Sustainability & Environment (Agri-Food Systems) · Foundations · also Water Resources and Environmental Hydraulics, Environmental Hydrology, Hydrology & Hydraulics
The water cycle, surface and groundwater resources and the hydraulics that GATE ES Section 4 examines. Agricultural engineering already teaches watershed hydrology, groundwater and canal hydraulics to design level (subjects watershed-hydrology and groundwater-wells-pumps); this subject covers what an environmental scientist needs and the environmental framing: water balance, threats to water resources and channel and pipe flow basics.
In the exams: GATE ES Section 4 (a large numerical section); ICAR AIEEA-PG Water Science & Technology group; ICAR AICE subject 71 Unit 1 (watersheds)
Hydrology is the science of water on and under the land: how it falls as precipitation, soaks in or runs off, evaporates, is stored in soils, lakes and aquifers, and flows through rivers to the sea. Water resources science applies this knowledge to the supply, use and protection of water for people, agriculture, industry and ecosystems. Environmental scientists need it to estimate how much water a catchment yields, how quickly a flood rises, how far a pollutant travels in groundwater, how much a reservoir can supply in a drought, and how pipes and channels carry water and effluent.
This subject covers what an environmental scientist needs and what GATE Environmental Science and Engineering examines in its water resources and environmental hydraulics section: global water resources and threats; the hydrological cycle and water balance; precipitation, infiltration, evapotranspiration and runoff; hydrographs and the unit hydrograph; stage-discharge relationships and reservoir capacity; reservoir and channel routing; aquifer properties; Darcy's law and well hydraulics; fluid properties and hydrostatics; the continuity, energy and momentum equations with pipe flow; open-channel flow; and flow measurement. Each chapter gives the standard equations with worked numbers, because this section of the examination is largely numerical.
Design-level engineering -- watershed hydrology for structures, well design and pump selection, canal design -- is taught in the Agricultural Engineering track (Watershed Hydrology; Groundwater, Wells and Pumps; Water Resources and Flood Management). Water use, scarcity, footprints and governance are in Agricultural Water Conservation, Water Footprint and Governance, and pollutant transport and water quality in Water Quality and Agricultural Non-point Source Pollution. This manual provides the shared physical foundation.
Use consistent SI units throughout: discharge in m3/s, rainfall intensity in mm/h, areas in km2 or m2, and g = 9.81 m/s2. Many examination errors are unit errors -- millimetres of rain over square kilometres, hours versus seconds -- so every worked example states its conversions. Coefficients such as runoff coefficients, Manning's n and discharge coefficients depend on site conditions and are given in problems; values quoted here are typical illustrations.
Almost all of Earth's water -- about 97.5% -- is saline, in the oceans and saline lakes and aquifers. Of the roughly 2.5% that is fresh, commonly cited estimates put about 69% in glaciers, ice caps and permanent snow, about 30% in groundwater, and only around 1% in surface and other fresh water: lakes, soil moisture, the atmosphere, wetlands, rivers and living things. Rivers hold a tiny fraction of the world's water at any moment but carry a large flow each year because their water is renewed quickly. The residence time of water varies enormously: days in the atmosphere, weeks in rivers, years to centuries in lakes, and centuries to many thousands of years in deep groundwater and ice sheets.
The resource that matters for human use is the renewable flow rather than the stock: the precipitation that becomes runoff and recharge each year. This is distributed very unevenly -- abundant in the humid tropics and high latitudes, scarce in arid zones -- and varies seasonally (monsoon regions receive most of their rain in a few months) and between years. Storage, natural (snowpack, aquifers, lakes) or built (reservoirs), bridges these gaps. India illustrates the pattern: most of its rain falls in the four monsoon months, its rivers carry very unequal flows between the Brahmaputra-Ganga system and the peninsular rivers, and its aquifers and reservoirs must carry water through eight dry months. Quantifying these flows and stores is the hydrologist's first task, and the rest of this manual provides the tools.
