Advanced Civil Engineering Handbook › Chapter 8

Chapter 8 of 14 · 40 formulas · 77 one-liners

Hydrology & Irrigation Engineering — formulas & one-liners

Rainfall, runoff, hydrographs, ground water, crop water requirement, canal design, weirs, gravity dams, spillways and water logging — key formulas and facts for AE/JE exams, with IS 4987 and IS 6512 references.

8.1 Precipitation & rain-gauge network

Optimum number of rain gauges

N = (\frac{C _{v}}{ε})^{2}

C_v = coefficient of variation of rainfall at existing gauges (%); ε = allowable error in mean rainfall (%), usually 10%

C_v = 100 σ_m−1/P̄, where σ_m−1 is the standard deviation of rainfall at the m existing gauges

Missing record – normal ratio method

P_{x} = \frac{N _{x}}{M} i = 1 \sum M \frac{P _{i}}{N _{i}}

P_i = rainfall at M index stations; N_i, N_x = normal annual rainfall of index stations and of station X

Used when normal annual rainfalls differ by more than 10%; otherwise simple arithmetic average

Mean areal rainfall – Thiessen polygon method

\overset{ˉ}{P} = \frac{\sum P _{i} A _{i}}{\sum A _{i}}

A_i = area of the polygon around gauge i; P_i = rainfall at gauge i

Mean areal rainfall – isohyetal method

\overset{ˉ}{P} = \frac{1}{A} \sum a_{i} \frac{P _{i} + P _{i + 1}}{2}

a_i = area between consecutive isohyets of values P_i and P_i+1; A = total area

The IMD standard non-recording rain gauge is the Symons type, read once daily at 08:30 IST.
Recording rain gauges are of tipping-bucket, weighing-bucket and natural-syphon (float) types; the natural-syphon gauge is the IMD standard recording gauge.
As per IS 4987, a rain-gauge network should have about one gauge per 520 km² in plains, one per 260–390 km² in moderately elevated areas and one per 130 km² in hilly, heavy-rainfall areas.
At least 10% of the rain-gauge stations in a network should be self-recording.
Isohyetal method is considered the most accurate method of computing mean areal rainfall; arithmetic mean is the simplest.
A hyetograph is a bar chart of rainfall intensity against time, while a mass curve of rainfall plots cumulative rainfall against time.
Intensity–duration–frequency (IDF) curves show that for a given return period, rainfall intensity decreases with increasing duration.

8.2 Runoff, hydrographs & infiltration

Rational formula for peak runoff

Q_{p} = \frac{C i A}{3.6}

Q_p in m³/s; C = runoff coefficient; i = rainfall intensity (mm/h) for duration = time of concentration; A = area (km²)

Suited to small catchments (generally < 50 km²)

Dickens formula (peak flood)

Q_{p} = C_{D} A^{3/4}

Q_p in m³/s; A in km²; C_D ≈ 6 for north Indian plains, 11–14 for north Indian hilly areas, 14–28 for central India

Ryves formula (peak flood)

Q_{p} = C_{R} A^{2/3}

Q_p in m³/s; A in km²; C_R = 6.8 within 80 km of the east coast, 8.5 for 80–160 km from the coast, 10.2 for limited areas near hills

Inglis formula (peak flood)

Q_{p} = \frac{124 A}{A + 10.4}

Q_p in m³/s; A in km²; for fan-shaped catchments of old Bombay State (Maharashtra)

Equilibrium discharge of an S-curve

Q_{e} = 2.778 \frac{A}{D}

Q_e in m³/s; A = catchment area (km²); D = duration of unit hydrograph (h)

φ-index

ϕ = \frac{P - R}{t _{e}}

P = total rainfall; R = total direct runoff (both cm); t_e = duration of rainfall excess (h)

W-index = (P − R − S)/t_e, where S = depression and initial-loss storage; W-index ≤ φ-index

Horton's infiltration equation

f_{p} = f_{c} + (f_{0} - f_{c}) e^{- k_{h} t}

f_p = infiltration capacity at time t; f₀ = initial; f_c = final (constant) capacity; k_h = decay constant

Muskingum storage equation (channel routing)

S = K [x I + (1 - x) Q]

S = storage; K = storage-time constant (≈ travel time); x = weighting factor (0 to 0.5); I, Q = inflow and outflow

