Groundwater & Well Hydraulics on the PE WRE Exam: Theis and Dupuit Walkthrough
Theis transient drawdown and Dupuit unconfined steady-state for the PE Civil WRE exam — two worked NCEES-style problems plus the Theis well-function reference table.
Well-hydraulics problems on the PE Civil Water Resources & Environmental exam blend tidy physics (Darcy's law in radial coordinates) with messy curve-fitting from real field data. The pain point most candidates report is the same: under time pressure, it's easy to grab the wrong equation. Theis is for transient drawdown in a confined aquifer; Dupuit is for steady-state pumping in an unconfined aquifer; Thiem is for steady-state confined; Cooper–Jacob is just a small-u simplification of Theis. Pick the wrong one and the answer family is wrong before any arithmetic.
Groundwater is also one of the topics where the NCEES PE Civil Reference Handbook does most of the heavy lifting if you can find what you need. The Theis well-function tabulation, the Dupuit equation, the Cooper–Jacob approximation, and the radial-flow assumptions are all reproduced. The formulas themselves are right there in the handbook; the skill the exam tests is recognizing which regime applies to the problem in front of you, then plugging the right inputs into the right equation in the right units.
This post walks through the four well-hydraulics problem types NCEES tests on the WRE exam, with two fully solved worked examples (Theis transient drawdown and Dupuit unconfined steady-state) and a Theis well-function reference table. Every formula traces to its section in the handbook (§6.6 for groundwater) and to its original USGS publication where applicable.
Why groundwater & wells matters on the PE WRE exam
Per the April 2024 NCEES PE Civil WRE specification, Topic 8 (Groundwater and Wells) carries 4–6 questions out of 80. That makes it a smaller block than open-channel hydraulics or hydrology, but well-hydraulics questions tend to be calculation-heavy and reward candidates who have practiced the radial-flow setup specifically. A confident 4-of-6 here is meaningful margin on a pass-or-fail exam.
Beyond the direct Topic-8 questions, well hydraulics shows up indirectly in a couple of places. Drinking-water treatment problems often start with a well source — you need to verify the well can deliver the design flow at sustainable drawdown before you size a treatment train. Hydrology questions about base flow and stream-aquifer interaction draw on the same Darcy-law fundamentals. Mastering Topic 8 also unlocks better instincts on those adjacent questions.
Core concepts you must master
Confined vs. unconfined aquifers
A confined aquifer sits between two impermeable layers and is fully saturated. The piezometric surface (the level water would rise to in an open well) sits above the top of the aquifer. Pumping causes pressure (head) to drop, but the aquifer stays fully saturated — no dewatering. This is the regime where the Theis equation applies for transient drawdown and the Thiem equation applies for steady state.
An unconfined (water-table) aquifer has a free water table at the top. Pumping physically lowers the water table, which lowers the saturated thickness near the well. Because the saturated thickness changes with drawdown, the math is non-linear — the Dupuit equation handles this with a key simplifying assumption (described below).
Transmissivity (T) and storage coefficient (S)
Two parameters define an aquifer's hydraulic response:
S = storage coefficient (dimensionless)
where K is hydraulic conductivity and b is the saturated thickness of a confined aquifer. Storage coefficient values are very different by aquifer type: S ≈ 10−5 to 10−3 for confined aquifers (water release is mostly compression of the aquifer matrix); S ≈ 0.05 to 0.30 for unconfined aquifers (water release is mostly drainage of the pore space, so S approximately equals specific yield Sy). Confused which to use? Read the question for "confined" or "water table" and check the magnitude of the given S: small (≪ 1) means confined, large (0.05+) means unconfined.
The Theis equation (transient confined)
For a fully penetrating well in a confined aquifer with constant pumping rate Q, the drawdown s at radial distance r after time t is (handbook §6.6):
u = r2·S / (4·T·t)
W(u) is the Theis well function (also called the exponential integral E1). It's tabulated in the handbook and in the reference table later in this post.
The Cooper–Jacob approximation
For small u (typically u < 0.01 — achieved at large times or small distances), the Theis well function simplifies to a closed form:
s ≈ (Q / 4πT) · ln(2.25·T·t / (r2·S))
The constant 0.5772 is the Euler–Mascheroni constant. The combined form on the second line is what most practical groundwater software uses.