Water resources face threats of quantity and quality. Over-abstraction depletes aquifers and dries rivers; dams and diversions alter flow regimes and fragment rivers; land-use change alters infiltration, runoff and erosion; and climate change shifts precipitation, increases evaporation, reduces glaciers and snowpack that feed rivers such as the Indus and Ganga, and intensifies floods and droughts. Pollution from sewage, industry and agriculture -- nutrients, pathogens, salts, metals, pesticides and emerging contaminants -- reduces usable supply. Groundwater faces geogenic contamination (arsenic, fluoride), seawater intrusion and slow-to-reverse pollution. Wetlands and floodplains, which store floods and purify water, have been widely lost.
These threats are covered in depth in Water Quality and Agricultural Non-point Source Pollution and Agricultural Water Conservation, Water Footprint and Governance. For the hydrologist, they translate into changed inputs to the water balance and changed parameters in the equations of the following chapters: lower infiltration and higher runoff coefficients after urbanisation, falling water tables, reduced dry-season baseflow, and reservoirs losing capacity to sediment.
The hydrological cycle is driven by solar energy and gravity. Water evaporates from oceans, lakes and soil and is transpired by plants (together, evapotranspiration), condenses in the atmosphere and falls as precipitation. On land, precipitation may be intercepted by vegetation and evaporate, infiltrate into the soil, or run off over the surface. Infiltrated water may be stored as soil moisture and used by plants, flow laterally as interflow to streams, or percolate to groundwater, which discharges to springs, rivers, lakes and the sea as baseflow. Rivers return water to the oceans, completing the cycle. Globally, more water evaporates from the oceans than falls on them; the excess is carried over land by the atmosphere and returns as river and groundwater discharge.
A catchment (watershed, drainage basin) is the area draining to a point on a river, bounded by a topographic divide. It is the natural unit for hydrological analysis because, apart from groundwater flows that may cross divides and inter-basin transfers, all water leaving it passes the outlet.
Conservation of mass gives the water balance for any control volume over a period: inputs - outputs = change in storage. For a catchment: P = ET + Q + G + dS, where P is precipitation, ET evapotranspiration, Q surface runoff at the outlet, G net groundwater outflow across the boundary, and dS the change in storage (soil moisture, groundwater, lakes, snow). Over long periods storage changes average out, so mean annual runoff is approximately precipitation minus evapotranspiration.
Depths and volumes convert easily: 1 mm of water over 1 km2 is 1,000 m3, and 1 mm over 1 ha is 10 m3. For example, a 500 km2 catchment receiving 900 mm of rain a year with 600 mm of evapotranspiration and negligible storage change and groundwater export yields about 300 mm of runoff, or 300 x 500 x 1,000 = 150 million m3 per year, a mean flow of 150,000,000 / 31,536,000 = 4.8 m3/s. The runoff ratio (runoff divided by precipitation) here is one-third. Water balances are used to estimate unmeasured components, check data consistency, assess the yield available for abstraction and evaluate the effects of land-use or climate change. Reservoir and lake balances add inflow, outflow, direct rainfall and open-water evaporation.
Precipitation is measured by rain gauges (standard non-recording and recording types) and estimated by weather radar and satellites. Average rainfall over a catchment can be estimated by the arithmetic mean of gauges (adequate for flat areas with evenly spread gauges), the Thiessen polygon method (each gauge weighted by the area closer to it than to any other gauge), or the isohyetal method (contours of equal rainfall, weighting by the area between isohyets). Missing records are commonly filled by the normal ratio method, P_x = (N_x / n) x sum of (P_i / N_i), where N are normal annual rainfalls; the consistency of a gauge record is checked with a double mass curve.
Infiltration is the entry of water into the soil; its rate is highest when the soil is dry and declines towards a steady value as the soil wets. Horton's equation describes this: f = fc + (f0 - fc) e^(-kt), where f0 is the initial and fc the final infiltration capacity and k a decay constant; cumulative infiltration is F = fc t + [(f0 - fc) / k](1 - e^(-kt)). The phi-index is the constant rate of loss that, subtracted from a storm hyetograph, leaves the observed runoff volume. Green-Ampt and Philip equations are physically based alternatives.
