1. 83 Energy or Energy Head Elevation head Velocity head Total head Energy Head -Elevation Head -Elevation Head -Velocity Head -Velocity Head -Total Head -Total Head Momentum Open Channel

2. 84 Energy or Energy Head Energy Head -Elevation Head -Elevation Head -Velocity Head -Velocity Head -Total Head -Total Head Momentum Open Channel total headThe total energy of water moving through a channel is expressed in total head in feet of water. This is simply the sum of the the elevation above a datum (elevation head), the pressure head and the velocity head. elevation headThe elevation head is the vertical distance from a datum to a point in the stream. The velocity head is expressed by:

3. 85 Energy Head Energy Grade Line Hydraulic Grade Line (water surface) Channel Bottom headloss Elevation Head Depth 1 Datum Depth 2 Energy Head -Elevation Head -Elevation Head -Velocity Head -Velocity Head -Total Head -Total Head Momentum Open Channel Veloctiyhead Graphical depiction of elevation head, velocity head, and total head. Total head is the sum of velocity head, depth and elevation head.

4. 86 Momentum Equation Energy HeadMomentum -Equation -Equation -Forces -Forces Open Channel Hydrostatic Forces Friction Forces Weight External Forces

5. 87 Hydrostatic Forces Hydrostatic Forces Hydrostatic Forces Friction Forces Weight External Forces Control Volume Hydrostatic Forces Hydrostatic ForcesHydrostatic Forces are the forces placed on a control volume by the surrounding water. The strength of the force is based on depth and can be seen in the following relationship: P=  H H Energy HeadMomentum -Equation -Equation -Forces -Forces Open Channel

6. 88 Friction Forces Friction Forces Hydrostatic Forces Friction Forces Weight External Forces Friction Force friction force The friction force on a control volume is due to the water passing the channel bottom and depends on the roughness of the channel. Control Volume Energy HeadMomentum -Equation -Equation -Forces -Forces Open Channel

7. 89 Weight Weight Hydrostatic Forces Friction Forces Weight External Forces Control Volume weight The weight of a control volume is due to the gravitational pull on the its mass. Weight Weight = mg Energy HeadMomentum -Equation -Equation -Forces -Forces Open Channel

8. 90 External Forces External Forces Hydrostatic Forces Friction Forces Weight External Forces Top View of Control Volume Streamflow direction FdFd External Forces (F d ) External Forces (F d ) the forces created by a control volume striking a stationary object. External Forces can be explained by the following equation: F d =1/2C d  Av 2 Energy HeadMomentum -Equation -Equation -Forces -Forces Open Channel

9. 91 Steady vs. Unsteady Flow Energy Head Momentum Open Channel -Steady -vs- Unsteady -Steady -vs- Unsteady -Uniform -vs- Nonuniform -Supercitical -vs- subcritical -Equations Fluid properties including velocity, pressure, temperature, density, and viscosity vary in time and space. steadyA fluid it termed steady if the depth of flow does not change or can be assumed constant during a specific time interval. unsteadyFlow is considered unsteady if the depth changes with time.

10. 92 Uniform and Nonuniform Flow Energy Head Momentum Open Channel -Steady -vs- Unsteady -Uniform -vs- Nonuniform -Uniform -vs- Nonuniform -Supercitical -vs- subcritical -Equations Uniform FlowUniform Flow is an equilibrium flow such that the slope of the total energy equals the bottom slope. Nonuniform FlowNonuniform Flow is a flow of water through a channel that gradually changes with distance.

