BULACAN STATE UNIVERSITY

COLLEGE OF ENGINEERING

CE463 HYDROLOGY

PROJECT NO. 3

DIFFERENT FORMULA’S USED IN HYDROLOGY

SCORE

SUBMITTED BY: SUBMITTED TO:

STO. TOMAS, EMMANUEL DL. ENGR2. GILBERT C. PASCUAL

JANUARY 24, 2014

1. The Rational Method estimates the peak rate of runoff at a specific watershed location as a function of the drainage area, runoff coefficient, and mean rainfall intensity for a duration equal to the time of concentration, tc. The tc is the time required for water to flow from the most remote point of the basin to the location being analyzed.

The Rational Method is expressed as:

Q = CIAWhere:

Q = maximum rate of runoff (cfs)

C = runoff coefficient representing a ratio of runoff to rainfall

I = average rainfall intensity for a duration equal to the tc (in/hr)

A = drainage area contributing to the design location (acres)

2. Time of Concentration use of the Rational Method requires calculating the time of concentration (tc) for each design point within the drainage basin. The duration of rainfall is then set equal to the time of concentration and is used to estimate the design average rainfall intensity (I). The basin time of concentration is defined as the time required for water to flow from the most remote part of the drainage area to the point of interest for discharge calculations. The time of concentration is computed as a summation of travel times within each flow path as follows:

tc = tt1 + tt2 + tm

Where:

Tc = time of concentration (hours)

Tt = travel time of segment (hours)

m = number of flow segments

Knox County policies regarding the calculation of tc are as follows: • The tc shall be the longest sub-basin travel time when all flow paths are considered. • The minimum tc for all computations shall be five (5) minutes. Time of concentration calculations are subject to the following limitations: 1. The equations presented in this section should not be used for sheet flow on impervious land uses where the flow length is longer than 50 feet; and 2. In watersheds

with storm sewers, use care to identify the appropriate hydraulic flow path to estimate tc.

3. In engineering hydrology, the Hydrologic Budget is a quantitative accounting technique linking the components of hydrologic cycle. It is a form of a continuity equation that balances the gains and losses of water with the amount stored in a region. The components of water budget are inflow, outflow and storage.

INFLOWS - OUTFLOWS = STORAGES, or, in mathematical term,

I – O = ∂S / ∂t

Breaking system into individual component as shown in schematic figure:

P – E + ([(Rin + Gin)] – (Rout + Gout)] – T = dS/dt

Where:

P: Areal mean rate of precipitation (L/T)E: Evaporation (L/T)Rin, Gin: Inflow from surface and groundwater (L/T)

Rout, Gout: Outflow from surface and groundwater (L/T) S: Storage (L) and, T: Transpiration (evaporation from plants, L/T)

4. Balance Equation for Open Bodies with Short Duration:

And, for an open water bodies, short duration,

I - O = S/ t Where,

I = inflow volume per unit time O = Outflow per unit time

5. Balance Equation for Urban Drainage:

For urban drainage system, ET (evapotranspiration) is often neglected,

P - I - R - D = 0Where:

P = precipitation

I = infiltration

R = direct runoff

D = Combination of interception and depression storage

6. Thiessen Polygon: Each gage is assigned an area bounded by a perpendicular bisect between the station and those surrounding it. The polygon represent their respective areas of influence (see figure). The average precipitation is calculated as:

Average PPT = ∑ Ai pi / AT

7. Inverse Distance Method: The method is based on the assumption that the precipitation at a given point is influenced by all stations. The method of solution is to subdivide the watershed area into m rectangular areas. The mean precipitation is calculated using the following formula:

m n n

P = 1/A ∑ Aj ( ∑ dij –b )-1 ∑ d ij –b P i

j = 1 i = 1 i = 1Where: m is the number of the subareas, A j is the area of the jth sub area, A is the total area, dij is the distance from the center of the jth area to the ith precipitation gage, n is the number of gages and b is a constant and in most applications, it is taken as equal to 2. Note that if b is 0, the equation is reduced to the following: P = 1/A ∑ Aj Pi

Methods of Estimating Missing Precipitation:

8. Normal Ratio Method:

The missing precipitation at station x is calculated by using weights for precipitation at individual stations. The precipitation at station x is,

nPx = ∑ wi Pi

i=1where n is the number of stations and w i designates the weight for station i and computed as

wi = Ax / n Ai

Where Ai is the average annual rainfall at gage i, Ax is the average annual rainfall at station x in question.