Evapotranspiration combines evaporation from soil and water surfaces with transpiration by plants. Reference evapotranspiration (ET0) for a standard grass surface is estimated by the FAO-56 Penman-Monteith equation from radiation, temperature, humidity and wind, or approximately from pan evaporation (ET0 = Kp x Epan, with a pan coefficient Kp typically around 0.6-0.85). Crop evapotranspiration is ETc = Kc x ET0, where the crop coefficient Kc varies with crop and growth stage.
Runoff from a storm depends on rainfall intensity and duration, soil infiltration, antecedent moisture, land use and slope. The rational method estimates peak runoff from small catchments: Q = C I A / 3.6, with Q in m3/s, C the runoff coefficient, I the rainfall intensity in mm/h for a duration equal to the time of concentration, and A the area in km2 (equivalently Q = 0.278 C I A). The SCS (now NRCS) curve number method estimates runoff depth from storm rainfall: S = 25,400 / CN - 254 (mm), initial abstraction Ia = 0.2 S, and Q = (P - 0.2 S)^2 / (P + 0.8 S) for P > 0.2 S. The curve number (0-100) reflects soil hydrologic group, land use and antecedent moisture. For example, with CN = 80, S = 63.5 mm and Ia = 12.7 mm; a 75 mm storm gives runoff of (75 - 12.7)^2 / (75 + 50.8) = 30.9 mm.
A hydrograph plots discharge against time at a point on a river. A storm hydrograph has a rising limb, a peak and a falling (recession) limb. Total flow is separated into baseflow (groundwater contribution, slowly varying) and direct runoff (quickflow from the storm). Simple separation methods draw a straight line from the start of the rise to a point on the recession, or extend the pre-storm recession; the area under the direct runoff hydrograph equals the volume of effective (excess) rainfall over the catchment. Hydrograph shape depends on catchment area, shape, slope, drainage density, land use and storm characteristics: small, steep, urban catchments produce sharp, early peaks; large, flat, forested ones produce broad, delayed peaks. Lag time is the time from the centroid of effective rainfall to the peak; the time of concentration is the time for water to travel from the hydraulically most distant point to the outlet. Urbanisation shortens lag times and raises peaks because impervious surfaces and drains speed runoff; wetlands, floodplains and forests lengthen them. Comparing pre- and post-development hydrographs is the basis of stormwater management requirements.
The unit hydrograph (UH), introduced by Sherman in 1932, is the direct runoff hydrograph produced by one unit of effective rainfall (1 cm, or 1 mm in some texts) falling uniformly over the catchment at a constant rate during a specified duration D (the D-hour unit hydrograph). It assumes linearity (ordinates proportional to effective rainfall depth) and time invariance (the same response to the same input at any time). Therefore the direct runoff from a storm of several blocks of effective rainfall is obtained by multiplying the UH ordinates by each block's depth, lagging each by D hours, and summing (convolution). Baseflow is added to give total flow.
The volume under a UH must equal 1 cm over the catchment: for a 100 km2 catchment, 0.01 m x 100,000,000 m2 = 1,000,000 m3, so the sum of ordinates (m3/s) times the time interval (s) should equal this. A UH of one duration can be converted to another using the S-curve method (the response to continuous effective rainfall at 1 cm per D hours) or, for integer multiples, by superposition of lagged UHs. Synthetic unit hydrographs (Snyder, SCS dimensionless) are derived from catchment characteristics where no flow records exist. Example: if a 3-hour UH has a peak of 50 m3/s, a 3-hour storm with 2.5 cm of effective rainfall produces a direct runoff peak of 125 m3/s.