11. 93 Super -vs.- Sub Critical Super -vs.- Sub Critical Energy Head Momentum Open Channel -Steady-vs.-Unsteady -Uniform-vs. Nonuniform -Sub/Supercritical -Sub/Supercritical -Equations

12. 94 Critical flow: a demonstration No velocity If a stone is dropped into a body of water, with no velocity, the waves formed by the water are fairly circular. This is similar to sub-critical flow. Energy Head Momentum Open Channel -Steady-vs.-Unsteady -Uniform-vs. Nonuniform -Sub/Supercritical -Sub/Supercritical -Equations

13. 95 Critical flow: a demonstration Small velocity Now, if a velocity is added to the body of water, the waves become unsymmetrical, increasing to the downstream side. This happens as the velocity approaches critical flow. Notice that the wave still moves upstream, though slower than the downstream wave. Energy Head Momentum Open Channel -Steady-vs.-Unsteady -Uniform-vs. Nonuniform -Sub/Supercritical -Sub/Supercritical -Equations

14. 96 Critical flow: a demonstration Large velocity Now if a large velocity is added to the body of water, the wave patterns only go in one direction. This represents the point when flow has gone beyond critical, into the supercritical region. Energy Head Momentum Open Channel -Steady-vs.-Unsteady -Uniform-vs. Nonuniform -Sub/Supercritical -Sub/Supercritical -Equations

15. 97 Froude number Froude number The Froude number is a numerical value that describes the type of flow present (critical, supercritical, subcritical), and is represented by the following equation for a rectangular channel: N F = Froude number v = mean velocity of flow g = acceleration of gravity d m = mean (hydraulic) depth Energy Head Momentum Open Channel -Steady-vs.-Unsteady -Uniform-vs. Nonuniform -Sub/Supercritical -Sub/Supercritical -Froude number -Froude number -Equations

16. 98 Froude number Froude number The generalized formula for the Froude number is as follows: Fr = Froude number Q = Flow rate in the channel B = Top width of water surface A = Area of the channel Energy Head Momentum Open Channel -Steady-vs.-Unsteady -Uniform-vs. Nonuniform -Sub/Supercritical -Sub/Supercritical -Froude number -Froude number -Equations

17. 99 Froude number - mean depth B=width of the free water surface A=cross-sectional area of the channel Mean depth is a ratio of the width of the free water surface to the cross-sectional area of the channel. Energy Head Momentum Open Channel -Steady-vs.-Unsteady -Uniform-vs. Nonuniform -Sub/Supercritical -Sub/Supercritical -Froude number -Froude number -Equations

18. 100 Froude number Froude number The Froude number can then be used to quantify the type of flow. If the Froude number is less than 1.0, the flow is subcritical. The flow would would be characterized as tranquil. If the Froude number is equal to 1.0, the flow is critical. If the Froude number is greater than 1.0, the flow is supercritical and would be characterized as rapid flowing. This type of flow has a high velocity which can be potentially damaging. Energy Head Momentum Open Channel -Steady-vs.-Unsteady -Uniform-vs. Nonuniform -Sub/Supercritical -Sub/Supercritical -Froude number -Froude number -Equations

19. 101 Super-vs.-Subcritical Critical depth can also be determined by constructing a Specific Energy Curve. The critical depth is the point on the curve with the lowest specific energy. Any depth greater than critical depth is subcritical flow and any depth less than is supercritical flow. Energy Head Momentum Open Channel -Steady-vs.-Unsteady -Uniform-vs. Nonuniform -Sub/Supercritical -Sub/Supercritical -Equations

20. 102Super-vs.-Subcritical Critical depth Subcritical depth Supercritical depth

21. 103 Open Channel Equations Chezy Equation Manning’s Equation Bernoulli Equation St. Venant Equations Energy Head Momentum Open Channel -Steady -vs- Unsteady -Uniform -vs- Nonuniform -Supercitical -vs- subcritical -Equations: -Equations:ChezyManningBernoulli St. Venant

22. 104 Chezy Equation Energy Head Momentum Open Channel -Chezy Equation -Chezy Equation -Manning’s -Bernoulli -St. Venant In 1769, the French engineer Antoine Chezy developed the first uniform-flow formula. The formula was derived based on two assumptions. First, Chezy assumed that the force resisting the flow per unit area of the stream bed is proportional to the square of the velocity (KV 2 ), with K being a proportionality constant.

