4-2.9- 000061.94.4-2.5- 030095.44.411 x 3 x 3600 = 10,800 06001211.34.46.96.9 x 3 x 3600 = 74,520 09001514.14.49.79.7 x 3 x 3600 = 104,760 12001816.44.41212 x 3 x 3600 = 129,600 15002117.74.413.313.3 x 3 x 3600 = 143,640 18002424.54.420.120.1 x 3 x 3600 = 217,080 21002713.34.48.98.9 x 3 x 3600 = 96,120 0000309.54.45.15.1 x 3 x 3600 = 55,080 0300337.34.42.92.9 x 3 x 3600 = 31,320 0600365.94.41.51.5 x 3 x 3600 = 16,200 0900395.24.40.80.8 x 3 x 3600 = 8,640 1200424.74.40.30.3 x 3 x 3600 = 3,240 1500454.54.40.10.1 x 3 x 3600 = 1,080 1800484.44.400 Total892,080 Hence the total volume of surface runoff during the flood is 892,080m3.
Engineering Hydrology16 c)Total average depth of net rain on the catchment The total average depth of net rain on the catchment is obtained by dividing the total volume of surface runoff during the flood by the catchment area, as follows Totalaveragedepth=Totalvolumeofsurfacerunoff Catchmentarea Total volume = 892,080m3 Catchment area = 28.6km2= 28.6 x 106m2 Totalaveragedepth=892,080m3 28.6x106m2=0.0312m=31.2mm The value obtained for the total average depth of net rain on the catchment is 31.2mm. This is a reasonable depth of the rain and it basically means that the flood was not devastating because damage is directly proportional to the depth of the flood(Pistrika, et al., 2014);(Win, et al., 2018).Therefore the accuracy of the value obtained for total average depth of net rain on the catchment is high. d)Runoff from a design storm of 6 hours Net rainfall intensity = 6 mm/h Direct runoff from the storm is calculated using equation 4 below(Ramirez, (n.d.)) Directrunoff(cm)=0.36x (∑ i N DROi)xΔt A ………………………………..……. (4) Where DRO = direct runoff ordinates (m3/s), Δt = time interval from one ordinate to the other (h), and A = catchment area (km2).
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Engineering Hydrology17 DRO is obtained by subtracting base flow from the corresponding discharge. The values of DRH for the 6-hour interval are provided in Table 4 below Table 4: Calculated values of DRH Hence direct runoff (cm) = 0.36x114x6 28.6=8.61cm=86.1mm Rainfall intensity = 6 mm/h Time interval = 6 h Rainfall (mm) can be obtained by multiplying rainfall intensity (mm/h) by the time interval (h). Index, ϕ (mm/h) can then be determined using the following formula(Shiksha, 2016) (36 – 6ϕ) + (36 – 6ϕ) + (36 – 6ϕ) + (36 – 6ϕ) + (36 – 6ϕ) + (36 – 6ϕ) + (36 – 6ϕ) + (36 – 6ϕ) = 86.1 288 – 48ϕ = 86.1 48ϕ = 201.9 ϕ = 4.2 mm/h TimeCumulativ e time (hr.) Q (m3/s)Base flow (m3/s) Direct runoff ordinate (DRO) (m3/s) 180001.64.4-2.8 0000654.40.6 06001216.74.412.3 12001830.54.426.1 18002442.24.437.8 00003022.84.418.4 06003613.24.48.8 1200429.94.45.5 1800488.94.44.5 ∑DRO=114
Engineering Hydrology18 Excess rainfall intensity is calculated as follows: Excess rainfall intensity = rainfall intensity – index The values obtained are provided in Table 5 below Table 5: Excess rainfall intensity Therefore average excess rainfall intensity is 1.8 mm/h. The total hours for the flood period is 48 hours (6 x 8). This means that total average excess rainfall intensity is 1.8 mm/h x 48 h = 86.4 mm Total runoff is calculated by multiplying the total average excess rainfall intensity by the catchment area as follows Catchment area = 28.6 km2= 28.6 x 106m2 Total average excess rainfall intensity = 86.4 mm = 0.0864 m Total runoff = 28.6 x 106m2x 0.0864 m = 2.5 x 106m3 Bibliography Time interva l (h) Rainfall intensity (mm/h) Rainfall (mm) Index (mm/h) Excess rainfall intensity (mm/h) 66364.21.8 66364.21.8 66364.21.8 66364.21.8 66364.21.8 66364.21.8 66364.21.8 66364.21.8
Engineering Hydrology19 Almedia, A. & Souza, V., 2017. An alternative method for measuring velocities in open-channel flows: perfomance evaluation of a Pitot tube compared to an acoustic meter.Brazilian Journal of Water Resources,22(1), pp. 26-37. Alvisi, S., Barbetta, S., Franchini, M., Melone, F. & Moramarco, T., 2014. Comparing grey formulations of the velocity-area method and entropy method for discharge estimation with uncertainty.Journal of Hydroinformation,16(1), pp. 797-811. Bolognesi, M., Farina, G., Alvisi, S., Franchini, M., Pellegrinelli, A. & Russo, P., 2017. Measurement of surface velocity in open channels using a lightweight remotely piloted aircraft system.Geomatics, Natural Hazards and Risk,8(1), pp. 73-86. Bosch, D.D., Arnold, J.G., Allen, P.G., Lim, K.J. & Park, Y.S.,2017. Temporal variations in baseflow for the Little River experimental watershed in South Georgia, USA.Journal of Hydrology: Regional Studies,10(C), pp. 110-121. Christiansen, D., 2009.Dye tracer tests to determine time-of-Travel in Iowa streams, 1990–2006,Reston, VA: U.S. Geological Survey. Czachorski, R., 2018.Flow Metering 101: A Guide to Measuring Flow.