The Dupuit equation (steady unconfined)
For steady-state radial flow to a fully penetrating well in an unconfined aquifer, the Dupuit equation gives the relationship between pumping rate Q and the saturated thicknesses at two radial distances (handbook §6.6):
The Dupuit–Forchheimer assumptions: (a) flow is horizontal (vertical components negligible), (b) the hydraulic gradient equals the slope of the water table, (c) flow is steady, (d) the aquifer is homogeneous and isotropic. These are reasonable approximations everywhere except very near the well screen, where the assumption breaks down (this is why Dupuit overestimates drawdown right at the well face for shallow aquifers, and why the equation is most accurate at the observation-well distance).
Thiem equation (steady confined)
For comparison, steady-state radial flow in a confined aquifer follows the Thiem equation:
Note that Thiem uses h2 − h1 (linear in head) while Dupuit uses h22 − h12 (quadratic). The difference is the variable saturated thickness in the unconfined case.
Image-well method and superposition
Linear groundwater equations let you superpose solutions. For a well near a recharge boundary (river, lake), an "image well" of equal pumping rate at the mirrored location across the boundary recreates the constant-head boundary condition. For a well near a no-flow boundary (impermeable wall), an image well of opposite sign (pumping into the aquifer) creates the no-flow boundary. Multi-well interference is the same idea: total drawdown at any point is the sum of drawdowns from each pumping well.
The 4 types of well-hydraulics problems on the PE WRE exam
Type 1: Unconfined steady-state pumping (Dupuit)
Given pumping rate, hydraulic conductivity, initial saturated thickness, well radius, and either a radius of influence or an observation-well measurement. Solve for drawdown at the well or at the observation point using the Dupuit equation. Worked below.
Type 2: Confined transient drawdown (Theis)
Given pumping rate, transmissivity, storage coefficient, distance to observation well, and elapsed time since pumping started. Compute u, look up or compute W(u), then compute drawdown. Often combined with a "is the Cooper–Jacob approximation valid?" check (yes if u < 0.01). Worked below.
Type 3: Multi-well interference (superposition)
Given two or more wells operating simultaneously, find drawdown at a point or at one of the wells due to the combined effect. Sum the individual drawdown contributions. Each contribution uses Theis (or Cooper–Jacob) with the radial distance from the relevant well to the point of interest.
Type 4: Recovery test analysis
After shutting off a pumping well, the residual drawdown s′ at time t′ after shutoff (with original pumping for time t) is given by the Theis recovery method: s′ = (Q/4πT) · ln(t/t′). Plotting s′ versus log(t/t′) gives a straight line whose slope yields T. The exam version usually asks you to back out T from one or two recovery measurements.
Worked example: Theis equation transient drawdown
Worked example 1 — Theis transient drawdown. A fully penetrating well in a confined aquifer pumps at Q = 0.1 ft3/s. The aquifer transmissivity is T = 1,000 ft2/day and storage coefficient is S = 5 × 10−4. Compute the drawdown at an observation well 100 ft from the pumping well after t = 1 day of continuous pumping.
Step 1 — Convert Q to consistent units (ft3/day to match T).
Step 2 — Compute u.
= 5 / 4,000 = 1.25 × 10−3
Step 3 — Cooper–Jacob check. u = 0.00125 < 0.01, so the Cooper–Jacob approximation is valid. Use it for speed:
= −ln(1.25 × 10−3) − 0.5772
= 6.685 − 0.5772 = 6.108
Cross-check from a Theis function table: at u = 1.25 × 10−3, the tabulated W(u) ≈ 6.10 (interpolating between u = 10−3, W = 6.332 and u = 5 × 10−3, W = 4.726). The two values agree.
Step 4 — Apply the Theis equation.
= (8,640 / (4π × 1,000)) · 6.108
= (8,640 / 12,566) · 6.108
= 0.6877 · 6.108
= 4.20 ft
Answer: s ≈ 4.2 ft drawdown at the observation well after 1 day. If the question instead asked for drawdown at the well face, you'd use a smaller r (the well radius) and a much larger W(u); drawdown at the well is always larger than at any observation well.