23. 105 Chezy Equation Energy Head Momentum Open Channel -Chezy Equation -Chezy Equation -Manning’s -Bernoulli -St. Venant The second assumption was that the channel was undergoing uniform flow. The difficulty with this formula is determining the value of C, which is the Chezy resistance factor. There are three different formulas for determining C, the G.K. Formula, the Bazin Formula, and the Powell Formula.

24. 106 Chezy Equation Energy Head Momentum Open Channel -Chezy Equation -Chezy Equation -Manning’s -Bernoulli -St. Venant Later on, when Manning's equation was developed in 1889, a relationship between Manning’s “n” and Chezy’s “C” was established. Finally in 1933, the Manning equation was suggested for international use rather than Chezy’s Equation.

25. 107 Manning’s Equation Energy Head Momentum Open Channel - -Chezy Equation -Manning’s -Manning’s -Bernoulli -St. Venant In 1889 Robert Manning, an Irish engineer, presented the following formula to solve open channel flow. V = mean velocity in fps R = hydraulic radius in feet S = the slope of the energy line n = coefficient of roughness The hydraulic radius (R) is a ratio of the water area to the wetted perimeter.

26. 108 Manning’s Equation Energy Head Momentum Open Channel -Chezy Equation -Manning’s -Manning’s -Bernoulli -St. Venant This formula was later adapted to obtain a flow measurement. This is done by multiplying both sides by the area. Manning’s equation is the most widely used of all uniform-flow formulas for open channel flow, because of its simplicity and satisfactory results it produces in real-world applications.

27. 109 Manning’s Equation Energy Head Momentum Open Channel -Chezy Equation -Manning’s -Manning’s -Bernoulli -St. Venant Note that the equation expressed in the previous slide was the English version of Manning’s equation. There is also a metric version of Manning’s equation, which replaces the 1.49 with 1. This is done because of unit conversions. The metric equation is:

28. 110 Bernoulli Equation Energy Head Momentum Open Channel -Chezy Equation -Manning’s -Bernoulli -Bernoulli -St. Venant The Bernoulli equation is developed from the following equation: This equation states that the elevation (z) plus the depth (y) plus the velocity head (V 1 2 /2g) is a constant. The difference being the headlosses - h L

29. 111 Bernoulli Equation Energy Head Momentum Open Channel -Chezy Equation -Manning’s -Bernoulli -Bernoulli -St. Venant This equation was then adapted by making a few assumptions. First, the head loss due to friction is equal to zero. This means the channel is perfectly frictionless surface. Second, that alpha 1 is equal to alpha 2 which is equal to 1. The alpha’s are in the original equation to account for a non-uniform velocity distribution. In this case we will assume a uniform distribution which produces the following equation:

30. 112 Bernoulli Equation Energy Head Momentum Open Channel -Chezy Equation -Manning’s -Bernoulli -Bernoulli -St. Venant A simplified version of the formula is given below:

31. 113 Bernoulli Equation Some comments on the Bernoulli equation Energy only Headloss in terms of energy Cannot calculate forces Limited Effect in “rapidly varying flow” Energy Head Momentum Open Channel -Chezy Equation -Manning’s -Bernoulli -Bernoulli -St. Venant

32. 114 St. Venant Equations Energy Head Momentum Open Channel -Chezy Equation -Manning’s -Bernoulli -St. Venant -St. Venant The two equations used in modeling are the continuity equation and the momentum equation. Continuity equation Momentum Equation

33. 115 St. Venant Equations Energy Head Momentum Open Channel -Chezy Equation -Manning’s -Bernoulli -St. Venant -St. Venant Unsteady -Nonuniform Steady - Nonuniform Diffusion or noninertial Kinematic The Momentum Equation can often be simplified based on the conditions of the model.

