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Engineering Hydrology – Complete Study Notes

GATE ESE / IES SSC JE State PSC CWC / WRIS

Comprehensive chapter-wise notes covering all 9 topics of Engineering Hydrology — hydrological cycle, precipitation measurement, evapotranspiration, infiltration, runoff estimation, stream flow measurement, unit hydrograph theory, flood routing, and groundwater hydraulics. All key formulae, derivations, worked examples, and exam-focused tables included.

Ch 1 · Hydrological Cycle Ch 2 · Precipitation Ch 3 · Evapotranspiration Ch 4 · Infiltration Ch 5 · Runoff & Stream Flow Ch 6 · Unit Hydrograph Ch 7 · Flood Frequency Analysis Ch 8 · Flood Routing Ch 9 · Groundwater ★ Quick Revision

1Hydrological Cycle►

1.1 The Water Cycle

The hydrological cycle describes the continuous circulation of water on, above, and below the surface of the Earth. Solar energy drives evaporation; gravity drives precipitation and runoff. The cycle has no beginning or end — it is a closed system.

Component	Description	Key Parameter
Precipitation	All forms of water falling from atmosphere — rain, snow, sleet, hail	Rainfall depth (mm), intensity (mm/hr)
Interception	Precipitation caught by vegetation canopy, evaporated before reaching ground	Interception loss (mm)
Infiltration	Water entering soil surface	Infiltration rate f (mm/hr)
Evapotranspiration	Evaporation from water bodies + transpiration by plants	PET, AET (mm/day)
Runoff	Water flowing over land surface into streams	Direct runoff, base flow
Groundwater flow	Sub-surface flow through aquifers	Hydraulic head, Darcy flux

1.2 Water Balance Equation

P = R + ET + ΔS
P = Precipitation; R = Runoff; ET = Evapotranspiration; ΔS = Change in storage

For a catchment over a long period (ΔS → 0):
P ≈ R + ET

Annual runoff coefficient: C = R/P (dimensionless; typically 0.1–0.6 for Indian catchments)

1.3 Catchment Characteristics

Form factor: F = A / L² (A = area, L = axial length)
Compactness coefficient: Kc = P / (2√(πA)) (P = perimeter)
Drainage density: D_d = ΣL / A (ΣL = total stream length)
Stream frequency: F_s = N / A (N = number of streams)

Time of concentration (Kirpich): t_c = 0.0195 × L^0.77 / S^0.385 (L in m, S = slope, t_c in min)

📝 GATE Tip: Water balance P = R + ET + ΔS is the most fundamental equation. For long-term annual averages ΔS = 0. Drainage density = total channel length / catchment area — a measure of how well drained a basin is.

2Precipitation►

2.1 Types of Precipitation

Type	Mechanism	Characteristics
Frontal	Warm moist air lifted over cold air mass at a front	Large area, moderate intensity, long duration
Orographic	Moist air forced upward over mountains	Windward slope heavy rain; leeward (rain shadow) dry
Convective	Local surface heating causes air mass to rise	Small area, very high intensity, short duration; thunderstorms
Cyclonic	Convergence of air into low-pressure system	Very large area; tropical cyclones / depressions

2.2 Measurement — Rain Gauges

Type	Working	Key Feature
Symon's gauge	Cylindrical collector, manual daily reading	Standard gauge in India; 127 mm dia orifice
Tipping bucket	0.25 mm tips recorded electronically	Continuous automatic record; suited for intensity
Weighing type	Spring balance / float records weight of collected water	Continuous record; measures snow also
Float type	Rising float records depth on chart drum	Self-recording; 24-hour chart

2.3 Optimum Number of Rain Gauges

N = (Cv / ε)²
Cv = Coefficient of variation of existing stations = (σ/x̄) × 100 (%)
ε = Allowable % error in mean rainfall estimate (typically 10%)
σ = standard deviation; x̄ = mean of existing gauge readings

Additional gauges needed = N − n (n = existing gauges)