Discharge is rarely measured continuously; instead, stage (water level) is recorded continuously and converted to discharge with a stage-discharge relationship (rating curve) developed from periodic discharge measurements. The velocity-area method divides the cross-section into verticals; velocity is measured with a current meter (often at 0.6 of the depth from the surface, or averaged at 0.2 and 0.8 of depth) or an acoustic Doppler instrument, and discharge is the sum of area times mean velocity for each segment, Q = sum (a_i v_i). Rating curves are often fitted as Q = C_r (h - a)^n, where h is stage, a the stage of zero flow and C_r and n constants. They must be checked regularly because channels change by scour, deposition and vegetation; hysteresis (higher discharge on the rising than on the falling limb at the same stage) occurs in unsteady flood flows. Dilution gauging adds a tracer at a known rate and measures its downstream concentration, useful in turbulent mountain streams.
Reservoirs store water in wet periods for release in dry ones. Storage zones include dead storage (below the lowest outlet, partly for sediment), live or useful storage (between minimum and full supply levels), and flood-control storage above full supply level. The capacity needed to meet a given demand is found by the mass curve (Rippl) method -- plotting cumulative inflow against time and drawing demand lines tangent to the curve's peaks; the largest vertical gap between the demand line and the following trough is the storage required -- or by the equivalent sequent peak algorithm, which accumulates deficits of demand over inflow. Conversely, the safe yield of a given reservoir is the demand that can be met through the critical dry period of record.
Reservoirs lose capacity to sediment. Trap efficiency -- the fraction of incoming sediment retained -- depends on the capacity-inflow ratio (Brune's curve): large reservoirs relative to annual inflow trap nearly all sediment. A reservoir's useful life can be estimated by dividing the capacity (or the share that can be lost before function fails) by the annual sediment volume retained. Evaporation from reservoir surfaces is a significant loss in hot climates.
Flood routing predicts how a flood wave changes as it passes through a reservoir or down a channel. Storage attenuates the peak and delays it. For a reservoir with a horizontal water surface, the continuity equation over a time step dt is (I1 + I2)/2 x dt - (O1 + O2)/2 x dt = S2 - S1, where I is inflow, O outflow and S storage at the start (1) and end (2) of the step. Because outflow from an uncontrolled spillway depends on water level, and storage also depends on level, the equation is solved step by step using the storage-outflow relationship; the modified Puls (storage indication) method rearranges it as (2S2/dt + O2) = (I1 + I2) + (2S1/dt - O1). In an uncontrolled reservoir, the outflow peak occurs where the falling limb of the inflow hydrograph crosses the outflow hydrograph, because storage is at its maximum when inflow equals outflow.
In a river reach the water surface is not level: storage depends on both inflow and outflow. The Muskingum method models reach storage as S = K [x I + (1 - x) O], where K is a storage constant approximately equal to the travel time through the reach and x a weighting factor between 0 and 0.5 (usually 0.1-0.3 for natural rivers; x = 0 gives reservoir-like routing). Combining this with continuity gives O2 = C0 I2 + C1 I1 + C2 O1, with C0 = (0.5 dt - K x) / D, C1 = (0.5 dt + K x) / D and C2 = (K - K x - 0.5 dt) / D, where D = K - K x + 0.5 dt; the coefficients sum to 1. For example, with K = 12 h, x = 0.2 and dt = 6 h, D = 12 - 2.4 + 3 = 12.6, C0 = (3 - 2.4)/12.6 = 0.048, C1 = (3 + 2.4)/12.6 = 0.429 and C2 = (12 - 2.4 - 3)/12.6 = 0.524. The time step should lie between about 2Kx and K to avoid negative coefficients.
Hydraulic routing solves the full Saint-Venant equations of continuity and momentum and is used where backwater, tides or steep waves matter. Routing underpins flood forecasting, design of spillways and detention basins, and assessment of how wetlands and floodplains attenuate floods.
An aquifer is a saturated geological formation that stores and transmits enough water to supply wells and springs: sands, gravels, fractured or karstic limestone, sandstone and fractured crystalline rock. An aquitard transmits water slowly (silty clay); an aquiclude is effectively impermeable (dense clay). An unconfined (water-table) aquifer has the water table as its upper boundary, at atmospheric pressure, and is recharged directly from above. A confined aquifer is bounded above and below by low-permeability layers and is under pressure; water in a well rises above the top of the aquifer to the piezometric (potentiometric) surface, and if that surface is above ground the well flows (artesian). A perched aquifer is a local saturated zone above a lens of low-permeability material in the unsaturated zone. Leaky aquifers receive water through semi-confining layers. In India, alluvial aquifers of the Indo-Gangetic plains are thick, productive and largely unconfined near the surface with deeper semi-confined layers, while hard-rock aquifers of the peninsula store water mainly in weathered and fractured zones with low storage, so wells there fail quickly when over-pumped. Knowing the aquifer type is the first step in interpreting water-level data and pumping tests.