Combining the above two equations,

nPx = (AX / n) ∑ Pi / Ai

i =19. Quadrant Method:

To account for the closeness of gage stations to the missing data gage, quadrant method is employed. The position of the station of the missing data is made to be the origin of the four quadrants containing the rest of stations. The weight for station i is computed as:

4wi = ∑ ( 1 / d2

i ) i=1

and the missing data is calculated as,

n nPx = ∑wi . Pi / ∑ wi

i=1 i=110. Wind Speed: Wind speed varies with the height above water surface. It can be

calculated using the empirical formula,

V/VO = (Z/ZO)0.15

Where V is the wind speed in mi/hr at Z height, VO is the wind speed at height ZO

measured in ft of the anemometer (instruments designed to measure total wind speed)

Empirical Formulas:

Using mass transfer, evaporation takes the following general form:

E = C f(u)(e –ea)

Where K is constant, f(u) is a function of wind speed at a given height, e is the actual vapor pressure at a given height, and es is the saturated vapor pressure at water surface,

The rate of evaporation from a lake can be calculated using empirical laws,

11. By Meyer (1944)

Where, E: Lake evaporation (inches / day)es – e: Water vapor deficit (difference between saturated vapor pressur and actual vapor pressure of atmosphere in-Hg)

C: Constant (0.36 for open water, 0.5 for wet soil) W: Wind Speed 25 ft above water level (mph)

12. By Dunne (1978)

Where, Rh: The relative humidity in %

e: Vapor pressure of air (mill bars) u2: Wind speed 2 m above water in km/day

13. Coaxial Chart: Penman (1948)

E = C (es – e) (1

E = (0.013 + 0.00016 u2) e

Penman developed an equation based on aerodynamic and energy balance equations for daily evaporation E and later Kohler (1955) developed an expression for lake evaporation in inches per day that is based on Penman’s theory, If EL designates average daily lake evaporation (in/day), then,

EL = 0.7 [EP + 0.00051 P αP (0.37 + 0.0041 uP) (T0 –Ta)0.88

To Outer face Temp of pan O F Lake Evap. in/day Ta Air Temp. in o F

Windspeed in mi/day Advected Energy (For given Temp & wind speed ,use chart to obtain αP) Atmospheric pressure in in-Hg

αP = 0.13 + 0.0065T0 – (6.0 x 10-8 T03) + 0.016 uP

0.36

14. Horton Infiltration Formula:

K: Constant depending on soil & surface & cover conditions, t is time

f = f∞ + (f0 – f∞) e-Kt

Initial infiltration capacity (L/T)Final infiltration capacity (equilibrium capacity)

Infiltration capacity at a given time t

The assumption inherent is that water is always “ponded” Horton equation requires evaluation of f0, f∞ and K .which can be derived from infiltration tests

15. Porosity: It is the ratio of the volume of interconnected voids within the soil to the total volume of the soil. If VV designates the volume of “interconnected” voids, VS is the bulk volume and VT is the total volume of the soil media, the porosity θ is defined as:

θ = VV / VT = (VT-VS) / VT = 1 – VS/VT

16. Volumetric Water content: It is the ratio of the volume of water to total volume of soil sample.

θW = VW / VT

17. Degree of Saturation: It is the ratio of the volume of water to the volume of voids. The soil is said to be saturated when the degree of saturation is 100%.

