Question 5 of 5: Leaky and Non-Leaky Confined Aquifer Transient Drawdown

Question 5: Leaky and Non-Leaky Confined Aquifer Transient Drawdown (equal value)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Confined aquifer $T=10^{-2}\ \text{m}^2/\text{s}$, $S=10^{-4}$, $K=10^{-3}$ cm/s (aquifer material property, not needed once $T$, $S$ are given directly); aquitard thickness $b'=6.2$ m above the aquifer; $Q=60\ \text{m}^3/\text{hr}$; $t=24$ h. (a) aquitard impermeable, observation well $r=90$ m. (b) aquitard $K'=10^{-6}$ cm/s (leaky), observation well $r=80$ m.

Find. (a) drawdown at $r=90$ m, $t=24$ h, treating the aquitard as a no-flow boundary (ordinary Theis). (b) drawdown at $r=80$ m, $t=24$ h, accounting for leakage through the now-permeable aquitard (Hantush-Jacob). (c) a qualitative log-log sketch comparing four characteristic drawdown-vs-time signatures.

Approach. Part (a) is the standard Theis non-equilibrium solution — the "aquifer $K$" given in the preamble is redundant once $T$ and $S$ are stated directly and is not needed for either calculation. Part (b) replaces the ordinary well function $W(u)$ with the Hantush-Jacob leaky well function $W(u,r/B)$, where $B=\sqrt{Tb'/K'}$ is the leakage factor; $W(u,r/B)$ is evaluated here by direct numerical integration of its defining integral (equivalent to, and checked against, Table 5.2). Part (c) is a standard qualitative comparison of the four canonical Theis-family drawdown signatures.

1. Part (a) — non-leaky Theis drawdown. $Q=60/3600=0.01667\ \text{m}^3/\text{s}$, $t=24\times3600=86{,}400$ s, $r=90$ m: $$u=\frac{r^2S}{4Tt}=\frac{(90)^2(10^{-4})}{4(10^{-2})(86{,}400)}=2.34\times10^{-4},\qquad s=\frac{Q}{4\pi T}W(u)=\boxed{1.03\ \text{m}}.$$
2. Part (b) — leakage factor. With the aquitard now permeable ($K'=10^{-6}\ \text{cm/s}=10^{-8}\ \text{m/s}$, $b'=6.2$ m): $$B=\sqrt{\frac{Tb'}{K'}}=\sqrt{\frac{(10^{-2})(6.2)}{10^{-8}}}=\boxed{2490\ \text{m}}.$$
3. Leaky drawdown at r=80 m. $u=r^2S/(4Tt)=1.85\times10^{-4}$, $r/B=80/2490=0.0321$; evaluating the Hantush-Jacob well function at this $(u,r/B)$ pair (cross-checked against Table 5.2, e.g. $W(0.005,0.10)=4.296$ matches the printed table exactly): $$s=\frac{Q}{4\pi T}W(u,r/B)=\frac{0.01667}{4\pi(10^{-2})}(6.99)=\boxed{0.927\ \text{m}}.$$ This is noticeably less than the non-leaky Theis value at the same $u$ would give ($\approx1.07$ m) — the leaky aquitard supplies additional water from storage above it, so the pumped aquifer draws down less than it would if truly confined.

Part (c). The four scenarios share the same early-time behaviour (before any boundary or leakage effect has propagated to the well, all four plot as the same Theis curve) but diverge at later time as shown below.

(i) The ideal, homogeneous, non-leaky, isotropic confined aquifer follows the Theis type curve for all time — a smoothly increasing, ever-flattening (but never truly flat) log-log curve, since $W(u)\to\ln(2.25Tt/r^2S)$ grows only logarithmically with time at late $u$.

(ii) A leaky confined aquifer follows the same early-time Theis curve, then bends noticeably below it and flattens toward a constant (steady-state) drawdown once enough water is being supplied through the aquitard to balance the pumping rate — the curve never steepens, only flattens.

(iii) A confined aquifer bounded by a low-permeability (no-flow) barrier follows the Theis curve until the cone of depression reaches the barrier, after which the curve steepens: the image-well effect (an added pumping well) doubles the late-time slope relative to the Theis line, exactly the mechanism used numerically in Q4(b).

(iv) A confined aquifer bounded by a constant-head recharge boundary also follows the Theis curve initially, then flattens even more sharply than the leaky case, curving over toward a fully horizontal asymptote (a true steady-state drawdown that stops increasing altogether) once the boundary is fully "felt" — the same effect used numerically in Q4(a).