2.4 Estimating Missing Rainfall Data

Normal Ratio Method (when Normal Annual Precipitation of adjacent stations differs by >10% from missing station):
P_x = (N_x/3) × [P_A/N_A + P_B/N_B + P_C/N_C]
P_x = missing rainfall; N_x = normal annual rainfall at x; P_A, P_B, P_C = adjacent station data; N_A, N_B, N_C = their normals

Arithmetic Mean (when normals differ by <10%):
P_x = (P_A + P_B + P_C) / 3

2.5 Areal Rainfall Estimation

Arithmetic Mean: P̄ = (1/n) Σ P_i (uniform distribution of gauges)

Thiessen Polygon Method:
P̄ = Σ(A_i × P_i) / A_total
A_i = area of Thiessen polygon around station i; accounts for non-uniform gauge distribution
Weightage: w_i = A_i / A_total

Isohyetal Method (most accurate):
P̄ = Σ(a_i × P̄_i) / A_total
a_i = area between adjacent isohyets; P̄_i = mean of two bounding isohyet values

2.6 Intensity–Duration–Frequency (IDF) Curves

Sherman formula: i = kT^x / (t + a)^n
i = rainfall intensity (mm/hr); T = return period (years); t = duration (min); k, x, a, n = constants

Return period (recurrence interval): T = 1/P (P = probability of exceedance in any year)
Probability of non-occurrence in n years: q = (1 − 1/T)^n
Risk of at least one occurrence in n years: R = 1 − (1 − 1/T)^n

📝 GATE Tip: Thiessen polygon method is the most frequently asked areal rainfall technique. Memorise the Normal Ratio formula — commonly asked as a numerical. IDF formula and return period / risk formula are ESE favourites.

3Evapotranspiration►

3.1 Definitions

Term	Definition
Evaporation	Water loss from open water / soil surface to atmosphere (liquid → vapour)
Transpiration	Water loss through plant stomata
Evapotranspiration (ET)	Combined evaporation + transpiration from a catchment
Potential ET (PET)	ET from an extensive green surface with unlimited water supply
Actual ET (AET)	ET under actual soil moisture conditions; AET ≤ PET

3.2 Penman Method (PET)

E_0 = [Δ/(Δ+γ)] × H_n + [γ/(Δ+γ)] × E_a
E_0 = PET (mm/day); H_n = net radiation expressed in mm of evaporation
Δ = slope of saturation vapour pressure vs temperature curve at mean T
γ = psychrometric constant (≈ 0.49 mm Hg/°C at sea level)
E_a = aerodynamic term = 0.35(e_s − e_a)(1 + u_2/16)
e_s = saturation VP; e_a = actual VP; u_2 = wind speed at 2 m height (km/day)

3.3 Pan Evaporation

Class A Evaporation Pan: standard pan, 1.22 m dia, 25.5 cm deep, mounted on wooden frame
Lake evaporation = pan coefficient × pan evaporation
Pan coefficient C_p ≈ 0.6–0.8 (typically 0.7 for Indian conditions)
E_lake = C_p × E_pan

3.4 Blaney–Criddle Method (Consumptive Use)

CU = k × f
f = p × (0.46T + 8.13) / 40.6 [monthly consumptive use factor]
p = % daytime hours of the year for the month; T = mean monthly temperature (°C)
k = empirical crop coefficient (varies by crop and growth stage)

📝 GATE Tip: Penman formula and pan evaporation with pan coefficient are the most tested. Remember C_p ≈ 0.7. AET ≤ PET always — AET equals PET only when soil moisture is at field capacity.

4Infiltration►

4.1 Infiltration Capacity

Infiltration capacity f is the maximum rate at which soil can absorb water (mm/hr). Actual infiltration rate equals f when rainfall intensity ≥ f; otherwise it equals the rainfall intensity. It decreases with time and approaches a constant rate f_c (final / ultimate infiltration capacity).

