Concept:

Hydrograph: It is a continuous plot of instantaneous discharge (runoff) v/s time.

It results from a combination of physiographic and meteorological conditions in a watershed and represents the integrated effects of climate, hydrologic losses, surface runoff, interflow, and groundwater flow.

Factors that influence the hydrograph shape and volume

i. Meteorological factors

ii. Physiographic or watershed factors and

iii. Human factors

Flood hydrograph has three characteristic regions:

(i) Rising limb AB: Curve Joining point A, the starting point of the rising curve and point B, the point of inflection. It is concave in shape and the shape depends both upon the rainfall and the catchment parameters.

(ii) Crest segment BC: Between the two points of inflection B and C with a peak P. For large catchments it generally occurs sometimes after the rainfall has ended.

(iii) Falling limb/ Recession Limb: Curve CD starting from the second point of inflection C and the shape depends only on catchment parameters as the rainfall has ceased by then.

Concept:

S curve is a hydrograph produced due to continuous effective rainfall at a constant rate for an infinite period .

The maximum ordinate of S-curve is the equilibrium discharge. It has an initial steep portion and reaches a maximum equilibrium discharge at a time equal to the time base of the first unit hydrograph. The average intensity of ER producing the S-curve is 1/D cm/h and the equilibrium discharge,

\({Q_e} = 2.778\frac{A}{D}{m^3}/s\) 

Where,

A = Area of the catchment in km²

D = Duration of rainfall in hour.

Calculation:

A = 600 km², D = 4 hour

\(\therefore {Q_e} = 2.778 \times \frac{{600}}{4} = 416.7\;{m^3}/s\) 

Concept:

Direct Runoff Hydrograph: It is a plot of discharge vs time past a specific point due to the direct runoff coming from the catchment without including any contribution from Baseflow.

Hydrograph from a catchment with a broader end near the outlet will have a higher and early peak with compared to a catchment with narrow end towards the outlet.

Also, a catchment with two separate broader ends will have two peaks.

Important Point:

The fan shaped catchment gives high peak and narrow hydrograph as compared to the fern shaped catchment.

Concept:

→ Estimation of flood of a particular Return period is done using :

(a) Empirical Probability Distribution:

(b) Theoretical Probability distribution:

(i) Log – Pearson’s type III Method

(ii) Gumbel’s Method: Flood discharge X_T for a return period T years is statistically calculated as below,

X_T = X̅ + K_T × σ_n-1

Where,

X_T = Flood discharge for T year return period.

X̅ = Mean of annual data series.

σ_n-1 = Standard Deviation of Annual data series.

K_T = Frequency factor.

Calculation:

x̅ = 30000 m³/s, k_T = 0.2, σ = 300 m³/s

∴ X_T = X̅ + K_T σ

⇒ X₁₀₀ = 30000 + 0.2 × 300 = 30060 m³/s

Hydrograph for the given catchment:

Direct until hydrograph ordinate at 15^th hour = 15 m³/seconds

Total rainfall intensity = 3.2 in hours

Excess rainfall = 3.2 – 0.4 = 2.8 cm

For 2.8 cm excess:

Ordinate of direct until hydrograph = 2.8 × 25 = 70 m³/sec

∴ Ordinate of flood hydrograph at 15^th hour = 70 + base flow = 70 + 10 = 80 m³/sec

Runoff depth = \({5\; hr\;\times\;Sum \;of \;DRH}\over{Area\;of\;the\;catchment}\)

\(={{5\times3600\times2625}\over{56700\times10^4}}\) = 0.0833 m = 8.33 cm

Concept:

A hydrograph is a plot between flood discharge and the duration.

If the hydrograph contains only direct runoff then it is called direct runoff hydrograph (DRH). But if it also contains the base flow then it is called flood hydrograph.

Calculation:

∴ Peak ordinate of 4 hour DRH with excess rainfall of 2 cm is 5.25 m³/s.

Concept:

If the probability of an event occurring is P, the probability of the event not occurring in a given year is q = (1 - P).

The binomial distribution can be used to find the probability of occurrence of the event r times in n successive years.

\({{\rm{P}}_{\left( {{\rm{r}},{\rm{n}}} \right)}} = {}_{}^{\rm{n}}{{\rm{C}}_{\rm{r}}}{{\rm{P}}_{\rm{r}}}{{\rm{q}}^{{\rm{n}} - {\rm{r}}}} = \frac{{{\rm{n}}!}}{{\left( {{\rm{n}} - {\rm{r}}} \right)!{\rm{r}}!}}{{\rm{P}}^{\rm{r}}}{{\rm{q}}^{{\rm{n}} - {\rm{r}}}}\)

Where

P_(r,n) = Probability of a random hydrologic event (rainfall) of given magnitude and exceedance probability P occurring r times in n successive years.

E.g.

a) The probability of an event of exceedance probability P occurring 2 times in n successive years is

\({{\rm{P}}_{\left( {2,{\rm{n}}} \right)}} = \frac{{{\rm{n}}!}}{{\left( {{\rm{n}} - 2} \right)!2!}}{{\rm{P}}^2}{{\rm{q}}^{{\rm{n}} - 2}}\)

b) The probability of the event not occurring at all in n successive years is

P_(0,n) = qⁿ = (1 - P)ⁿ

c) The probability of the event occurring at least once in n successive years

P₁ = 1 – qⁿ = 1 – (1 - P)ⁿ

Calculation:

Given: P = 1/50 = 0.02

For rainfall occurring two times in 15 successive years.

n = 15, r = 2

\({{\rm{P}}_{\left( {2,15} \right)}} = \frac{{15!}}{{13!2!}} \times {\left( {0.02} \right)^2} \times {\left( {0.98} \right)^{13}} = 15 \times \frac{{14}}{2} \times 0.0004 \times 0.769 = 0.0323 = 3.23\% \)

Concept:

Unit hydrograph (UH):

A unit hydrograph can be defined as the hydrograph of direct runoff resulting from one unit depth (1 cm) of rainfall excess occurring uniformly over the basin for a specified duration.The unit hydrograph is the unit pulse response function of a linear hydrologic system.

The unit hydrograph is a simple linear model that can be used to derive the hydrograph of any duration resulting from any amount of excess rainfall for the given area.

Peak of UH = \( \frac{{Peak\;of\;DRH}}{{Rainfall\;Excess}}\)

Calculation:

Peak of flood hydrograph = 180 m³/s

Base flow = 30 m³/s

∴ Peak of DRH = 180 – 30 = 150 m³/s

Total rainfall = 2.7 cm

Infiltration loss = 0.3 cm/hour

∴ Rainfall excess = 2.7 – 0.3 × 3 = 1.8 cm

∴ Peak of 3 – h unit hydrograph \(= \frac{{Peak\;of\;DRH}}{{Rainfall\;Excess}} = \frac{{150}}{{1.8}} = 83.33\;{m^3}/s\)

Run-off produced by 12 hr storm,

R = P – ϕt

= 16 – 0.5 × 12

= 10 cm

∴ Peak flow resulting from direct Run-off hydrograph = Run-off × Ordinate of UH = 10 × 22.5 = 225 cumec