34. 116 Simulating the Hydrologic Response Model Types Precipitation Losses Modeling Losses Model Components

35. 117 Model Types Model Types Empirical Lumped Distributed Model Types Precipitation Losses Modeling Losses Model Components

36. 118 Precipitation … magnitude, intensity, location, patterns, and future estimates of the precipitation. … In lumped models, the precipitation is input in the form of average values over the basin. These average values are often referred to as mean aerial precipitation (MAP) values. … MAP's are estimated either from 1) precipitation gage data or 2) NEXRAD precipitation fields. Model TypesPrecipitation -Thiessen -Isohyetal -Nexrad Losses Modeling Losses Model Components

37. 119 Precipitation (cont.) … If precipitation gage data is used, then the MAP's are usually calculated by a weighting scheme. … a gage (or set of gages) has influence over an area and the amount of rain having been recorded at a particular gage (or set of gages) is assigned to an area. … Thiessen method and the isohyetal method are two of the more popular methods. Model TypesPrecipitation -Thiessen -Isohyetal -Nexrad Losses Modeling Losses Model Components

38. 120 Thiessen Thiessen methodThiessen method is a method for areally weighting rainfall through graphical means. Model TypesPrecipitation -Thiessen -Thiessen -Isohyetal -Nexrad Losses Modeling Losses Model Components

39. 121 Isohyetal Isohyetal methodIsohyetal method is a method for areally weighting rainfall using contours of equal rainfall (isohyets). Model TypesPrecipitation -Thiessen -Isohyetal -Isohyetal -Nexrad Losses Modeling Losses Model Components

40. 122 NEXRAD NexradNexrad is a method of areally weighting rainfall using satellite imaging of the intensity of the rain during a storm. Model TypesPrecipitation -Thiessen -Isohyetal -Nexrad -Nexrad Losses Modeling Losses Model Components

41. 123 Losses … modeled in order to account for the destiny of the precipitation that falls and the potential of the precipitation to affect the hydrograph. … losses include interception, evapotranspiration, depression storage, and infiltration. … Interception is that precipitation that is caught by the vegetative canopy and does not reach the ground for eventual infiltration or runoff. … Evapotranspiration is a combination of evaporation and transpiration and was previously discussed. … Depression storage is that precipitation that reaches the ground, yet, as the name suggests, is stored in small surface depressions and is generally satisfied during the early portion of a storm event. Model Types PrecipitationLosses Modeling Losses Model Components

42. 124 Modeling Losses … simplistic methods such as a constant loss method may be used. … A constant loss approach assumes that the soil can constantly infiltrate the same amount of precipitation throughout the storm event. The obvious weaknesses are the neglecting of spatial variability, temporal variability, and recovery potential. Other methods include exponential decays (the infiltration rate decays exponentially), empirical methods, and physically based methods. … There are also combinations of these methods. For example, empirical coefficients may be combined with a more physically based equation. (SAC-SMA for example) Model Types Precipitation Losses Modeling Losses -SAC-SMA Model Components

43. 125 Simulating Watershed Response Infiltration Long Term –vs.- Short Infiltration Evapotranspiration Unit Hydrograph TimingRouting Infiltration or “losses” - this section describes the action of the precipitation infiltrating into the ground. It also covers the concept of initial abstraction, as it is generally considered necessary to satisfy the initial abstraction before the infiltration process begins.

44. 126 Simulating Watershed Response Infiltration Long Term –vs.- Short Infiltration Evapotranspiration Unit Hydrograph TimingRouting Initial Abstraction - It is generally assumed that the initial abstractions must be satisfied before any direct storm runoff may begin. The initial abstraction is often thought of as a lumped sum (depth). Viessman (1968) found that 0.1 inches was reasonable for small urban watersheds. Would forested & rural watersheds be more or less?

45. 127 Simulating Watershed Response Infiltration Long Term –vs.- Short Infiltration Evapotranspiration Unit Hydrograph TimingRouting Forested & rural watersheds would probably have a higher initial abstraction. The Soil Conservation Service (SCS) now the NRCS uses a percentage of the ultimate infiltration holding capacity of the soil - i.e. 20% of the maximum soil retention capacity.

