(a) Explain an aquifer. Deduce the expression for discharge (Q) through a well in an unconfined aquifer taking usual symbols. If…

(a) An aquifer is a saturated geological formation that can both store water and yield it to wells and springs in usable quantity, because it is porous and permeable. Examples are unconsolidated sand and gravel, sandstone, and fractured or cavernous limestone.

The main types of aquifer are:

• Unconfined (water-table) aquifer: its upper surface is the water table, at atmospheric pressure, and it is recharged directly from above.
• Confined (artesian) aquifer: it lies between two impervious layers, and water is under pressure above atmospheric; the piezometric surface lies above the top of the aquifer.
• Perched aquifer: a local lens of saturated material resting on a small impervious layer above the main water table.
• Leaky aquifer: it is bounded by an aquitard through which some water leaks.

Related formations are the aquiclude (porous but not permeable, e.g. clay), the aquitard (slightly permeable) and the aquifuge (neither porous nor permeable, e.g. massive granite).

Discharge of a well in an unconfined aquifer (Dupuit's theory)

Assumptions (Dupuit–Forchheimer):

• The aquifer is homogeneous, isotropic and of infinite areal extent, resting on a horizontal impervious base.
• Flow is steady, radial and horizontal. The velocity is uniform over the depth and proportional to the slope of the water table, dz/dx.
• The well penetrates the full depth of the aquifer, and Darcy's law applies.

Consider a cylinder of radius x around the well, where the height of the drawdown curve above the base is z.

• Flow area: A = 2πxz
• Darcy velocity: v = K (dz/dx)
• Q = Av = 2πxzK (dz/dx)

Separating the variables: Q (dx/x) = 2πK z dz

Integrating from the well face (x = r, z = h) to the edge of the cone of depression (x = R, z = H): Q ln(R/r) = 2πK (H² − h²)/2

Q = πK(H² − h²)/ln(R/r) = 1.36K(H² − h²)/log₁₀(R/r)

Numerical: K = 10⁻⁴ m/s, R = 500 m, r = 5.0 m, H = 30 m, h = 10 m

• H² − h² = 900 − 100 = 800 m²
• ln(R/r) = ln 100 = 4.605
• Q = π × 10⁻⁴ × 800/4.605 = 0.2513/4.605

Q = 0.0546 m³/s ≈ 54.6 L/s ≈ 4716 m³/day

(b) Given flow = 10 × 10⁶ L/day, t = 45 s, G = 450 s⁻¹, μ = 0.89 × 10⁻³ N·s/m², and water depth : diameter = 2 : 1.

Flow: Q = 10⁷ L/day = 10⁴ m³/day = 10⁴/86400 = 0.1157 m³/s

Volume of tank: V = Qt = 0.1157 × 45 = 5.21 m³

With water depth H = 2D:

• V = (πD²/4)(2D) = πD³/2
• D³ = 2 × 5.21/π = 3.316
• D = 1.49 m (say 1.5 m)
• Water depth H = 2 × 1.49 = 2.98 m (say 3.0 m), plus a freeboard of about 0.3 m, so the overall depth is about 3.3 m.

Power from G = √(P/μV): P = μG²V = 0.89 × 10⁻³ × 450² × 5.21 = 0.89 × 10⁻³ × 202500 × 5.21

P = 939 W ≈ 0.94 kW (this is power delivered to the water; the motor is rated higher, allowing for drive efficiency)

Check: Gt = 450 × 45 ≈ 2.0 × 10⁴, which lies within the usual range for rapid (flash) mixing. Baffles on the wall prevent vortexing so that the input power produces shear rather than rotation of the whole water mass.

(c)(i) The 5 R's are a ranked set of actions that keep waste from being produced, or keep materials in use, so that only a small residue needs disposal.

• Refuse: say no to what is not needed, such as single-use plastic bags, cutlery and excess packaging.
• Reduce: use less and generate less, for example by buying in bulk, making durable products, and using less material in design and packaging.
• Reuse: use an item again for the same or another purpose, such as refillable bottles, repairing goods, or reusing construction formwork.
• Recycle: reprocess segregated waste into raw material, for example paper, PET, metals, glass, and construction and demolition waste into aggregates.
• Recover: extract value from what remains, by composting and biomethanation of wet waste, refuse-derived fuel, or waste-to-energy.

How they reduce the ill-effects of waste:

• Less waste reaches dumpsites and landfills. This cuts leachate that pollutes groundwater, methane emissions and dumpsite fires such as the one in part (ii). It also saves scarce urban land.
• Resources and energy are saved, because recycling aluminium or paper needs far less energy than making them from ore or pulp. Lower emissions follow.
• Less littering and choking of drains, so fewer floods and vector-borne diseases, and a smaller marine plastic load.
• Income and jobs are created in segregation, recycling and composting, including for informal waste pickers.

The order matters. The R's follow the waste-management hierarchy, in which prevention (refuse and reduce) is the most preferred option, then reuse, recycling and recovery, with safe disposal in a sanitary landfill as the last resort. In India this is built into the Solid Waste Management Rules, 2016 (segregation at source, composting of wet waste), the Plastic Waste Management Rules with extended producer responsibility, and the Swachh Bharat Mission (Urban) 2.0 goal of garbage-free cities, and together they support a circular economy.

(c)(ii) The dumpsite fire is a ground-level source (effective height H = 0). With complete reflection of the plume at the ground, the Gaussian plume equation is:

C(x, y, z) = [Q/(π u σᵧ σ_z)] · exp(−y²/2σᵧ²) · exp(−z²/2σ_z²)

This is the general form Q/(2πuσᵧσ_z)[e^(−(z−H)²/2σ_z²) + e^(−(z+H)²/2σ_z²)] with H = 0, where the two reflection terms add up to give π in place of 2π.

Given Q = 4 g/s, u = U₁₀ = 5 m/s, and at x = 2.0 km σᵧ = 150 m and σ_z = 50 m (stability D):

• Q/(πuσᵧσ_z) = 4/(π × 5 × 150 × 50) = 4/117810 = 3.395 × 10⁻⁵ g/m³ = 33.95 µg/m³
• 2σᵧ² = 45000 m² and 2σ_z² = 5000 m²

Equation at x = 2.0 km: C(y, z) = 33.95 exp(−y²/45000) exp(−z²/5000) µg/m³ (y, z in metres)

Maximum at ground (on the plume centre-line, y = 0, z = 0): C = 33.95 µg/m³

Maximum at 50 m above ground (y = 0, z = 50 m): C = 33.95 × exp(−2500/5000) = 33.95 × e^(−0.5) = 33.95 × 0.6065 C = 20.6 µg/m³

For a ground-level source the concentration is greatest at the ground and falls with height. At z = σ_z it is 61% of the ground value. The ground value is below the 24-hour NAAQS for NO₂ (80 µg/m³), though a dumpsite fire also releases particulates and toxic organics. Assumptions: steady emission and wind, flat terrain, no chemical reaction or deposition, and wind speed taken as U₁₀ for the near-ground plume.