Groundwater: Flow, Wells and Drawdown (FE Civil)

NCEES FE Civil spec: Water Resources and Environmental Engineering, H. Groundwater (e.g., flow, wells, drawdown). FE Reference Handbook: Civil Engineering chapter, Hydrology/Water Resources section (Darcy, well drawdown).

What the FE Civil Spec Covers

Section H covers groundwater flow, wells and drawdown. Problems usually ask for a flow rate from Darcy's law, travel time of a contaminant using seepage velocity, hydraulic conductivity from a pumping test, or the drawdown at a given distance from a pumping well. Layered soil permeability and flow nets also show up, and they overlap with the geotechnical seepage topics.

Key Concepts

Aquifers. A confined aquifer is bounded above and below by low permeability layers and is under pressure; its piezometric surface can be above the aquifer top. An unconfined (water table) aquifer has a free water surface at atmospheric pressure.
Hydraulic conductivity K (L/T) describes how easily water moves through the soil. Gravels and clean sands are high; clays are many orders of magnitude lower.
Hydraulic gradient i is head loss per unit length of flow path, i = Δh/L.
Porosity versus effective porosity. Only the connected pores that drain carry flow, so transport calculations use effective porosity n_e.
Cone of depression. Pumping lowers the water table or piezometric surface around a well. Drawdown s is the drop from the original level and decreases with distance until the radius of influence.
Storativity and specific yield describe how much water is released per unit area per unit head decline; they matter for unsteady pumping, while the FE mostly tests steady state.

Core Formulas

Darcy's law, Darcy velocity and average seepage velocity:

Q = K i A v = K i v_{s} = \frac{K i}{n _{e}}

Transmissivity of a confined aquifer of thickness b:

T = K b

Steady flow to a well in a confined aquifer (Thiem), with heads h measured at radii r:

Q = \frac{2 π K b ( h _{2} - h _{1} )}{ln ( r _{2} / r _{1} )}

Steady flow to a well in an unconfined aquifer (Dupuit), heads measured above the aquifer bottom:

Q = \frac{π K ( h _{2}^{2} - h _{1}^{2} )}{ln ( r _{2} / r _{1} )}

Equivalent conductivity of layered soil, for flow parallel and perpendicular to the layers:

K_{h} = \frac{\sum K _{i} H _{i}}{\sum H _{i}} K_{v} = \frac{\sum H _{i}}{\sum H _{i} / K _{i}}

Seepage from a flow net with N_f flow channels and N_d equipotential drops:

q = K H \frac{N _{f}}{N _{d}}

Travel time questions use seepage velocity. With K = 10 m/day, i = 0.005 and n_e = 0.25, the Darcy velocity is 0.05 m/day and the seepage velocity is 0.05/0.25 = 0.2 m/day, so a conservative contaminant takes about 100/0.2 = 500 days to travel 100 m. Using total porosity or skipping the porosity step gives the wrong answer by a large factor, and the wrong choices on the exam are usually built from exactly those errors.

When several wells pump from the same confined aquifer, drawdowns add (superposition), so the drawdown at a point is the sum of the drawdown each well would cause alone. Both well equations assume steady state, a well that fully penetrates the aquifer, and a uniform, isotropic aquifer.

Where It Is in the FE Reference Handbook

Search the Civil Engineering chapter, Hydrology/Water Resources section, for Darcy's law, the aquifer definitions and the well drawdown equations (Dupuit for unconfined, Thiem for confined). Flow nets and layered permeability are in the Geotechnical section of the same chapter.

Worked Example: Confined Aquifer Pumping Test

A well fully penetrates a confined aquifer 15 m thick with K = 30 m/day. At steady state, an observation well 20 m away shows 3.0 m of drawdown and one 100 m away shows 1.2 m. Find the pumping rate, the transmissivity, and the drawdown 50 m from the well.

Step 1: Head difference. The farther well has less drawdown, so its head is higher by

h_{2} - h_{1} = s_{1} - s_{2} = 3.0 - 1.2 = 1.8 m

Step 2: Thiem equation.

Q = \frac{2 π ( 30 ) ( 15 ) ( 1.8 )}{ln ( 100/20 )} = \frac{5089}{1.609} = 3162 m^{3} / day \approx 0.0366 m^{3} / s

Step 3: Transmissivity.

T = K b = (30) (15) = 450 m^{2} / day

Step 4: Drawdown at 50 m. Apply Thiem between r = 20 m and r = 50 m:

s_{50} = s_{20} - \frac{Q}{2 π T} ln \frac{50}{20} = 3.0 - \frac{3162}{2 π ( 450 )} (0.916) = 3.0 - 1.02 = 1.98 m

Check: the same expression from 20 m to 100 m gives 3.0 minus 1.80, which is the measured 1.2 m.

Common Mistakes

Using drawdowns in the Dupuit equation. It needs saturated thickness h measured from the aquifer bottom, then squared.
Using log base 10 instead of the natural log, or flipping r₂/r₁ relative to h₂ and h₁.
Using total porosity for travel time when effective porosity is given.
Mixing K in cm/s with distances in m or times in days. Convert first.
Averaging K arithmetically for vertical flow through layers; the low K layer controls.

Practice Problems

Try these from the free PolyCalc FE Civil problem bank. Each one has a full worked solution.

Using Darcy's law, what is the groundwater discharge for hydraulic conductivity k = 1 × 10⁻⁴ m/s, hydraulic gradient i = 0.02, and gross cross-sectional flow area A = 100 m²? See the worked solution or practice the Soil Mechanics deck
An unconfined aquifer has a saturated thickness of 20 m above an impermeable layer. A well pumped at a steady 0.04 m³/s produces a drawdown of 8.0 m in an observation well 10 m away and 2.0 m in a second observation well 100 m away. The hydraulic conductivity is most nearly: See the worked solution or practice the Environmental & Water Quality deck
A 2 m thick sand layer with hydraulic conductivity 5 × 10⁻⁴ m/s overlies a 1 m thick clay layer with hydraulic conductivity 1 × 10⁻⁷ m/s. For vertical flow through both layers, the equivalent hydraulic conductivity is most nearly: See the worked solution or practice the Soil Mechanics deck
Seepage beneath a sheet pile is analysed with a flow net having 4 flow channels and 12 equipotential drops. The hydraulic conductivity is 2 × 10⁻⁵ m/s and the total head loss is 6 m. The seepage per metre of wall is most nearly: See the worked solution or practice the Soil Mechanics deck

Related Calculators and Examples

Frequently Asked Questions

What is the difference between Darcy velocity and seepage velocity?

Darcy velocity (specific discharge) is flow divided by the whole cross-sectional area, v = Ki. Water actually moves only through the pores, so the average seepage velocity is larger: Darcy velocity divided by the effective porosity.

When do I use the Thiem equation versus the Dupuit equation?

Use Thiem for a confined aquifer, where the saturated thickness b stays constant and drawdown is linear in head. Use Dupuit for an unconfined aquifer, where the saturated thickness itself drops, so the equation uses the difference of the squared heads above the aquifer bottom.

What is transmissivity?

Transmissivity is hydraulic conductivity times the saturated thickness of a confined aquifer, T = Kb. It has units of area per time, such as m2/day, and measures how much water the full aquifer thickness can transmit under a unit gradient.

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