18-Env-A3 Geotechnical and Hydrogeological Engineering · December 2017
Question 4 of 6: Confined Aquifer Pump Test — Hydraulic Conductivity and Tracer Travel Time
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exams — December 2017 — 04-Env-A3 / Geotechnical & Hydrogeological Engineering. 3 hours duration; open book exam, any non-communicating calculator permitted. FIVE (5) questions constitute a complete exam paper (the first five as they appear in the answer book are marked, 20 marks each, 100 marks total); all six printed questions are solved below for completeness.
Reference texts. Braja M. Das, Principles of Geotechnical Engineering (9th ed.) — weight–volume relations, compaction, seepage/flow nets, and consolidation chapters; Craig & Knappett, Craig's Soil Mechanics (8th ed.) — cross-reference for flow-net theory and finite-difference seepage; Freeze & Cherry, Groundwater (1979) — Darcy's law, the Thiem confined-flow equation, and radial travel time.
Question 4: Confined Aquifer Pump Test — Hydraulic Conductivity and Tracer Travel Time (20 marks)
Aquifer thickness (confined, beneath 11 m aquitard)
$b$
19 m
Static depth to water (all wells)
—
16.9 m below grade
Radial distances (pumping well → A → B → C)
$r_A,r_B,r_C$
3 m, 30 m, 55 m
Drawdown at A, B, C
$s_A,s_B,s_C$
6.6 m, 1.2 m, 0 m
Find. (a) hydraulic conductivity of the aquifer; (b) travel time for a conservative tracer from well A to the pumping well.
Approach. The aquitard above the aquifer makes this a confined (Thiem) radial-flow problem; with three observation wells, "best available data" means fitting the confined-flow head-vs-$\ln r$ line through all three by least squares rather than using only one well pair. Travel time then follows by integrating the radially-varying Darcy velocity from $r_A$ to the well.
Fig. Q4 — confined aquifer pump test: pumping well plus observation wells A, B, C at r = 3, 30, 55 m.
Part (a) — best-fit hydraulic conductivity. For confined radial flow, head varies linearly in $\ln r$: $h(r)=h_0+m\ln r$, with $m=Q/(2\pi k b)$. Using the three observation-well drawdowns (pumping-well drawdown is excluded — well losses make it unreliable for $k$) and a least-squares fit through $(\ln r_A,-s_A)$, $(\ln r_B,-s_B)$, $(\ln r_C,-s_C)$ gives slope $m=2.290\ \text{m}$, so
$$k=\frac{Q}{2\pi b\,m}=\frac{0.020}{2\pi\times19\times2.290}=\boxed{7.32\times10^{-5}\ \text{m/s}}\ \ (6.32\ \text{m/day}),$$
consistent with the individual pairwise Thiem checks ($k_{AB}=7.14\times10^{-5}$, $k_{AC}=7.38\times10^{-5}$, $k_{BC}=8.46\times10^{-5}$ m/s), and with the "medium-to-coarse silty sand" texture given.
Part (b) — travel time from well A. At radius $r$ the specific discharge through the cylindrical flow surface is $v(r)=Q/(2\pi r b)$; the actual (seepage) velocity is $v(r)/n_e$. Integrating $dt=n_e\,dr/v(r)=\dfrac{2\pi b\,n_e}{Q}r\,dr$ from the pumping-well radius $r_w\approx0.15\ \text{m}$ (typical small test-well radius, not given — its effect below is negligible) to $r_A=3\ \text{m}$, with an assumed effective porosity $n_e=0.30$ (typical clean-to-silty sand, Freeze & Cherry Table 2.4):
$$t=\frac{\pi n_e b\,(r_A^2-r_w^2)}{Q}=\frac{\pi\times0.30\times19\times(3.0^2-0.15^2)}{0.020}=8038\ \text{s}=\boxed{2.23\ \text{hours}}.$$
Check: effective porosity $n_e=0.30$ and pumping-well radius $r_w=0.15$ m are not given in the source and are assumed (typical values for a medium-to-coarse silty sand and a small-diameter test well); travel time scales linearly with $n_e$, and dropping $r_w$ entirely changes $t$ by only 0.25%, so the $r_w$ assumption is immaterial while the $n_e$ assumption should be read as a transparent, order-of-magnitude-safe estimate.