4.2 Horton's Equation

f(t) = f_c + (f_0 − f_c) × e^(−kt)
f(t) = infiltration capacity at time t (mm/hr)
f_0 = initial infiltration capacity (t = 0)
f_c = final (minimum) infiltration capacity
k = decay constant (hr⁻¹), depends on soil type and cover

Cumulative infiltration: F(t) = f_c × t + (f_0 − f_c)(1 − e^(−kt)) / k

4.3 Green–Ampt Model

f = K_s × [1 + (ψ × Δθ) / F]
K_s = saturated hydraulic conductivity (mm/hr)
ψ = suction head at wetting front (mm)
Δθ = θ_s − θ_i = change in volumetric moisture content
F = cumulative infiltration (mm)

More physically-based than Horton; useful for variable rainfall

4.4 φ-index (Phi-index)

The φ-index is the uniform rate of infiltration (mm/hr) such that the volume of rainfall above the index equals the observed direct runoff volume.

Procedure: assume a φ value → compute rainfall excess (P_i − φ) for each time interval → sum = observed runoff
Trial and error or graphical method.

W-index: average rate of infiltration during the entire rainfall period (including initial abstraction):
W = (P − Q − I_a) / t_r
P = total rainfall; Q = direct runoff; I_a = initial abstraction; t_r = storm duration

📝 GATE Tip: Horton's equation is most frequently asked numerically — given f_0, f_c, k, find F or f at time t. φ-index numerical is a staple in ESE. Remember: φ-index applies uniformly; below it, all rainfall infiltrates.

5Runoff & Stream Flow►

5.1 Components of Runoff

Component	Description
Surface runoff (overland flow)	Precipitation excess flowing over land surface; quickest response
Interflow (subsurface storm flow)	Lateral flow through upper soil layers; intermediate response time
Base flow (groundwater runoff)	Groundwater discharge to streams; slow, sustained flow
Direct runoff	Surface runoff + interflow; the rapidly responding portion
Total runoff / streamflow	Direct runoff + base flow

5.2 Factors Affecting Runoff

Rainfall characteristics (intensity, duration, distribution); catchment characteristics (area, shape, slope, soil type, land use, vegetation); climate (temperature, humidity, antecedent moisture condition).

5.3 SCS Curve Number Method

Direct Runoff Q = (P − I_a)² / (P − I_a + S)
S = potential maximum retention = (25400/CN) − 254 [S in mm]
I_a = initial abstraction = 0.2S (standard assumption)
CN = Curve Number (0–100); depends on soil group (A, B, C, D), land use, AMC condition

CN = 100 → impervious (all rainfall becomes runoff)
CN = 0 → infinite storage (no runoff)

AMC I: dry; AMC II: normal (standard); AMC III: wet (antecedent soil moisture conditions)

5.4 Stream Flow Measurement

Area-velocity method: Q = A × V̄ (cross-sectional area × mean velocity)
Mean velocity measured by current meter at 0.6d (one-point method) or (0.2d + 0.8d)/2 (two-point method)
d = total depth

Stage-discharge relationship (rating curve): Q = a(H − H_0)^b
H = gauge height; H_0 = gauge height at zero flow; a, b = constants

Dilution gauging (salt dilution): Q = q × (C_1 − C_2) / (C_2 − C_0)
q = injection rate; C_1 = tracer concentration injected; C_2 = plateau concentration; C_0 = background

📝 GATE Tip: SCS-CN method is extremely important. Memorise S = 25400/CN − 254 and Q = (P − 0.2S)²/(P + 0.8S). Current meter position: 0.6d for one point; average of 0.2d and 0.8d for two points.

6Unit Hydrograph►

6.1 Definition and Assumptions

A Unit Hydrograph (UH) is the direct runoff hydrograph resulting from 1 cm (or 1 unit) of effective rainfall distributed uniformly over the entire catchment at a uniform rate for a specified duration D.

Assumptions: (1) Time-invariance — same UH for same D regardless of when the storm occurs; (2) Linearity — runoff is directly proportional to effective rainfall; (3) Superposition — hydrographs from successive storms can be added.