Porosity n is the ratio of void volume to total volume. Not all of it yields water: specific yield Sy is the volume of water that drains by gravity per unit volume of aquifer, and specific retention Sr the volume retained against gravity, so n = Sy + Sr. Clays have high porosity but low specific yield; sands and gravels release much of their pore water. Storativity (storage coefficient) S is the volume of water released per unit surface area of aquifer per unit decline in head. For unconfined aquifers S is approximately Sy (typically 0.01-0.3); for confined aquifers it is far smaller (typically 10^-5 to 10^-3), because water is released only by expansion of water and compression of the aquifer skeleton -- which is why pumping a confined aquifer lowers heads over large areas quickly.
Hydraulic conductivity K (m/day or m/s) measures the ease with which a porous medium transmits water; it depends on both the medium (grain size, sorting) and the fluid (viscosity, density). Transmissivity T = K b, where b is the saturated thickness, is the rate at which water is transmitted through a unit width of the full aquifer thickness under unit hydraulic gradient (m2/day). Example: a 25 m thick confined sand aquifer with K = 20 m/day has T = 500 m2/day. Volume of water released from storage = S x area x head decline: a confined aquifer with S = 2 x 10^-4 over 10 km2 releases 2 x 10^-4 x 10,000,000 m2 x 5 m = 10,000 m3 for a 5 m decline, whereas an unconfined aquifer with Sy = 0.15 would release 7.5 million m3.
Henry Darcy (1856) found that the discharge through a column of saturated sand is proportional to the cross-sectional area and the head difference and inversely proportional to the length: Q = K A (h1 - h2) / L = K i A, where i = dh/dl is the hydraulic gradient. The Darcy flux (specific discharge) q = Q / A = K i is a volumetric flux per unit total area, not the actual velocity of water; because water flows only through the connected pores, the average linear (seepage) velocity is v = K i / n_e, where n_e is the effective porosity. This seepage velocity determines how fast a solute or contaminant plume advances. Example: with K = 10 m/day, i = 0.002 and n_e = 0.25, q = 0.02 m/day and v = 0.08 m/day, so water travels about 29 m in a year.
Darcy's law holds for laminar flow, typical of groundwater; it breaks down at high velocities (near wells in coarse gravels, in karst conduits), where a Reynolds number based on grain diameter exceeds roughly 1-10. Combined with continuity, it yields the groundwater flow equations used in models.
When a well is pumped, heads fall around it, forming a cone of depression; drawdown s is the fall in head at a point. Under steady-state conditions (the cone has stabilised), the Thiem equation for a fully penetrating well in a confined aquifer is Q = 2 pi K b (h2 - h1) / ln(r2 / r1) = 2 pi T (s1 - s2) / ln(r2 / r1), where h1, h2 (or drawdowns s1, s2) are heads at radial distances r1 and r2 from the well. For an unconfined aquifer, the Dupuit-Thiem equation is Q = pi K (h2^2 - h1^2) / ln(r2 / r1), with h the saturated thickness above the base. The radius of influence R is the distance at which drawdown becomes negligible.
Example: two observation wells at 10 m and 100 m from a well pumping a confined aquifer show drawdowns of 3.0 m and 1.0 m; with Q = 0.02 m3/s (1,728 m3/day), T = Q ln(r2/r1) / [2 pi (s1 - s2)] = 1,728 x 2.303 / (2 pi x 2.0) = 317 m2/day. Unsteady (transient) conditions are analysed with the Theis equation and its Cooper-Jacob approximation, taught in the Agricultural Engineering track. Wells pumped close together interfere: drawdowns superpose, a key consideration in well-field design and in regulating spacing of farm wells.