Used with continuity: I − Q = dS/dt

Direct runoff consists of surface runoff and prompt interflow; base flow is the delayed groundwater contribution to streamflow.
A flood hydrograph has a rising limb, a crest segment and a recession (falling) limb; the shape of the recession limb depends only on catchment characteristics.
A unit hydrograph is the direct-runoff hydrograph from 1 cm of rainfall excess falling uniformly over the catchment in a specified duration D.
Unit hydrograph theory rests on two assumptions: time invariance and linear response (proportionality and superposition).
Unit hydrograph theory is generally not applied to catchments larger than about 5000 km².
Infiltration capacity is measured with flooding-type (single- or double-ring) infiltrometers or with rainfall simulators.
Pan coefficient is about 0.80 for the ISI standard pan and about 0.70 for the US Class A land pan.
In reservoir routing with an uncontrolled spillway, the peak outflow occurs at the point where the outflow hydrograph cuts the falling limb of the inflow hydrograph.
In the Muskingum method x = 0 corresponds to reservoir-type storage depending only on outflow.
Gumbel's extreme-value distribution and Log-Pearson Type III distribution are used for flood frequency analysis.
Spillways of large dams are designed for the Probable Maximum Flood (PMF).

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8.3 Ground water & wells

Darcy's law

V = k i, Q = k i A

k = coefficient of permeability (m/s); i = hydraulic gradient; A = gross cross-sectional area

Valid for laminar flow, Reynolds number below about 1

Steady flow to a well in a confined aquifer (Thiem)

Q = \frac{2 π k b ( s _{1} - s _{2} )}{ln ( r _{2} / r _{1} )}

b = aquifer thickness; s₁, s₂ = drawdowns at observation wells at radii r₁, r₂

Transmissivity T = kb

Steady flow to a well in an unconfined aquifer (Dupuit)

Q = \frac{π k ( H ^{2} - h _{w}^{2} )}{ln ( R / r _{w} )}

H = saturated thickness before pumping; h_w = depth of water in well; R = radius of influence; r_w = well radius

Radius of influence – Sichardt

R = 3000 s k

R, s (drawdown) in m; k in m/s

Yield of an open well (recuperation test)

Q = C A H

A = well area; H = depression head; C = specific yield of well per hour ≈ 0.25 (clay), 0.5 (fine sand), 1.0 (coarse sand)

Porosity equals the sum of specific yield and specific retention.
An aquifer stores and transmits water freely; an aquiclude (e.g. clay) holds water but cannot transmit it; an aquifuge (e.g. solid granite) neither holds nor transmits water.
An aquitard is a semi-pervious layer that transmits water only very slowly.
Storage coefficient of a confined aquifer is typically 5 × 10⁻⁵ to 5 × 10⁻³, whereas that of an unconfined aquifer is practically equal to its specific yield.
A flowing artesian well occurs where the piezometric surface of a confined aquifer lies above ground level.
A perched water table forms above a local impervious lens lying above the main water table.
Theis and Cooper–Jacob methods analyse unsteady (non-equilibrium) flow to a well.

8.4 Water requirement of crops & soil–water relations

Duty–delta relation

Δ = \frac{8.64 B}{D}

Δ = delta (m); B = base period (days); D = duty (ha/cumec)

Depth of water to bring root zone to field capacity

d_{w} = \frac{γ _{d}}{γ _{w}} d (F_{c} - m_{0})

γ_d = dry unit weight of soil; d = root-zone depth; F_c = field capacity; m₀ = existing moisture content (both as fractions)

Frequency of irrigation

f_{w} = \frac{d _{w}}{C _{u}}

d_w = readily available water depth stored per irrigation; C_u = daily consumptive use (mm/day); f_w in days

Duty is the area irrigated per cumec of water flowing continuously for the base period; delta is the total depth of water a crop needs over its base period.
Duty increases (more area per cumec) as one moves downstream from the canal head towards the field, because conveyance losses reduce.
Base period is the time between the first and last watering before harvest, and is slightly less than the crop period.
Gross commanded area (GCA) includes unculturable land; culturable commanded area (CCA) is only the cultivable part, irrigated or not.
Intensity of irrigation is the percentage of CCA proposed to be irrigated in a season.
Kharif crops are sown with the monsoon (June–July) and harvested about October; rabi crops are sown about October and harvested about March.
Approximate delta values as per common Indian textbook tables: rice about 120 cm, sugarcane about 90 cm and wheat about 40 cm (figures vary between texts).
Paleo is the pre-sowing watering, while kor watering is the first watering after the crop has grown a few centimetres.
Kor depth is about 19 cm for rice and about 13.5 cm for wheat.
Time factor of a canal is the ratio of days the canal actually ran to the days it was meant to run; capacity factor is mean supply divided by full supply capacity.
Available water (moisture) is the difference between field capacity and permanent wilting point; hygroscopic water is not available to plants.
Consumptive use (evapotranspiration) is estimated by Blaney–Criddle, Hargreaves or Penman methods.
Field irrigation requirement equals the net irrigation requirement (consumptive use minus effective rainfall) divided by the water application efficiency.

