Question 3 of 5: Confined-Aquifer Drawdown Near a Recharge Boundary, Leaky-Aquitard Response, and Exploration Methods

Question 3: Confined-Aquifer Drawdown Near a Recharge Boundary, Leaky-Aquitard Response, and Exploration Methods (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. (a)/(b) Confined aquifer thickness $b=100$ m, $S_s=10^{-6}\ \text{m}^{-1}$, $K=10^{-3}$ cm/sec, aquitard thickness $b'=9$ m, pumping rate $Q=24$ L/s, observation well 150 m north of the pumping well. (a) constant-head (recharge) boundary running N–S, 150 m due east of both wells; aquitard impermeable; drawdown wanted 24 h after pumping starts. (b) aquifer instead infinite in areal extent, aquitard hydraulic conductivity $K'=10^{-6}$ cm/sec, negligible aquitard storage.

Find. (a) drawdown at the observation well after 24 h, using image-well theory for the recharge boundary. (b) drawdown at the same observation well and elapsed time if the aquifer is unbounded but leaky, plus a qualitative statement of the effect of significant aquitard storage. (c) two advantages and two disadvantages each of geophysics, remote sensing, and borehole investigations for groundwater exploration.

Approach. Part (a) places an image recharge (injection) well at the mirror position of the real well across the constant-head boundary and superposes two Theis drawdowns (pumping minus injection). Part (b) instead treats the aquifer as infinite but leaky, using the Hantush-Jacob leaky well function $W(u,r/B)$, evaluated here by direct numerical integration rather than table lookup. Part (c) is a qualitative comparison of exploration methods.

1. Part (a) — aquifer properties and image-well geometry. $T=Kb=(10^{-3}/100)(100)=\boxed{1.0\times10^{-3}\ \text{m}^2\text{/s}}$, $S=S_sb=(10^{-6})(100)=\boxed{1.0\times10^{-4}}$. Placing the pumping well at the origin, the observation well is at $r_1=150$ m north; the boundary runs N–S 150 m east, so its mirror image well (injection, $-Q$ in the superposition sense) sits 300 m east of the pumping well, giving $r_2=\sqrt{300^2+150^2}=\boxed{335.4\ \text{m}}$ from the observation well.
2. Superposed Theis drawdown. At $t=24\ \text{h}=86{,}400$ s, $u_1=r_1^2S/(4Tt)=0.006510$ and $u_2=r_2^2S/(4Tt)=0.032552$, so: $$s=\frac{Q}{4\pi T}\big[W(u_1)-W(u_2)\big]=\frac{0.024}{4\pi(1.0\times10^{-3})}\big[W(0.006510)-W(0.032552)\big]=\boxed{3.02\ \text{m}}.$$ The image (recharge) well's rise partially offsets the real well's drawdown, so this is smaller than an ordinary Theis drawdown for an unbounded aquifer at the same $r,t$.
3. Part (b) — leaky-aquifer parameters. With the aquitard now leaky ($K'=10^{-6}$ cm/s $=10^{-8}$ m/s) instead of impermeable: leakage factor $$B=\sqrt{\frac{Tb'}{K'}}=\sqrt{\frac{(1.0\times10^{-3})(9)}{10^{-8}}}=\boxed{948.7\ \text{m}}.$$ At $r=150$ m, $t=86{,}400$ s: $u=r^2S/(4Tt)=0.006510$ (same $u$ as before, since $T,S,r,t$ are unchanged) and $r/B=150/948.7=0.1581$.
4. Hantush-Jacob drawdown. Evaluating $W(u,r/B)$ by numerical integration of $W(u,r/B)=\int_u^\infty y^{-1}\exp[-y-(r/B)^2/4y]\,dy$: $$s=\frac{Q}{4\pi T}W(u,r/B)=\frac{0.024}{4\pi(1.0\times10^{-3})}(3.724)=\boxed{7.11\ \text{m}}.$$ This is larger than the recharge-boundary case in part (a) (a constant-head boundary just 150 m away is a much stronger, immediate source than diffuse leakage spread across the whole aquifer), but smaller than a purely non-leaky, unbounded Theis drawdown at the same $r,t$ ($\approx8.52$ m) — consistent with leakage supplying part of the pumped water.
5. Effect of significant aquitard storage. The no-storage Hantush-Jacob curve used above assumes leakage responds instantaneously to the drawdown in the aquifer below. If the aquitard instead has significant storage of its own, the aquitard releases water from its own elastic storage as the head in it declines. This is an ADDITIONAL source on top of the water transmitted through from the overlying bed, and at early-to-intermediate time the head disturbance has only penetrated the lower part of the aquitard, so the vertical gradient (and hence the flux) at its base is steeper than the linear $s/b'$ that the no-storage model assumes. Both effects feed the aquifer more water, so the drawdown would be SMALLER than the 7.11 m computed above (Hantush, 1960: the storage-inclusive function $H(u,\beta)$ lies below the no-storage curve), with the two solutions converging at late time as the aquitard storage is exhausted. For illustration only, a compressible clay aquitard with $S_s'=10^{-3}\ \text{m}^{-1}$ ($S'=9\times10^{-3}$, $\beta=\tfrac{r}{4}\sqrt{K'S'/(b'TS)}=0.119$, early-time formula valid here since $t<b'S'/10K'$) gives about 3.9 m instead of 7.11 m.

Part (c). Geophysics (e.g. resistivity, seismic refraction, gravity/EM surveys) — advantages: (1) covers large areas quickly and non-invasively, mapping subsurface geometry (aquifer depth/thickness, salt-water interfaces) between sparse boreholes; (2) relatively low cost per unit area compared with drilling. Disadvantages: (1) results are indirect and non-unique — a given resistivity or velocity profile can correspond to more than one geological interpretation, requiring ground-truthing; (2) resolution and depth of investigation are limited by noise, terrain, and the physical contrast between the target and its surroundings. Remote sensing (satellite/aerial imagery, thermal IR, InSAR) — advantages: (1) extremely broad regional coverage, useful for identifying lineaments, springs, vegetation anomalies, and land-use change tied to groundwater; (2) repeatable over time at low marginal cost, supporting change detection (e.g. land subsidence from aquifer compaction). Disadvantages: (1) only detects surface or near-surface expressions of groundwater conditions, not the aquifer itself; (2) interpretation can be confounded by vegetation cover, cloud cover, or surface conditions unrelated to the aquifer. Borehole investigations (drilling, logging, pumping tests) — advantages: (1) direct, unambiguous measurement of lithology, water levels, and aquifer hydraulic properties ($T$, $S$, $K$) at the specific location; (2) allows water-quality sampling and long-term monitoring instruments to be installed. Disadvantages: (1) high cost per data point and strictly point-scale information, requiring many holes to characterize a heterogeneous aquifer; (2) slow, and can locally disturb the aquifer (e.g. mixing water from different depths) if not properly constructed and sealed.