S = VW / VV

18. Moisture Content: It is the ratio of the weight of water to the weight of solid.

W = WW / WS

19. Distribution of Pressure:

The force that balances the gravity force and holds water in equilibrium is,

∂P / ∂ z = - γ

Where P is the pressure and γ is the unit weight of water. Integrating,

P = - γz + C

20. Darcy’s law shown above is limited to flow that is one–dimensional and homogeneous incompressible medium. Introducing the concept of specific discharge q, Darcy’s law becomes,

q = K Δh / L

In the above, q is also referred to as the Darcy flux. It is fictitious form of velocity because the equation assumes that the discharge occurs throughout the cross sectional area of soil in spite of the fact that solid particles constitute a major portion of the cross sectional area. The portion of the area available to flow is equal to θA, where θ is the porosity (as defined as the volume of pore relative to the total volume of the medium). The average “real” velocity is, therefore is,

v = q / θ

The energy loss Δh is due to the friction between the moving water and the walls of the solid. The hydraulic gradient is simply the difference between the head at inlet and head at the outlet divided by the distance between the inlet and the outlet. Note that the flow prescribed by Darcy’s law,

1) Takes place from higher head to lower head and not necessarily from higher pressure to lower one.2) Darcy’s law specifies linear relationship between velocity and hydraulic head. This relationship is valid only for small Reynolds number. At high Reynolds number, viscous forces do not govern the flow and the hydraulic gradient will have higher order terms.

21. Groundwater Flow Equation:

The derivation of groundwater equation is based on the conservation of mass between one point in flow to another coupled with the application of Darcy’s law. For incompressible flow, the continuity equation states that,

∂2h / ∂x2 + ∂2h / ∂y2 + ∂2h / ∂z2 = 0,

and for in one-dimensional flow, the equation is,

∂2h / ∂x2 = 022. Dissipation of energy

Mechanical energy = elevation + pressure + kinetic energy

Energy contained in a unit mass of stream water:

= g(z + d) + u2/2 = potential + u2/2

g: gravitational acceleration (= 9.8 m s-2)

Mechanical energy is dissipated by friction and turbulence.

23. Area-velocity method A cross-section is divided into strips. Why? Average velocity (v) in each strip represented by the velocity measured at “six-tenth” depth.

Q = A1 v1 + A2 v2 ……..

24. Float methodMeasurement of surface velocity (vs) using a float combined with a survey of cross-sectional area (A).

Q = 0.8vsA

Assumption: average velocity (u) is about 80 % of the surface velocity in the central part of stream.

25. V-notch weirV-notch weir has a well-defined stage-discharge relation:

Q = Cw g1/2 tan(θ/2) h5/2

Cw: weir coefficient (= 0.43 for a perfect weir, but should be determined by field calibration).

26. Snowmelt energy balance

Latent heat of fusion (Lf) = 334 kJ kg-1

The temperature of snowpack is nearly at 0 °C when snow is melting. Under the constant-temperature condition,

Qm = Qs(1 - α) + Qlw + Qh + Qe + Qp

Qm: energy available for snowmelt, Qs: incoming shortwave radiation, Qlw: long wave radiation into the snowpack, Qh: sensible heat transfer into the snowpack, Qe: latent heat transfer into the snowpack, Qp: energy input by rain on the snowpack.

27. Drainage density

= Total channel length (km) / Basin area (km2)

28. Relief ratio:

= elevation difference, highest –lowestlength of the basin, parallel to the main stream

Relief ratio indicates the average slope of the basin.

29. Water balance of a drainage basin

∆SM + ∆GWS = P -I - AET - OF – GWRWhere:

∆SM: soil moisture storage change

∆GWS: groundwater storage change

P: precipitation I: interception

AET: evapotranspiration OF: Overland flow

GWR: groundwater runoff → base flow

30. Storativity (S) is defined as:

S = volume of water pumped per area (m 3 /m 2 = m) Amount of pressure head drop in the aquifer (m)

S is proportional to the thickness(y, m) and compressibility (α, m2 N-1) of the aquifer.

S = ρgαy ρ = 1000 kg m-3 g = 9.8 m s-2

Typical values of α ranges from 10-6 (soft clay) to 10-8 (sands).