Density rho of water is about 1,000 kg/m3 (maximum near 4 degrees C; about 998 kg/m3 at 20 degrees C); specific weight gamma = rho g, about 9.81 kN/m3; specific gravity is density relative to water. Viscosity resists shear: dynamic viscosity mu of water is about 1.0 x 10^-3 Pa s at 20 degrees C and decreases with temperature; kinematic viscosity nu = mu / rho, about 1.0 x 10^-6 m2/s at 20 degrees C. Newtonian fluids have shear stress proportional to velocity gradient (tau = mu du/dy). Surface tension (about 0.073 N/m for water at 20 degrees C) produces capillary rise h = 4 sigma cos(theta) / (rho g d) in a tube of diameter d, the basis of water retention in soils. Water is nearly incompressible (bulk modulus about 2.2 GPa). Vapour pressure matters for cavitation in pumps. Viscosity matters for the environmental scientist because it controls the settling velocity of particles in water (Stokes' law, used in sedimentation tanks and particle-size analysis) and the hydraulic conductivity of aquifers, which falls in cold water.
In a fluid at rest, pressure increases linearly with depth: p = p0 + rho g h. Gauge pressure is measured relative to atmospheric pressure (about 101.3 kPa at sea level). Pressure head h = p / (rho g); a manometer measures pressure differences from column heights. The total hydrostatic force on a plane submerged surface is F = rho g h_c A, where h_c is the depth of the centroid; it acts at the centre of pressure, at depth h_p = h_c + I_G / (A h_c) for a vertical surface, where I_G is the second moment of area about the centroidal axis -- always below the centroid. Example: a vertical rectangular gate 2 m wide and 3 m high with its top at the water surface has h_c = 1.5 m, A = 6 m2, so F = 1,000 x 9.81 x 1.5 x 6 = 88.3 kN, acting at h_p = 1.5 + (2 x 3^3 / 12) / (6 x 1.5) = 2.0 m below the surface (two-thirds of the depth).
Archimedes' principle states that a body immersed in a fluid experiences an upward buoyant force equal to the weight of fluid displaced, acting through the centre of buoyancy. A floating body is stable if its metacentre lies above its centre of gravity; the metacentric height GM = BM - BG, with BM = I / V (I the second moment of the waterplane area, V the displaced volume). These principles govern the stability of floating structures, pontoons, and the design of gates and dams.
Continuity (conservation of mass) for steady incompressible flow: Q = A1 V1 = A2 V2. Energy (Bernoulli's equation along a streamline for steady, incompressible, frictionless flow): p/(rho g) + V^2/(2g) + z = constant, the sum of pressure head, velocity head and elevation head; for real flow between two sections, H1 = H2 + head losses (+ pump head added or turbine head removed). Momentum: the net force on a control volume equals the rate of change of momentum flux, F = rho Q (V2 - V1) in each direction, used for forces on bends, nozzles, vanes and hydraulic jumps.
Example: water flows at 0.05 m3/s through a pipe narrowing from 0.2 m to 0.1 m diameter. V1 = 0.05 / (pi x 0.1^2) = 1.59 m/s and V2 = 0.05 / (pi x 0.05^2) = 6.37 m/s; ignoring losses on a horizontal pipe, the pressure drop is rho (V2^2 - V1^2) / 2 = 1,000 x (40.5 - 2.5) / 2 = 19.0 kPa.
Flow regime is given by the Reynolds number Re = V D / nu: laminar below about 2,000, turbulent above about 4,000, transitional between. Frictional head loss is given by the Darcy-Weisbach equation hf = f L V^2 / (2 g D); for laminar flow f = 64 / Re, and for turbulent flow f depends on Re and relative roughness (Moody diagram, Colebrook-White equation). The empirical Hazen-Williams formula is widely used for water supply. Minor losses at entries, bends, valves and expansions are expressed as K V^2 / (2g). Example: 500 m of 0.2 m pipe carrying water at 1.5 m/s with f = 0.02 loses hf = 0.02 x 500 x 1.5^2 / (2 x 9.81 x 0.2) = 5.73 m of head.