Worked example: Dupuit unconfined steady-state
Worked example 2 — Dupuit unconfined steady pumping. A fully penetrating well in an unconfined aquifer pumps at Q = 100 gpm under steady-state conditions. Hydraulic conductivity K = 50 ft/day. Initial saturated thickness h0 = 50 ft. The radius of influence is estimated at R = 1,000 ft, and the well radius is rw = 0.5 ft. Compute the steady-state water level at the well face and the drawdown at the well.
Step 1 — Convert Q to consistent units (gpm → ft3/day to match K):
= 100 × 192.5 = 19,250 ft3/day
Step 2 — Apply the Dupuit equation with h2 = h0 = 50 ft at r2 = R = 1,000 ft, and h1 = hw at r1 = rw = 0.5 ft:
Step 3 — Solve for hw2.
h02 − hw2 = Q · ln(R/rw) / (π·K)
= 19,250 × 7.601 / (π × 50)
= 146,319 / 157.08 = 931.5 ft2
Step 4 — Back out hw and the drawdown sw.
hw = √1,568.5 = 39.6 ft
sw = h0 − hw = 50 − 39.6 = 10.4 ft
Answer: Water level at the well face hw ≈ 39.6 ft; drawdown at the well sw ≈ 10.4 ft. Cross-check: plug hw = 39.6 ft back into the Dupuit equation: Q = π(50)(502 − 39.62)/ln(2,000) = π(50)(931.84)/7.601 = 19,253 ft3/day ≈ 100 gpm ✓.
Common errors that cost points
Using Dupuit on a confined aquifer (or Thiem on unconfined)
The two most common steady-state equations have different forms because the geometry is different. Dupuit's h22 − h12 term comes from the variable saturated thickness in unconfined flow. Thiem's h2 − h1 term assumes constant saturated thickness b in the confined case (it's already absorbed into T = K·b). Mixing them is a category error that lands you on the wrong answer family by 30–50%.
Wrong sign convention on drawdown
Drawdown is the decrease in head, so s = h0 − h(r, t) is positive for pumping and negative for injection. The Theis equation as written above gives positive s for pumping. If your problem describes recharge or injection (Q negative), drawdown becomes a "buildup" with opposite sign. Watch for this on superposition problems involving image wells across recharge boundaries.
Forgetting to convert head to drawdown (or the other way around)
Some prompts give you the water-level elevation h; others give you drawdown s. The Dupuit equation needs h (saturated thickness above the impermeable base); the Theis equation outputs s directly. If the question gives drawdown but you need head: h(r, t) = h0 − s. If the question gives head but you need drawdown: s = h0 − h(r, t). Read carefully which the question wants.
Misinterpreting W(u) outside the Cooper–Jacob range
Cooper–Jacob is valid only for u < 0.01 (some sources allow u < 0.05). For larger u — typically achieved at small times or large distances — you must use the full Theis well-function table or numerical integration of the exponential integral. Applying Cooper–Jacob at u = 0.5, for example, would give a negative W(u) which is physically impossible.
Unit-system inconsistency
Groundwater problems mix gpm, cfs, ft3/day, m3/day, and ft2/day vs. gpd/ft for transmissivity. The Theis and Dupuit equations don't care which units you use as long as they're consistent. The most defensible approach: convert everything to ft3/day and ft2/day at the start, run the equation, then convert back at the end. Carrying units through the calculation catches a lot of arithmetic errors.
How to study well hydraulics for the PE WRE exam
Phase 1 — Equation fluency (Week 1)
Read handbook §6.6 (groundwater) end-to-end. Practice writing Theis, Cooper–Jacob, Thiem, and Dupuit equations from a blank page until you can identify which regime to apply (confined vs. unconfined, transient vs. steady) in under 30 seconds from the problem statement. The four equations look similar but have meaningfully different forms.
