GATE Environmental Engineering Water Resources and Environmental Hydraulics Syllabus

What does the principle of superposition allow you to do with a unit hydrograph?

It allows the direct runoff from a complex storm (multiple rainfall blocks) to be obtained by lagging and scaling the unit hydrograph for each rainfall increment and summing the ordinates.

What is an S-curve (S-hydrograph) and what is it used for?

An S-curve is the hydrograph from a continuous effective rainfall of constant intensity $1/D$ (cm/h) applied indefinitely. It is used to derive a unit hydrograph of one duration from a UH of a different duration.

Define an instantaneous unit hydrograph (IUH).

The IUH is the direct runoff hydrograph from $1\ \mathrm{cm}$ of effective rainfall applied instantaneously (duration $\to 0$) and uniformly over the catchment. It depends only on basin characteristics and is the limit of the UH as duration approaches zero.

What is a stage-discharge (rating) curve and the typical form of its equation?

A rating curve relates river stage (water-surface elevation) to discharge. Its common form is $$Q = C_r (G - a)^{\beta}$$ where $G$=gauge stage, $a$=stage of zero flow, and $C_r$, $\beta$ are constants found from measured data.

What is loop rating (hysteresis) in a stage-discharge relationship?

During a flood wave, discharge for a given stage is greater on the rising limb than on the falling limb (due to unsteady, non-uniform flow and varying water-surface slope), producing a looped rating curve rather than a single-valued one.

How is reservoir capacity determined from inflow data, and what graphical tool is used?

Reservoir storage capacity to meet a demand is found using the mass curve (Rippl) method: a cumulative inflow (mass) curve is plotted; the maximum cumulative departure between demand lines and the mass curve gives the required storage capacity.

On a mass (Rippl) curve, what do the slopes of the inflow curve and the demand line represent?

The slope of the cumulative inflow mass curve at any point equals the inflow rate; the slope of the straight demand line equals the (constant) draft/demand rate. Storage is needed where demand exceeds inflow.

What is the fundamental difference between reservoir routing and channel routing?

Reservoir (level-pool) routing assumes storage is a unique function of outflow/water level alone (horizontal surface). Channel routing storage depends on both inflow and outflow because the water surface is sloped (wedge + prism storage).

Write the lumped continuity (storage) equation used in flood routing.

$$\bar{I} - \bar{Q} = \frac{dS}{dt}$$ In finite-difference form over $\Delta t$: $$\frac{I_1+I_2}{2}\Delta t - \frac{Q_1+Q_2}{2}\Delta t = S_2 - S_1$$

State the Muskingum storage equation and the meaning of $K$ and $x$.

$$S = K\left[x I + (1-x)Q\right]$$ where $K$=storage-time constant (travel time of flood wave), and $x$=weighting factor ($0 \leq x \leq 0.5$; $x=0$ gives reservoir-type storage, $x=0.5$ gives pure translation).

Write the Muskingum routing equation with its coefficients.

$$Q_2 = C_0 I_2 + C_1 I_1 + C_2 Q_1,\quad C_0+C_1+C_2=1$$ with $C_0=\dfrac{-Kx+0.5\Delta t}{K-Kx+0.5\Delta t}$, $C_1=\dfrac{Kx+0.5\Delta t}{K-Kx+0.5\Delta t}$, $C_2=\dfrac{K-Kx-0.5\Delta t}{K-Kx+0.5\Delta t}$.

What is the rational method for estimating peak surface runoff, and what does each term mean?

$$Q_p = \frac{C\,i\,A}{360}\ (\mathrm{m^3/s},\ A\ \text{in ha})\quad\text{or}\quad Q_p = C\,i\,A$$ where $C$=runoff coefficient, $i$=rainfall intensity for duration equal to time of concentration, and $A$=catchment area. It assumes peak flow occurs when the whole catchment contributes.

Define the time of concentration $t_c$.

The time of concentration is the time required for runoff to travel from the hydraulically most distant point of the catchment to the outlet. At $t=t_c$ the entire catchment contributes simultaneously, producing the peak in the rational method.

What is the SCS Curve Number method and its runoff equation?

The SCS-CN method estimates direct runoff depth $Q$ from rainfall $P$ using the curve number $CN$ (function of soil group, land use, antecedent moisture): $$Q = \frac{(P - 0.2S)^2}{P + 0.8S},\quad S = \frac{25400}{CN} - 254\ (\mathrm{mm})$$ valid for $P > 0.2S$.

List key strategies of integrated surface water management.

Watershed/catchment management, construction of dams and reservoirs, flood control and floodplain zoning, inter-basin transfers, water quality protection, demand management, environmental flow maintenance, and stakeholder-based integrated water resources management (IWRM).

What is rainwater harvesting and what are its two broad categories?