Pipes in series carry the same discharge and their head losses add; pipes in parallel share the same head loss and their discharges add. Networks of loops are solved iteratively, classically by the Hardy Cross method, which corrects assumed loop flows until head losses around each loop sum to zero, and today by computer models. Pumping power is P = rho g Q H / efficiency. Water hammer -- pressure surges from sudden valve closure -- must be considered in design.
Open-channel flow has a free surface at atmospheric pressure, as in rivers, canals, drains and partly full sewers. In uniform flow, depth and velocity are constant along the channel and the energy slope equals the bed slope. Manning's equation gives the velocity: V = (1/n) R^(2/3) S^(1/2) in SI units, where n is Manning's roughness coefficient (about 0.012-0.015 for concrete, 0.025-0.035 for earth channels, higher for vegetated rivers), R = A / P the hydraulic radius (area divided by wetted perimeter) and S the bed slope; the Chezy equation V = C (R S)^(1/2) is equivalent with C = R^(1/6) / n. Example: a rectangular concrete channel 3 m wide flowing 1 m deep (A = 3 m2, P = 5 m, R = 0.6 m), with n = 0.015 and S = 0.001, has V = (1/0.015) x 0.6^(2/3) x 0.001^(1/2) = 1.50 m/s and Q = 4.5 m3/s.
Specific energy is the energy relative to the channel bed: E = y + V^2 / (2g), or for a rectangular channel with discharge per unit width q, E = y + q^2 / (2 g y^2). For a given q, E has a minimum at critical depth y_c = (q^2 / g)^(1/3), where E_min = 1.5 y_c. Depths greater than y_c are subcritical (tranquil, Froude number Fr = V / (g y)^(1/2) < 1); smaller depths are supercritical (rapid, Fr > 1). Critical flow occurs at controls such as weir crests and free overfalls, which is the basis of many flow-measuring structures. Example: q = 2 m2/s gives y_c = (4 / 9.81)^(1/3) = 0.74 m and E_min = 1.11 m.
A hydraulic jump is an abrupt transition from supercritical to subcritical flow, with intense turbulence that dissipates energy -- used below spillways and gates to protect channels from scour, and for mixing chemicals in treatment plants. For a rectangular channel, the sequent (conjugate) depths are related by y2 / y1 = 0.5 [(1 + 8 Fr1^2)^(1/2) - 1], derived from the momentum equation, and the energy loss is dE = (y2 - y1)^3 / (4 y1 y2). Example: y1 = 0.3 m and V1 = 6 m/s give Fr1 = 6 / (9.81 x 0.3)^(1/2) = 3.50, y2 = 0.3 x 0.5 x [(1 + 8 x 12.23)^(1/2) - 1] = 1.34 m, and an energy loss of about 0.70 m.
Pipe flow meters exploit Bernoulli's equation. A venturi meter has a converging section, a throat and a gradual diverging section; the pressure difference between inlet and throat, h (as head of flowing fluid), gives Q = C_d A1 A2 (2 g h)^(1/2) / (A1^2 - A2^2)^(1/2), with a discharge coefficient C_d typically about 0.95-0.99 and low permanent head loss. An orifice meter (a plate with a sharp-edged hole) uses the same equation with a much lower C_d (around 0.6) and higher head loss but is cheap. A Pitot tube measures point velocity from the stagnation pressure: V = C (2 g dh)^(1/2). Other meters include electromagnetic, ultrasonic, turbine and rotameter types; bucket-and-stopwatch volumetric measurement suits small flows.
Example: a venturi with inlet diameter 0.3 m and throat 0.15 m (A1 = 0.0707 m2, A2 = 0.0177 m2) shows a differential head of 0.5 m of water; with C_d = 0.98, Q = 0.98 x 0.0707 x 0.0177 x (2 x 9.81 x 0.5)^(1/2) / (0.0707^2 - 0.0177^2)^(1/2) = 0.0560 m3/s.
