Topic 8 of 10 — free theory
Groundwater Flow
Darcy's law, confined and unconfined flow, and Thiem well equations.
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Darcy's law — the groundwater workhorse
Groundwater flow is slow, laminar, and beautifully linear. Darcy's law says the discharge is proportional to the hydraulic gradient:
Q = K · i · A with i = dh/dl
| Q | discharge through the porous medium |
| K | hydraulic conductivity (permeability) |
| i | hydraulic gradient — head loss per unit length, dimensionless |
| A | bulk cross-sectional area normal to flow (solids + voids) |
v = Q/A (Darcy flux) vs = v/n (seepage velocity)
| v | discharge per unit bulk area — not the actual pore-water speed |
| vs | true average velocity through the pores |
| n | porosity |
Contaminant travel time uses the seepage velocity vs, which is faster than the Darcy flux. Using v instead of vs underestimates how fast a plume moves.
One-dimensional aquifer flow
Confined (constant thickness b): q = T · (dh/dl), T = K·b
| T | transmissivity — the aquifer's headline property for confined flow |
| b | saturated thickness of the confined aquifer |
Unconfined (Dupuit): q = (K/2L)·(h1² − h2²) per unit width
| h | saturated thickness (water-table height above the impermeable base) |
| Assumes nearly horizontal flow — the Dupuit approximation; fine for gentle gradients. |
Wells — the Thiem equations (steady state)
Pumping a fully penetrating well in a homogeneous, isotropic aquifer, at steady state, with two observation wells:
Confined: Q = 2πT(h2 − h1)ln(r2/r1)
| h1, h2 | hydraulic heads at radial distances r1, r2 from the well |
| T = Kb | transmissivity |
Unconfined: Q = πK(h2² − h1²)ln(r2/r1)
| Heads h are water-table heights above the aquifer base — and they enter squared. |
Drawdown s = original head − pumped head. Either form can be rewritten in drawdowns: confined Q = 2πT(s1 − s2)/ln(r2/r1).
PE depth: transient drawdown and well interference
Before steady state develops, drawdown in a confined aquifer depends on time and storativity. The Theis solution is:
s = Q4πT W(u), u = r²S4Tt
| s | drawdown at radius r and elapsed pumping time t |
| S | dimensionless storativity; do not confuse with specific yield |
| W(u) | well function, normally supplied or evaluated from a table |
For late time (small u), the Cooper–Jacob approximation is convenient:
s ≈ 2.3Q4πT log10(2.25Tt / r²S)
Drawdowns from multiple pumping wells superpose in a linear confined aquifer: stotal = Σsj. Keep pumping and injection signs consistent.
Worked example Pumping rate from observation-well data
Given:
- Confined aquifer, transmissivity T = 500 ft²/day.
- Observation well 1: r1 = 50 ft, head h1 = 102.5 ft.
- Observation well 2: r2 = 150 ft, head h2 = 105.0 ft.
- Steady state reached.
Solution:
- Thiem (confined): Q = 2π·500·(105.0 − 102.5) / ln(150/50).
- Numerator: 2π × 500 × 2.5 = 7,854 ft³/day. Denominator: ln(3) = 1.0986.
- Q = 7,854 / 1.0986 = 7,149 ft³/day ≈ 53,500 gal/day.
Answer: Well discharge ≈ 7,150 ft³/day (≈ 53,500 gpd).