46. 128 Simulating Watershed Response Infiltration Long Term –vs.- Short Infiltration Evapotranspiration Unit Hydrograph TimingRouting Infiltration is a natural process that we attempt to mimic using mathematical processes. Some of the mathematical process or simulation methods are conceptual while others are more physically based.

47. 129 Simulating Watershed Response Infiltration Long Term –vs.- Short Infiltration Evapotranspiration Unit Hydrograph TimingRouting Constant Infiltration Rate : A constant infiltration rate is the most simple of the methods. It is often referred to as a phi-index or  -index. In some modeling situations it is used in a conservative mode. The saturated soil conductivity may be used for the infiltration rate. The obvious weakness is the inability to model changes in infiltration rate. The phi-index may also be estimated from individual storm events by looking at the runoff hydrograph.

48. 130 Simulating Watershed Response Infiltration Long Term –vs.- Short Infiltration Evapotranspiration Unit Hydrograph TimingRouting Constant Percentage Method : Another very simplistic approach - this method assumes that the watershed is capable of infiltrating or “using” a value that is proportional to rainfall intensity. The constant percentage rate can be “calibrated” for a basin by again considering several storms and calculating the percentage by :

49. 131 Constant Percentage Example Long Term –vs.- Short Infiltration Evapotranspiration Unit Hydrograph TimingRouting 0 1 2 77.5% infiltrates

50. 132 Simulating Watershed Response Infiltration Long Term –vs.- Short Infiltration Evapotranspiration Unit Hydrograph TimingRouting Exponential Decay: This is purely a mathematical function - of the following form:

51. 133 Simulating Watershed Response Infiltration Long Term –vs.- Short Infiltration Evapotranspiration Unit Hydrograph TimingRouting Exponential Decay: Effect of f o or f c

52. 134 Simulating Watershed Response Infiltration Long Term –vs.- Short Infiltration Evapotranspiration Unit Hydrograph TimingRouting Exponential Decay: Effect of K

53. 135 Simulating Watershed Response Infiltration Long Term –vs.- Short Infiltration Evapotranspiration Unit Hydrograph TimingRouting SCS Curve Number: Soil Conservation Service is an empirical method of estimating EXCESS PRECIPITATION We can imply that : P - P e = F

54. 136 SCS (NRCS) Runoff Curve Number The basic relationships used to develop the curve number runoff prediction technique are described here as background for subsequent discussion. The technique originates with the assumption that the following relationship describes the water balance of a storm event. where F is the actual retention on the watershed, Q is the actual direct storm runoff, S is the potential maximum retention, and P is the potential maximum runoff

55. 137 MoreModifications More Modifications At this point in the development, SCS redefines S to be the potential maximum retention SCS defines I a in terms of S as : I a = 0.2S and since the retention, F, equals effective precipitation minus runoff : F = (P-I a ) - Q Substituting gives the familiar SCS rainfall-runoff

56. 138 Estimating “S” The difficult part of applying this method to a watershed is the estimation of the watershed’s potential maximum retention, S. SCS developed the concept of the dimensionless curve number, CN, to aid in the estimation of S. CN is related to S as follows : CN ranges from 1 to 100 (not really!)

57. 139 Determine CN The Soil Conservation Service has classified over 8,500 soil series into four hydrologic groups according to their infiltration characteristics, and the proper group is determined for the soil series found. The hydrologic groups have been designated as A, B, C, and D. Group A is composed of soils considered to have a low runoff potential. These soils have a high infiltration rate even when thoroughly wetted. Group B soils have a moderate infiltration rate when thoroughly wetted, while group C soils are those which have slow infiltration rates when thoroughly wetted. Group D soils are those which are considered to have a high potential for runoff, since they have very slow infiltration rates when thoroughly wetted (SCS, 1972).