Weirs and flumes create a control at which discharge depends uniquely on upstream head. For a sharp-crested rectangular weir, Q = (2/3) C_d L (2 g)^(1/2) H^(3/2), where L is the crest length and H the head over the crest measured upstream (with C_d about 0.6, this gives the familiar Q of about 1.84 L H^(3/2) in SI units, modified for end contractions by Francis's formula). A triangular (V-notch) weir is more sensitive at low flows: Q = (8/15) C_d (2 g)^(1/2) tan(theta/2) H^(5/2); for a 90-degree notch with C_d about 0.58-0.6 this is roughly 1.38-1.42 H^(5/2). Broad-crested weirs pass flow at critical depth (Q = 1.705 L H^(3/2) for C_d = 1 in SI units). Parshall flumes and other critical-depth flumes measure flow with little head loss and pass sediment, suiting irrigation canals.
In rivers, the velocity-area method with current meters or acoustic Doppler current profilers, combined with rating curves (chapter 5), is standard; floats give rough surface velocities (mean velocity about 0.85 of surface velocity in many channels). Dilution gauging and slope-area methods (using Manning's equation with surveyed flood marks) estimate flows where direct measurement is impractical. Example: a 90-degree V-notch with H = 0.2 m and C_d = 0.6 passes Q = (8/15) x 0.6 x (19.62)^(1/2) x 1 x 0.2^(5/2) = 0.0254 m3/s.
Treat this subject as a numerical toolkit. Memorise and practise the standard equations -- water balance, Horton, rational and SCS-CN, unit hydrograph convolution, level-pool and Muskingum routing, Darcy and seepage velocity, Thiem and Dupuit-Thiem, hydrostatic force and centre of pressure, Bernoulli and momentum, Darcy-Weisbach, Manning, critical depth, hydraulic jump, venturi and weir formulas -- and always write units. Understand the assumptions behind each (linearity of the UH, laminar flow for Darcy, steady state for Thiem) so you can spot when they fail.
As an environmental consultant, water-resources officer or catchment scientist you estimate flows for ungauged streams, size culverts and detention ponds, check effluent dilution at low flow, interpret pumping tests, estimate travel times of contaminants in groundwater, check flow measurements at treatment plants and irrigation canals, and build simple water balances for EIAs and water-use permits. You decide which method is adequate for a question and document assumptions and uncertainty, and you must recognise when a problem needs a specialist hydrologist or engineer.
As a senior hydrologist or water-resources manager you oversee monitoring networks, flood forecasting, reservoir operation and groundwater management, review consultants' models, and translate hydrological uncertainty into decisions on dam safety, flood defences, allocation and drought plans. You ensure rating curves and data are maintained, and that climate change and land-use change are reflected in design values. You also communicate flood and drought risk to planners, emergency managers and the public, and set priorities for data collection, staffing and model development.
As a researcher you work on catchment processes, non-stationarity of hydrological extremes under climate change, remote sensing and data assimilation, groundwater-surface water interaction, distributed and physically based models, and the uncertainty of predictions in ungauged basins. You develop methods that practitioners use, maintain long-term observatories and train hydrologists and engineers. Your work feeds national design standards, IPCC assessments of hydrological change and the tools used for water allocation, drought management and flood warning in many countries and river basins around the world.
The manual gives the standard equations and worked examples but not design tables (Manning's n tables, curve-number tables, discharge coefficients), which vary with site conditions and should be taken from handbooks. Transient well hydraulics (Theis, Cooper-Jacob), unsaturated flow (Richards' equation), sediment transport, hydrological frequency analysis and design floods, Hardy Cross iterations and Saint-Venant hydraulic routing are introduced only briefly; design-level coverage is in the Agricultural Engineering track. Global water distribution figures are commonly cited estimates that differ slightly between sources.
Original practice questions written to the syllabus and pattern of the exams named in each question's examTags. They are not past papers and do not reproduce any board's questions. Every numerical answer was computed by script; Coefficients and site data are stated in each stem; every numerical answer was computed by script with g = 9.81 m/s2.
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