6.2 Derivation of UH from Observed Hydrograph

Step 1: Separate base flow from observed hydrograph to get direct runoff hydrograph (DRH)
Step 2: Determine effective rainfall (rainfall excess) from hyetograph using φ-index
Step 3: Volume of DRH = volume of effective rainfall (check: Q_total = P_eff × A)
Step 4: UH ordinate = DRH ordinate / rainfall excess (cm)
Sum of UH ordinates × time interval = catchment area (in consistent units)

6.3 S-Curve Method (Changing UH Duration)

S-curve = summation hydrograph derived by lagging and adding UHs of duration D
To obtain UH of duration D' from S-curve derived from D-hour UH:
1. Lag the S-curve by D' hours to get S'
2. Difference: S − S' (offset S-curve)
3. Scale: UH(D') ordinate = (S − S') × (D/D')

Note: if D' < D, more accuracy needed; if D' = nD (multiple), simpler lagging works

6.4 Synthetic Unit Hydrograph — Snyder's Method

t_p = C_t × (L × L_c)^0.3
t_p = basin lag (hr) = time from centroid of effective rainfall to peak of UH
L = length of main stream from outlet to divide (km)
L_c = length from outlet to point nearest to centroid of basin (km)
C_t = basin coefficient (1.35–1.65 for Indian rivers)

Q_p = C_p × A / t_p (Q_p = peak discharge, m³/s; A = area, km²; C_p = 2.75–4.0)

Standard duration: t_r = t_p / 5.5
Base time: t_b = 72 + 3t_p (hours)

6.5 SCS Dimensionless UH

Q_p = 2.08 × A / T_p
T_p = D/2 + t_lag (D = effective rainfall duration; t_lag = basin lag ≈ 0.6 × t_c)
Q_p = peak discharge (m³/s); A = basin area (km²)
Time to peak T_p (hours); t_c = time of concentration (hours)

📝 GATE Tip: Unit hydrograph numericals are the most asked in this subject — deriving UH ordinates from a given storm and DRH, and using linearity/superposition to compute DRH for a multi-period storm. S-curve for duration change is an ESE favourite.

7Flood Frequency Analysis►

7.1 Return Period and Probability

Return period T = 1/P (P = annual exceedance probability)
Probability of occurrence in n years: P_n = 1 − (1 − 1/T)^n
For design: T = 100 yr flood → P = 1% per year

Weibull plotting position: T = (N+1) / m (m = rank in descending order, N = record length)
Hazen plotting position: T = 2N / (2m − 1)

7.2 Gumbel's Extreme Value Distribution (EVI)

x_T = x̄ + K × σ
x_T = flood magnitude for return period T
x̄ = mean; σ = standard deviation of annual maximum flood series
K = frequency factor = (y_T − ȳ_n) / S_n
y_T = −ln[−ln(1 − 1/T)] = reduced variate for return period T
ȳ_n, S_n = mean and standard deviation of reduced variate (tabulated for sample size N)

7.3 Log-Pearson Type III

Take logarithms: y = log x (or ln x)
Compute: ȳ, σ_y, and skewness coefficient G_s
x_T = antilog(ȳ + K × σ_y)
K = frequency factor from Pearson Type III table for given G_s and T

Recommended by US Water Resources Council for flood frequency studies

7.4 Rational Formula (Peak Flood Estimation)

Q_p = C × i × A / 3.6
Q_p = peak discharge (m³/s); C = runoff coefficient (0.1–0.95)
i = rainfall intensity for duration = t_c and selected return period (mm/hr)
A = catchment area (km²)

Applicable for small urban catchments (< 50 km²)
Assumes: uniform rainfall, steady state (storm duration ≥ t_c)

📝 GATE Tip: Gumbel distribution numerical — given annual peak flood data, find flood for a given return period — is a very common GATE question. Memorise y_T = −ln[−ln(1 − 1/T)]. Rational formula for small catchment peak flow is frequently asked in SSC JE.

