Well drawdown (Thiem, Theis, Cooper-Jacob) on the PE Civil WRE exam

Pumping-well problems are the most equation-heavy part of Groundwater and Wells, which carries 4–6 of the 80 questions on the WRE exam. The relations are compact. Most lost points come from misreading the setup: confined or unconfined, steady or transient, head or depth.

What you actually need to own

• Aquifer properties. Transmissivity T = K·b, using the saturated thickness b of a confined aquifer. Storativity S is the volume released per unit area per unit head decline. In a confined aquifer it is small, because the water comes from elastic storage. In an unconfined aquifer it is essentially the specific yield, which is orders of magnitude larger.
• Steady confined flow (Thiem): Q = 2πT(h₂ − h₁)/ln(r₂/r₁). It is linear in head, so you can write it with drawdowns (s₁ − s₂) just as well. The Handbook prints it under the Confined Aquifer heading as "Uniform Flow (Thiem Equation)".
• Steady unconfined flow (Dupuit/Thiem): Q = πK(h₂² − h₁²)/ln(r₂/r₁). Here h is the saturated thickness measured up from the aquifer base. The squares are not optional.
• Transient flow (Theis): s = [Q/(4πT)]·W(u), with u = r²S/(4Tt). Drawdown keeps growing with time, and the well function W(u) comes from a table or a series.
• Cooper-Jacob: for small u, W(u) ≈ −0.5772 − ln u. That gives s = [2.3Q/(4πT)]·log(2.25Tt/(r²S)). Plotted as drawdown against log time, the data fall on a straight line:
  • T = 2.3Q/(4πΔs), where Δs is the drawdown change over one log cycle of time.
  • S = 2.25T·t₀/r², where t₀ is where the line crosses zero drawdown.
  • The same relation against log distance, at one instant, gives T = 2.3Q/(2πΔs). The 2π is not a 4π.

Where people lose points

• Head vs. depth to water. Head is an elevation: casing-top elevation minus depth to water. Differences in depth to water alone ignore casings set at different elevations. In Dupuit, h is measured from the aquifer base, not from the ground or the static water level.
• Using Cooper-Jacob outside its range. The straight line holds only once u is small (u < 0.01 is the usual rule). That means late time, or close to the pumped well. Early-time points curve away from the line, so fit the straight tail. Check u with your fitted T and S.
• Log cycle vs. any interval. Δs has to be read across a full log cycle (say 10 to 100 minutes). If you take it over a shorter span, divide that Δs by the span's Δlog t to get the drop per log cycle.
• Radius from the pumped well. Measure r₁ and r₂ from the pumping well, not from each other. Drawdown inside the pumped well also includes well loss, which the aquifer equations don't predict.
• Units. Pumping rates arrive in gpm and T often in gpd/ft. Convert everything to one system (1 gpm ≈ 192.5 ft³/day) and keep t₀ in the same time unit as T. S is dimensionless, so a strange value (say S = 40) usually means a units slip.
• Mixing storage terms. Putting a confined S near 10⁻⁴ into an unconfined problem, or the reverse, is off by orders of magnitude.

How to study it

Before choosing an equation, sort each problem on two questions: confined or unconfined, and steady or transient. Then drill a Cooper-Jacob workflow on tabulated pump-test data: plot s against log t, draw the line through the late data, read Δs and t₀, compute T and S, and check u. One useful fact: take the Cooper-Jacob drawdown at two radii at the same time and the difference reduces to Thiem. That is why the steady equations work on late-time observation-well pairs. For the Handbook, Cooper-Jacob is printed but neither the Theis equation nor a W(u) table is, so a Theis problem has to give you W(u) or a u small enough for Cooper-Jacob. A search for "Dupuit" misses the unconfined relation because of how the heading is spelled. Search "unconfined". Pumped-well layouts tie into dewatering on the earthwork and trench safety side of the exam.

Part of the Groundwater and Wells area of the PE Civil WRE exam. → Start practicing free — the whole practice bank, free with an account.