58. 140 Adjust CN’s

59. 141 SAC-SMA … The Sacramento Soil Moisture Accounting Model (SAC-SMA) is a conceptual model of soil moisture accounting that uses empiricism and lumped coefficients to attempt to mimic the physical constraints of water movement in a natural system. TensionFree TensionFree - Primary Free - Supplemental Upper Zone Lower Zone Model Types Precipitation Losses Modeling Losses -SAC-SMA -SAC-SMA Model Components

60. 142 Runoff … Runoff is essentially the excess precipitation - the precipitation minus the losses. … In the NWSRFS, runoff is modeled through the use of the SAC-SMA or an antecedent precipitation index (API) model. … Runoff is transformed to streamflow at the basin outlet via a unit hydrograph. … In actuality, all forms of surface and subsurface flow that reach a stream channel and eventually the outlet are modeled through the use of the unit hydrograph. Model Types Precipitation Losses Modeling Losses Model Components -Runoff -Runoff -Unit Hydrograph

61. 143 Unit Hydrograph The hydrograph that results from 1-inch of excess precipitation (or runoff) spread uniformly in space and time over a watershed for a given duration. The key points : 1-inch of EXCESS precipitation Spread uniformly over space - evenly over the watershed Uniformly in time - the excess rate is constant over the time interval There is a given duration Model Types Precipitation Losses Modeling Losses Model Components -Runoff -Unit Hydrograph -Unit Hydrograph

62. 144 Linearity of Unit Hydrograph … In addition, when unit hydrograph theory is applied, it is assumed that the watershed responds linearly. … Meaning that peak flow from 2 inches of excess will be twice that of 1 inch of excess

63. 145 Derived Unit Hydrograph

64. 146 Derived Unit Hydrograph

65. 147 Derived Unit Hydrograph Rules of Thumb : … the storm should be fairly uniform in nature and the excess precipitation should be equally as uniform throughout the basin. This may require the initial conditions throughout the basin to be spatially similar. … Second, the storm should be relatively constant in time, meaning that there should be no breaks or periods of no precipitation. … Finally, the storm should produce at least an inch of excess precipitation (the area under the hydrograph after correcting for baseflow).

66. 148 Synthetic Unit Hydrograph SCS Snyder Clark - (time-area)

67. 149 SCS - Dimensionless UHG

68. 150 SCS - Dimensionless UHG

69. 151 SCS - Dimensionless UHG

70. 152 Time-Area

71. 153 Time-Area

72. 154 Time-Area

73. 155 Stream Routing... stream routing is used to account for storage and translation effects as a runoff hydrograph travels from the outlet of one basin through the next downstream basin. … Most of the time, channels act as reservoirs and have the effect of attenuating the hydrograph. … 2 basic types of flow or channel routing : hydrologic hydraulic

74. 156 Typical Effect of Routing

75. 157 Lakes, Reservoirs, Impoundments,...have the effect of storing flow and attenuating hydrographs. … Reservoirs (and impoundments) are modeled with some form of routing. … hydrologic and hydraulic routing may be applicable; although most often, hydrologic routing is used in reservoir routing for normal flow conditions. … During failure scenarios an unsteady flow model (hydraulic routing) is usually necessary due to the nature of the flow, which is rapidly changing.

76. 158 Factors Affecting the Hydrologic Response Current Conditions Precipitation Patterns Land Use Channel Changes Others…..

77. 159 Current Conditions Wet Dry Update model states subjective

78. 160 Precipitation Patterns … The pattern is both temporal and spatial. … A storm moving away from an outlet will have a very different result than the identical storm pattern (spatially) moving towards the outlet. … Lumped hydrologic models have a very difficult time in simulating spatially and temporally varied storm events. … The very nature of MAP values - indicates one of the problems. … A forecaster must understand the potential of precipitation patterns to affect the forecast

79. 161 Land Use Urban Agricultural Anything that changes the infiltration, runoff, etc...

80. 162 Channel Changes Slopes Storage Rating Curve Ice!!!

81. 163 Rating Curves Rating curves establish a relationship between depth and the amount of flow in a channel.