[Online] Available at:https://www.ohm-advisors.com/insights/flow-metering-101-guide-measuring-flow [Accessed 15 April 2019]. Goel, A., Verma, D. & Sangwan, S., 2015. Open Channel Flow Measurement of Water by Using Width Contraction.World Academy of Science, Engineering and Technology,9(2), pp. 1557-1562. Grover, V., 2015. Impact of Climate Change on the Water Cycle. In: S. Shrestha, A. Anal, P. Salam & v. d. V. M., eds.Managing Water Resources under Climate Uncertainty.Cham, Switzerland: Pringer, pp. 3-30. IPCC, 2013.Climate Change 2013: The Physical Science Basis. Contribution of Working Group I to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change,New York: Cambridge University Press. Khan Academy, 2016.The water cycle.[Online] Available at:https://www.khanacademy.org/science/biology/ecology/biogeochemical-cycles/a/the- water-cycle [Accessed 14 April 2019]. Le, P., Kumar, P. & Drewry, D., 2011. Implications for the hydrologic cycle under climate change due to the expansion of bioenergy crops in the Midwestern United States.Proceedings of the National Academy Sciences of the U.S.A.,108(37), pp. 15085-15090. McQuade, T., 2018.Why Is the Water Cycle Important to Humans & Plants?.[Online] Available at:https://sciencing.com/water-cycle-important-humans-plants-7452871.html [Accessed 14 April 2019]. National Environmental Monitoring Standards (NEMS) Group, 2013.Open channel flow measurement, Wellington: NEMS.
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Engineering Hydrology20 Open Channel Flow, (n.d.).Methods of measuring flows in open channels.[Online] Available at:https://www.openchannelflow.com/blog/methods-of-measuring-flows-in-open-channels [Accessed 15 April 2019]. Pistrika, A., Tsakiris, G. & Nalbantis, I., 2014. Flood Depth-Damage Functions for Built Environment. environmental Processes,1(4), pp. 553-572. Ramirez, J., (n.d.).Unit hydrographs.[Online] Available at:http://www.engr.colostate.edu/~ramirez/ce_old/classes/cive322-Ramirez/CE322_Web/ Example_UnitHydrographs.htm [Accessed 15 April 2019]. Riya, P., (n.d.).Runoff Hydrograph: Meaning, Components and Factors | Geography.[Online] Available at:http://www.geographynotes.com/precipitation-2/runoff/runoff-hydrograph-meaning- components-and-factors-geography/6071 [Accessed 15 April 2019]. Rowinski, P. & Chrzanowski, M., 2011. Influence of selected fluorescent dyes on small aquatic organisms. Acta Geophysica,59(1), pp. 91-109. Samani, Z., 2017. Three Simple Flumes for Flow Measurement in Open Channels.Journal of Irrigation and Drainage Engineering,143(6), pp. 1-7. Science Education Resource Center at Carleton College , 2017.Baseflow Separation Using Straight Line Method.[Online] Available at:https://serc.carleton.edu/hydromodules/steps/baseflow_separa.html [Accessed 15 April 2019]. Shiksha, K., 2014.Stream Flow Hydrograph.[Online] Available at:http://ecoursesonline.iasri.res.in/mod/page/view.php?id=125278 [Accessed 15 April 2019]. Shiksha, K., 2016.Base flow separation.[Online] Available at:http://ecoursesonline.iasri.res.in/mod/page/view.php?id=2226 [Accessed 15 April 2019]. Sohoulande, C. & Singh, V., 2015. impact of climate change on the hydrologic cycle and implications for society.Environment and Social Psychology,1(1), pp. 9-16. Supriya, K., (n.d.).6 Main Components of Water Cycle.[Online] Available at:http://www.environmentalpollution.in/water-pollution/6-main-components-of-water- cycle/1312 [Accessed 14 April 2013]. Tauro, F., Petroselli, A. & Arcangeletti, E., 2015. Assessment of drone-based surface flow observations. Hydrological Processes,30(1), pp. 1114-1130. Tazioli, A., 2011. Experimental methods for river discharge measurements: comparison among tracers and current meter.Hydrological Sciences Journal,56(7), pp. 1314-1324.
Engineering Hydrology21 The Climate Reality Project, 2016.How is climate change impacting the water cycle?.[Online] Available at:https://www.climaterealityproject.org/blog/climate-change-impacting-water-cycle [Accessed 14 April 2019]. Win, S., Zin, W., Kawasaki, A. & San, Z., 2018. Establishment of flood damage function models: A case study in the Bago River Basin, Myanmar.International Journal of Disaster Risk Reduction,28(1), pp. 688- 700. Xu, B., Dong, P., Zhang, J. & Yao, J., 2016. Research on a novel flow rate inferential measurement method and its application in hydraulic elevators.Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science,231(2), pp. 372-386. Appendix 1: values of p, T, log T, q and y