Rainwater harvesting is the collection and storage of rainwater for use or recharge. Two categories: (1) storage for direct use (rooftop collection into tanks/cisterns), and (2) groundwater recharge (recharge pits, trenches, percolation ponds, recharge wells).

How do you estimate the volume of rooftop rainwater that can be harvested?

$$V = C \times A \times R$$ where $V$=harvestable volume, $C$=runoff coefficient (roof, $\approx 0.8\text{–}0.9$), $A$=catchment (roof plan) area, and $R$=rainfall depth over the period.

Which geologic formations act as aquifers, aquitards, aquicludes, and aquifuges?

Aquifer: saturated permeable formation yielding usable water (sand, gravel, fractured rock). Aquitard: low permeability, transmits water slowly (silt, clay). Aquiclude: saturated but essentially impermeable (clay). Aquifuge: neither porous nor permeable (solid granite).

Differentiate the vadose (unsaturated) zone from the saturated (phreatic) zone.

The vadose zone lies above the water table where pores contain both air and water (pressure $<$ atmospheric). The saturated zone lies below the water table where all pores are filled with water (pressure $>$ atmospheric); the boundary is the water table where pressure equals atmospheric.

Compare confined and unconfined aquifers.

An unconfined (water-table) aquifer is bounded above by the water table and recharged directly from the surface; its upper surface fluctuates freely. A confined (artesian) aquifer is bounded above and below by impermeable layers, holds water under pressure greater than atmospheric, and its potentiometric surface lies above the top of the aquifer.

Define porosity and specific yield, and state how they differ.

Porosity $n = \dfrac{V_v}{V}$ is the ratio of void volume to total volume. Specific yield $S_y$ is the volume of water released by gravity drainage per unit aquifer volume. They differ by the specific retention $S_r$: $$n = S_y + S_r$$

Define hydraulic conductivity (permeability) $K$ and its dependence.

Hydraulic conductivity $K$ is the volume of water flowing per unit time through a unit cross-sectional area under a unit hydraulic gradient. It depends on both the medium (intrinsic permeability $k$) and the fluid: $$K = \frac{k \rho g}{\mu}$$

Define transmissivity $T$ and give its formula.

Transmissivity is the rate of water transmitted through the full saturated thickness of an aquifer per unit width under a unit hydraulic gradient: $$T = K b$$ where $K$=hydraulic conductivity and $b$=saturated aquifer thickness.

Define the storage coefficient (storativity) $S$ and give typical ranges for confined vs unconfined aquifers.

Storativity is the volume of water released from storage per unit surface area per unit decline in head: $S = S_s b$. Confined aquifers: $S \approx 5\times10^{-5}$ to $5\times10^{-3}$ (elastic storage). Unconfined aquifers: $S \approx S_y \approx 0.05$ to $0.30$.

State Darcy's law and define each term.

$$Q = -KA\frac{dh}{dl}\quad\text{or}\quad v = -K\,i$$ where $Q$=discharge, $K$=hydraulic conductivity, $A$=cross-sectional area, $\dfrac{dh}{dl}=i$=hydraulic gradient, and $v$=Darcy (apparent) velocity. The negative sign indicates flow toward decreasing head.

Distinguish Darcy velocity from actual (seepage) velocity.

Darcy velocity $v = Q/A$ assumes flow through the full cross-section. Actual seepage velocity accounts for flow only through pores: $$v_s = \frac{v}{n}$$ where $n$=effective porosity, so $v_s > v$.

State the validity limit of Darcy's law.

Darcy's law is valid for laminar flow, generally for Reynolds number $Re = \dfrac{\rho v d}{\mu} \leq 1$ (some sources allow up to $10$). At higher $Re$ (coarse gravel, near well screens) flow becomes turbulent and Darcy's law breaks down.

Write the Thiem (Dupuit) steady-state discharge equation for a fully penetrating well in a confined aquifer.

$$Q = \frac{2\pi T (h_2 - h_1)}{\ln(r_2/r_1)} = \frac{2\pi K b (h_2 - h_1)}{\ln(r_2/r_1)}$$ where $h_1,h_2$ are heads at radial distances $r_1,r_2$ from the well, $K$=conductivity, and $b$=aquifer thickness.

Write the steady-state Dupuit equation for a fully penetrating well in an unconfined aquifer.

$$Q = \frac{\pi K (h_2^{2} - h_1^{2})}{\ln(r_2/r_1)}$$ where $h_1,h_2$ are saturated thicknesses (heads above the impervious base) at radii $r_1,r_2$. It uses the Dupuit assumptions of nearly horizontal flow and gradient equal to the slope of the water table.

Define the cone of depression and the radius of influence in steady well hydraulics.

The cone of depression is the inverted-cone-shaped lowering of the water table/potentiometric surface around a pumping well. The radius of influence $R$ is the radial distance from the well to where drawdown becomes negligible (effectively zero), marking the edge of the cone.