8.5 Canal design & canal structures

Kennedy's critical velocity

V_{0} = 0.55 m D^{0.64}

V₀ = critical (non-silting, non-scouring) velocity (m/s); D = depth of flow (m); m = critical velocity ratio (1.0 for Upper Bari Doab silt)

Lacey's silt factor

f = 1.76 d_{mm}

d_mm = mean particle size of bed material (mm)

Lacey's regime velocity

V = (\frac{Q f ^{2}}{140})^{1/6}

Q = discharge (m³/s); V in m/s

Lacey's wetted perimeter and hydraulic radius

P = 4.75 Q, R = \frac{5}{2} \frac{V ^{2}}{f}

P, R in m; Q in m³/s

Lacey's regime slope

S = \frac{f ^{5/3}}{3340 Q ^{1/6}}

S = bed slope (dimensionless)

Flexibility of a canal outlet

F = \frac{m}{n} \cdot \frac{y}{H}

m, n = exponents of head in outlet and channel discharge equations; y = depth in distributary; H = head on outlet

Proportional outlet when F = 1, i.e. setting H/y = m/n

In Kennedy's theory, m > 1 is used for coarser silt and m < 1 for finer silt than that of the Upper Bari Doab canal.
Kennedy's theory gives no equation for bed slope and leaves the width-to-depth ratio arbitrary, so design is by trial.
Lacey's silt factor is 1.0 for standard silt of 0.323 mm mean size.
Lacey distinguished true regime, initial regime and final regime of a channel.
A ridge (watershed) canal irrigates both sides and needs few cross-drainage works; a contour canal irrigates only one side.
Aqueduct: canal passes over the drain with drain HFL below canal trough; syphon aqueduct: drain HFL rises above the canal bed and the drain flows under pressure.
Super-passage: drain passes over the canal; canal syphon: canal passes under the drain under pressure.
Canal falls are provided where the natural ground slope is steeper than the designed canal bed slope; the Sarda fall is common in Uttar Pradesh.
A head regulator controls supply entering an offtaking canal, while a cross regulator raises the water level in the parent canal upstream of it.
A semi-modular (flexible) outlet's discharge depends only on the distributary water level; a rigid module (e.g. Gibb's module) gives a constant discharge within limits.
Canal lining reduces seepage losses, controls water logging and allows higher permissible velocity and smaller section.

8.6 Weirs & barrages on permeable foundations

Bligh's creep theory – safe creep length

L = C H

L = total creep length along the base profile (horizontal + vertical); C = Bligh's creep coefficient; H = seepage head

Bligh's floor thickness against uplift

t = \frac{4}{3} \cdot \frac{h}{G - 1}

h = residual uplift head at the section (m); G = specific gravity of floor material; 4/3 = factor of safety

Lane's weighted creep length

L_{w} = \frac{1}{3} \sum L_{h} + \sum L_{v}

L_h = horizontal creep lengths (and slopes flatter than 45°); L_v = vertical creep lengths (and slopes steeper than 45°)

Safe when L_w ≥ C_w H, C_w = Lane's coefficient

Khosla's exit gradient

G_{E} = \frac{H}{d} \cdot \frac{1}{π λ}

H = maximum seepage head; d = depth of downstream cut-off; b = total floor length

λ = [1 + √(1 + α²)]/2 with α = b/d

Bligh's creep coefficient C is about 15 for fine micaceous sand and about 12 for coarse-grained sand.
Bligh's theory treats horizontal and vertical creep as equally effective and does not consider exit gradient.
Lane's theory takes vertical creep as three times as effective as horizontal creep.
Khosla showed that failure by piping begins at the downstream end, so the exit gradient must be kept within safe limits by a deep downstream cut-off.
Safe exit gradients (Khosla): about 1/4 to 1/5 for shingle, 1/5 to 1/6 for coarse sand and 1/6 to 1/7 for fine sand.
Khosla's method of independent variables gives pressures at key points of simple profiles, then applies corrections for mutual interference of piles, floor thickness and floor slope.
Upstream floor of a weir needs only nominal thickness because the water above it balances the uplift; the downstream floor must resist the unbalanced uplift.
A barrage has gates over its full length with a low crest so that the pond level is controlled by gates, whereas a weir raises water mainly by its raised crest (with or without shutters).
Silt excluders are placed in the under-sluice pocket at the head works, while silt ejectors (extractors) are placed in the canal a short distance downstream of the head regulator.