8Flood Routing►

8.1 Concept of Flood Routing

Flood routing is the technique of determining the outflow hydrograph at a downstream point given the inflow hydrograph upstream. It accounts for storage effects that attenuate and delay the flood wave.

Type	Method	Applicability
Reservoir / Level pool routing	Modified Puls, Storage-indication method	Lakes, reservoirs — uniform water surface
Channel routing	Muskingum method, kinematic wave	River reaches — non-uniform, wedge storage

8.2 Continuity Equation for Routing

I − O = dS/dt (inflow − outflow = rate of change of storage)
Finite difference form:
(I_1 + I_2)/2 − (O_1 + O_2)/2 = (S_2 − S_1) / Δt
Rearranging: (2S_2/Δt + O_2) = (I_1 + I_2) + (2S_1/Δt − O_1)

8.3 Muskingum Method (Channel Routing)

Storage equation: S = K[x × I + (1−x) × O]
K = travel time of flood wave through reach (hours)
x = weighting factor (0 ≤ x ≤ 0.5); x = 0 → pure reservoir; x = 0.5 → pure translation
Typical: K ≈ travel time; x ≈ 0.2–0.3 for natural rivers

Routing equations:
O_2 = C_0 × I_2 + C_1 × I_1 + C_2 × O_1
C_0 = (−Kx + 0.5Δt) / (K − Kx + 0.5Δt)
C_1 = (Kx + 0.5Δt) / (K − Kx + 0.5Δt)
C_2 = (K − Kx − 0.5Δt) / (K − Kx + 0.5Δt)
Check: C_0 + C_1 + C_2 = 1

8.4 Reservoir Routing (Level Pool)

Storage-indication curve: plot O vs (2S/Δt + O)
Procedure for each time step Δt:
1. Compute RHS = (I_1 + I_2) + (2S_1/Δt − O_1)
[Note: 2S_1/Δt − O_1 = (2S_1/Δt + O_1) − 2O_1]
2. Read O_2 and (2S_2/Δt + O_2) from the storage-indication curve
3. 2S_2/Δt − O_2 = (2S_2/Δt + O_2) − 2O_2 for next step

📝 GATE Tip: Muskingum routing is the single most important topic in hydrology for GATE — numericals involving computing C_0, C_1, C_2 and routing an inflow hydrograph are asked almost every year. Always verify C_0 + C_1 + C_2 = 1.

9Groundwater►

9.1 Subsurface Water Zones

Zone	Description
Vadose zone (aeration zone)	Above water table; pores partly filled with air and water
Capillary zone	Just above water table; water held by capillarity; pressure < atmospheric
Saturated zone (phreatic zone)	Below water table; all pores saturated; pressure > atmospheric

9.2 Aquifer Types

Aquifer	Characteristics	Piezometric surface
Unconfined (phreatic)	Water table is the upper boundary; free surface	Water table itself
Confined (artesian)	Bounded above by impermeable layer (aquitard); water under pressure	Above top of aquifer; may be above ground → flowing artesian well
Leaky (semi-confined)	Semi-permeable layer above; leakage from adjacent layer	Between confined and unconfined behaviour
Perched	Local saturated zone above main water table on impermeable lens	—

9.3 Darcy's Law

Q = K × i × A
Q = discharge (m³/s); K = hydraulic conductivity (m/s); i = hydraulic gradient = dh/dl; A = cross-sectional area

Darcy velocity (specific discharge): v = Q/A = K × i
Seepage velocity: v_s = v / n (n = porosity; actual velocity through pores)

Validity: Re = v × d_50 / ν < 1–10 (laminar flow; Re based on median grain size)

Transmissivity: T = K × b (b = saturated thickness of aquifer; m²/day)
Storativity (Storage coefficient): S = specific storage × b (confined); S = specific yield (unconfined)

9.4 Well Hydraulics — Steady State

Confined aquifer (Thiem equation):
Q = 2πT(h_2 − h_1) / ln(r_2/r_1)
h_1, h_2 = heads at radii r_1, r_2; T = transmissivity