Phase 2 — Worked-problem drills (Weeks 2–3)
Work twelve problems across the four problem types: three Dupuit unconfined, three Theis transient, three superposition (multi-well or boundary effects), three recovery-test backouts of T. Time yourself: four to six minutes per problem on the exam. PEwise's Module 13 (Groundwater Flow) walks Theis, Dupuit, image-well superposition, and the cone-of-depression analysis NCEES tests, with worked NCEES-style problems and reference citations.
Phase 3 — Integration with adjacent topics (Week 4)
Solve five integration problems that combine well hydraulics with: (a) drinking-water source sizing (does the well deliver design flow at sustainable drawdown?), (b) base-flow contribution to a stream gauge, (c) contaminant transport between a leaking source and a downgradient well. The exam favors integration questions; standalone Theis or Dupuit problems are less common than chained problems that combine these with hydrology or water-quality content.
Quick reference: Theis function and Dupuit summary
Theis well function W(u)
| u | W(u) | Cooper–Jacob valid? |
|---|---|---|
| 1 × 10−4 | 8.633 | Yes |
| 5 × 10−4 | 7.024 | Yes |
| 1 × 10−3 | 6.332 | Yes |
| 5 × 10−3 | 4.726 | Yes |
| 1 × 10−2 | 4.038 | Borderline |
| 5 × 10−2 | 2.468 | No (use full Theis) |
| 1 × 10−1 | 1.823 | No |
| 5 × 10−1 | 0.5598 | No |
| 1.0 | 0.2194 | No |
Source: Standard Theis well-function tabulation (USGS / NCEES PE Civil Reference Handbook §6.6). For u values not in the table, interpolate on a log–log basis or use the Cooper–Jacob approximation when applicable.
Equation summary
| Regime | Equation | Outputs |
|---|---|---|
| Confined, transient | Theis: s = (Q/4πT)·W(u) | s(r, t) |
| Confined, transient (small u) | Cooper–Jacob: s ≈ (Q/4πT)·ln(2.25Tt/r2S) | s(r, t) |
| Confined, steady | Thiem: Q = 2πT(h2 − h1) / ln(r2/r1) | Q or h(r) |
| Unconfined, steady | Dupuit: Q = πK(h22 − h12) / ln(r2/r1) | Q or h(r) |
| Recovery test | Theis recovery: s′ = (Q/4πT)·ln(t/t′) | T from slope |
See the Cone of Depression Animated
PEwise's PE WRE course walks through Theis, Dupuit, and image-well superposition with the cone-of-depression evolving in real time on the screen — once you can SEE radial drawdown, the equations stop feeling abstract.
Connecting this to your overall PE WRE exam strategy
Topic 8 (Groundwater and Wells) sits between Topic 5/6 (Hydraulics — Closed Conduit and Open Channel) and Topic 9 (Surface Water and Groundwater Quality). Once well hydraulics is automatic, you'll recognize the radial-flow framing in contaminant-transport problems, well-source sizing for drinking water, and stream-aquifer interaction in hydrology. For the full WRE topic blueprint, our PE WRE exam topics breakdown walks all 12 topic areas. For the drinking-water treatment side — where well sources feed disinfection and CT-value calculations — see the drinking-water treatment and disinfection post.
Final thoughts
Well-hydraulics problems reward engineers who treat the regime identification as the first 30 seconds of the question: confined or unconfined? transient or steady? single well or multi-well? Once the regime is fixed, the equation is fixed, and the calculation is mechanical. The candidates who pass internalize the four-equation taxonomy and pick the right one without thinking. The candidates who don't second-guess between Theis, Cooper–Jacob, Thiem, and Dupuit at every problem and burn time on the wrong setup. Drill the regime check until it's automatic.
Master Well Hydraulics with PEwise
PEwise's Module 13 (Groundwater Flow) of the PE WRE course covers Theis transient drawdown, Dupuit unconfined steady-state, image-well superposition, and the cone-of-depression analysis NCEES tests — in 13 lessons with worked NCEES-style problems and detailed reference citations. Course author Mahdi Bahrampouri, Ph.D., Geotechnical Earthquake Engineer, built the curriculum directly against the NCEES PE Civil Reference Handbook §6.6 and the original USGS groundwater publications (Theis 1935; Dupuit–Forchheimer assumption).
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