Groundwater: Darcy, Theis & Cooper-Jacob

Quick Answer: Steady flow → Darcy. Pump test drawdown → Theis (or Cooper-Jacob at late time). Know T = Kb (transmissivity) and S (storativity) for confined aquifers.

Groundwater supplies drinking water, receives septic and landfill leachate, and carries contaminant plumes. FE Environmental items test Darcy calculations and conceptual pumping-test interpretation.

Darcy's Law

$$Q = K \, i \, A = K \, A \, \frac{h_1 - h_2}{L}$$

Valid for laminar flow in saturated porous media (Reynolds < ~10 in granular media).

Worked example: Sand column A = 2 m²; L = 50 m; h₁ − h₂ = 2 m; K = 1×10⁻⁴ m/s.

$$Q = 10^{-4} \times 2 \times \frac{2}{50} = 8 \times 10^{-6} \text{ m}^3/\text{s}$$

Permeability vs. Hydraulic Conductivity

Intrinsic permeability k (darcy) depends on media only. K = k ρg/μ includes fluid properties. Temperature affects μ — warmer water slightly higher K.

Aquifer Types

Transmissivity T = K × b (confined aquifer thickness b). Storativity S (confined) is small (~10⁻⁵–10⁻³); specific yield Sy (unconfined) is larger (~0.1–0.3).

Steady Radial Flow to a Well (Confined, Thiem)

$$Q = \frac{2\pi T (h_2 - h_1)}{\ln(r_2/r_1)}$$

Drawdown s = h₀ − h at distance r. Used for equilibrium pumping tests.

Theis Equation (Transient)

Drawdown in confined aquifer from constant discharge Q:

$$s(r,t) = \frac{Q}{4\pi T} W(u), \quad u = \frac{r^2 S}{4 T t}$$

W(u) is the well function — Handbook table or approximation.

Assumptions: Infinite aquifer, horizontal flow, constant T and S, fully penetrating well, instantaneous withdrawal from storage.

Cooper-Jacob Approximation

When u < ~0.01:

$$s = \frac{2.3 Q}{4\pi T} \log_{10}\!\left(\frac{2.25 T t}{r^2 S}\right)$$

Plot s vs. log t → straight line; slope Δs per log cycle gives T; intercept gives S.

Worked example (conceptual): Steeper drawdown curve at early time near well; flattening at distance — typical Theis behavior.

Unconfined Aquifer Notes

Delayed yield complicates early-time data — Jacob correction or numerical models. FE may stay conceptual: water table lowering increases gradient toward well.

Contaminant Transport (Brief)

Advection v = K i / n_e (effective porosity n_e). Dispersion spreads plume beyond mean advective front. Retardation R = 1 + ρ_b K_d / n_e slows solutes.

Well Design and Influence

Radius of influence R₀ where drawdown negligible — used in design estimates. Well efficiency losses (skin, partial penetration) raise drawdown in real wells.

FE Exam Checklist

1. Darcy: identify A, L, Δh, K.
2. Pumping: confined vs. unconfined; use T not K alone when Theis applies.
3. Match units — days vs. seconds in t ruins S.
4. Drawdown adds for multiple wells (superposition principle).

Exam trap: Using Sy in Theis confined equation — use storativity S, not specific yield.

Exam trap: Darcy velocity v = Q/A differs from seepage velocity v_s = Q/(n_e A).

Groundwater math pairs with water treatment when wells supply plants or when remediation extracts contaminated aquifers.

Specific Capacity and Well Tests

Specific capacity = Q/s (yield per unit drawdown) declines as wells clog or aquifer depletes. Step-drawdown tests separate aquifer loss from well loss (turbulence near screen).

Boundary Conditions

Image well theory handles constant head streams or no-flow boundaries — conceptual adjustment to Theis solutions near rivers and barriers.

Unconfined Delayed Yield

Early-time drawdown data may overestimate T if delayed yield not accounted for — late-time data more reliable for parameter fitting.

FE Problem Walkthrough

Given: Q = 0.05 m³/s, r = 30 m, t = 2 days, T = 0.01 m²/s, S = 0.0003 → compute u, lookup W(u), solve s. Units on t must match S and T (days vs. seconds — catastrophic if mixed).

Leaky Aquifer Concept

Leakance from confined aquifer through aquitard to adjacent aquifer adds water to pumped well — drawdown lower than ideal Theis predicts. Image wells model boundaries.

Well Field Interference

Multiple pumping wells within radius of influence superpose drawdowns — municipal well fields spaced to limit mutual interference during peak demand.

Full Theis Numeric Sketch

Q = 0.04 m³/s, r = 20 m, t = 1 day = 86400 s, T = 0.008 m²/s, S = 0.0002.

$$u = \frac{r^2 S}{4Tt} = \frac{400 \times 0.0002}{4 \times 0.008 \times 86400} = 2.9 \times 10^{-5}$$

For small u, W(u) ≈ -0.5772 - ln(u) ≈ 10.0. Then $s = \frac{Q}{4\pi T} W(u) = \frac{0.04}{0.1005} \times 10 \approx 4.0$ m drawdown at 20 m after one day.

Thiem Equilibrium Example

Confined aquifer: T = 50 m²/day, h₀ = 30 m at R₀ = 500 m, well r_w = 0.3 m, Q = 200 m³/day.

$$s_w = \frac{Q}{2\pi T}\ln\frac{R_0}{r_w} = \frac{200}{314}\ln(1667) = 0.637 \times 7.4 \approx 4.7\text{ m}$$

at the well (conceptual order-of-magnitude check).

Unit Conversion Trap

T in m²/day with t in seconds without conversion changes u by 86400² — always convert t to same time base as T before Theis.