Unconfined aquifer (Dupuit-Thiem):
Q = πK(h_2² − h_1²) / ln(r_2/r_1)
h_1, h_2 = water table depths at r_1, r_2 from well

Radius of influence: R = 3000 × s_w × √K [Sichardt formula; s_w = drawdown at well, m; K in m/s; R in m]

9.5 Theis Equation (Unsteady Pumping)

s = (Q / 4πT) × W(u)
u = r²S / (4Tt)
s = drawdown (m); Q = pumping rate (m³/s); T = transmissivity; S = storage coefficient
r = distance from well; t = time since pumping started
W(u) = well function = −0.5772 − ln(u) + u − u²/2·2! + u³/3·3! − …

For small u (<0.05): W(u) ≈ −0.5772 − ln(u) [Cooper-Jacob simplification]

📝 GATE Tip: Darcy's law, Thiem equation for steady-state pumping, and distinguishing confined vs unconfined aquifer behaviour are core topics. For numericals: always identify aquifer type first, then select the correct equation. Storativity S ≈ 10⁻⁵–10⁻³ (confined); S ≈ 0.05–0.3 (unconfined = specific yield).

★Quick Revision — Key Formulae►

Topic	Key Formula / Value
Water balance	P = R + ET + ΔS
Optimum rain gauges	N = (Cv/ε)²
Missing rainfall	P_x = (N_x/3)[P_A/N_A + P_B/N_B + P_C/N_C]
Pan evaporation	E_lake = 0.7 × E_pan
Horton's equation	f = f_c + (f_0 − f_c)e^(−kt)
SCS runoff	Q = (P − 0.2S)² / (P + 0.8S); S = 25400/CN − 254
Gumbel distribution	x_T = x̄ + Kσ; y_T = −ln[−ln(1 − 1/T)]
Muskingum routing	O_2 = C_0 I_2 + C_1 I_1 + C_2 O_1; C_0+C_1+C_2 = 1
Darcy's law	Q = KiA; T = Kb; v_s = v/n
Thiem (confined)	Q = 2πT(h_2 − h_1)/ln(r_2/r_1)
Thiem (unconfined)	Q = πK(h_2² − h_1²)/ln(r_2/r_1)
Rational formula	Q_p = CiA/3.6 (m³/s; i in mm/hr; A in km²)

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Engineering Hydrology – Complete Study Notes

1.1 The Water Cycle

1.2 Water Balance Equation

1.3 Catchment Characteristics

2.1 Types of Precipitation

2.2 Measurement — Rain Gauges

2.3 Optimum Number of Rain Gauges

2.4 Estimating Missing Rainfall Data

2.5 Areal Rainfall Estimation

2.6 Intensity–Duration–Frequency (IDF) Curves

3.1 Definitions

3.2 Penman Method (PET)

3.3 Pan Evaporation

3.4 Blaney–Criddle Method (Consumptive Use)

4.1 Infiltration Capacity

4.2 Horton's Equation

4.3 Green–Ampt Model

4.4 φ-index (Phi-index)

5.1 Components of Runoff

5.2 Factors Affecting Runoff

5.3 SCS Curve Number Method

5.4 Stream Flow Measurement

6.1 Definition and Assumptions

6.2 Derivation of UH from Observed Hydrograph

6.3 S-Curve Method (Changing UH Duration)

6.4 Synthetic Unit Hydrograph — Snyder's Method

6.5 SCS Dimensionless UH

7.1 Return Period and Probability

7.2 Gumbel's Extreme Value Distribution (EVI)

7.3 Log-Pearson Type III

7.4 Rational Formula (Peak Flood Estimation)

8.1 Concept of Flood Routing

8.2 Continuity Equation for Routing

8.3 Muskingum Method (Channel Routing)

8.4 Reservoir Routing (Level Pool)

9.1 Subsurface Water Zones

9.2 Aquifer Types

9.3 Darcy's Law

9.4 Well Hydraulics — Steady State

9.5 Theis Equation (Unsteady Pumping)