8.7 Gravity dams, spillways & energy dissipation

Base pressure on a gravity dam

p = \frac{\sum V}{B} (1 \pm \frac{6 e}{B})

ΣV = total vertical force; B = base width; e = eccentricity of resultant from base centre

No tension when e ≤ B/6 (resultant within middle third)

Factor of safety against sliding and shear friction factor

F S = \frac{μ \sum V}{\sum H}, S F F = \frac{μ \sum V + B q}{\sum H}

μ = coefficient of friction; ΣH = total horizontal force; q = average shear strength of joint

Wave height – Molitor (fetch F > 32 km)

h_{w} = 0.032 V F

h_w in m; V = wind velocity (km/h); F = fetch (km)

For F < 32 km: h_w = 0.032√(VF) + 0.763 − 0.271 F^1/4

Wave pressure on dam

P_{w} = 2 γ_{w} h_{w}^{2}

P_w in kN/m when γ_w in kN/m³; acts at 3h_w/8 above still water level

Hydrodynamic force due to earthquake (von Kármán)

P_{e} = 0.555 α_{h} γ_{w} H^{2}

α_h = horizontal seismic coefficient; acts at 4H/(3π) above base

Base width of elementary profile (no tension, with uplift)

B = \frac{H}{S _{c} - C}

S_c = specific gravity of dam material; C = uplift intensity coefficient (C = 0 when uplift ignored)

Discharge over an ogee spillway

Q = C L_{e} H_{e}^{3/2}

C = coefficient of discharge; L_e = effective crest length; H_e = total head over crest including velocity head

A gravity dam may fail by overturning, sliding, compression (crushing) of material, or tension at the heel.
Uplift on a gravity dam varies from full reservoir head at the heel to tailwater head at the toe; IS 6512 reduces it at the line of drains to tailwater head plus one-third of the head difference.
Factor of safety against overturning of a gravity dam is commonly required to be at least 1.5.
Wind pressure is usually neglected in gravity dam design, and silt pressure is computed with Rankine's active earth pressure coefficient using submerged weight.
Ogee spillway profile follows the lower nappe of a ventilated sharp-crested weir, so that sub-atmospheric pressure on the crest is avoided at design head.
A side-channel spillway is used in narrow gorges where the crest cannot be placed across the main valley; the flow turns about 90° after crossing the crest.
A shaft (morning glory) spillway is used where there is no room for a surface spillway; a siphon spillway passes flood automatically at a nearly constant reservoir level.
A hydraulic-jump stilling basin is used when tailwater depth matches the sequent depth; a ski-jump (trajectory) bucket is used when tailwater is too low, and a roller (submerged) bucket when tailwater is too high.
Radial (Tainter) gates are widely used as spillway crest gates because the water thrust passes through the trunnion, needing small hoisting force.

8.8 Water logging, drainage & reservoir sedimentation

Leaching requirement

L R = \frac{E C _{i w}}{E C _{d w}}

EC_iw = electrical conductivity of irrigation water; EC_dw = permissible EC of drainage water

Sodium adsorption ratio

S A R = \frac{N a ^{+}}{( C a ^{2 +} + M g ^{2 +} ) /2}

Ion concentrations in meq/L

Spacing of tile drains (ellipse equation)

S = \frac{4 k ( b ^{2} - a ^{2} )}{q}

k = permeability; b = max height of water table above impervious layer midway between drains; a = height of drain above impervious layer; q = drainage coefficient (m/day)

Land becomes waterlogged when the water table rises into the root zone so that soil aeration is cut off and crop yields fall.
Main causes of water logging are seepage from unlined canals, over-irrigation, inadequate natural drainage and obstruction of drainage by roads, railways and canal banks.
Water logging is controlled by lining canals, lowering canal FSL, intercepting drains along canals, surface and sub-surface (tile) drainage and economical use of water.
Saline (white alkali) soils have EC above 4 dS/m, ESP below 15 and pH below 8.5; sodic (black alkali) soils have ESP above 15 and pH above 8.5.
Saline soils are reclaimed by leaching with good-quality water, while sodic soils are reclaimed by adding gypsum followed by leaching.
Drainage coefficient is the depth of water to be removed from the drained area in 24 hours.
Trap efficiency of a reservoir is related to its capacity–inflow ratio (Brune's curve) and decreases as sediment fills the reservoir.
Dead storage is the volume below the minimum drawdown level and is meant mainly to receive sediment.
Live (useful) storage lies between minimum drawdown level and full reservoir level; flood (surcharge) storage lies between full reservoir level and maximum water level.
The required storage capacity of a reservoir for a given demand is found from the mass curve (Rippl) method.

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Values follow the current Indian Standards and IRC / Indian Railways documents named in each item. Always check the latest amendment of a code before using a value in design.