18-Env-A3 Geotechnical and Hydrogeological Engineering · December 2013
Question 3 of 6: Confined Aquifer — Steady Radial Flow to a Pumping Well
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exams — December 2013 — 04-Env-A3 / Geotechnical & Hydrogeological Engineering. 3 hours duration; open book exam, any non-communicating calculator permitted. The first five questions as they appear in the answer book are marked (20 marks each, 100 marks total); all six are solved below for completeness.
Reference texts. Braja M. Das, Principles of Geotechnical Engineering (9th ed.) — unit weight/compaction relations, permeability and seepage/flow nets, lateral earth pressure and retaining-wall stability chapters; Craig & Knappett, Craig's Soil Mechanics (8th ed.) — cross-reference for the falling-head permeability test and flow-net construction under a cutoff wall; Freeze & Cherry, Groundwater (1979) — Darcy's law, confined-aquifer (Thiem) flow and seepage velocity.
Question 3: Confined Aquifer — Steady Radial Flow to a Pumping Well (20 marks)
Figure 2 (idealised) — confined-aquifer cross-section: steady axisymmetric radial flow converges on the pumping well; the cone of depression is lower (h₁) at the nearer observation well and higher (h₂) at the farther one.
Given.
Given data
Quantity
Symbol
Value
Aquifer thickness
$b$
30 m
Porosity
$n$
0.15
Hydraulic conductivity
$K$
25 m/day
Radius, well 1
$r_1$
1000 m
Radius, well 2
$r_2$
2000 m
Head, well 1
$h_1$
52.35 m
Head, well 2
$h_2$
56.90 m
Find. (a) pumping rate $Q_w$ and Darcy flux $q$ at $r=2000$ m; (b) tracer travel time between the two observation wells.
Approach. Use the Thiem equation for steady confined radial flow to back out $Q_w$ from the two head observations, then Darcy's law (flux = $Q_w$ divided by the cylindrical flow area at $r$) for part of (a); for (b), integrate the SEEPAGE (linear, not Darcy) velocity – which varies with $r$ because the flow area grows with radius even though $Q_w$ is constant – along the radial path between the wells.
Part (a) — transmissivity and pumping rate (Thiem equation).
$$T=Kb=(25)(30)=750\ \text{m}^2/\text{day}.$$
$$Q_w=\frac{2\pi T (h_2-h_1)}{\ln(r_2/r_1)}=\frac{2\pi(750)(56.90-52.35)}{\ln(2000/1000)}=\boxed{30{,}933\ \text{m}^3/\text{day}}\ \left(\approx21.5\ \text{m}^3/\text{min},\ \approx358\ \text{L/s}\right).$$
Darcy flux at $r=2000$ m. By continuity, the same $Q_w$ crosses every concentric cylindrical surface of area $2\pi r b$:
$$q(r{=}2000)=\frac{Q_w}{2\pi r_2 b}=\frac{30{,}933}{2\pi(2000)(30)}=\boxed{0.0821\ \text{m/day}}.$$
Part (b) — seepage velocity as a function of radius. The linear (seepage) velocity is the Darcy flux divided by porosity, $v(r)=\dfrac{Q_w}{2\pi r b\,n}$, which is largest near the well (small $r$) and smallest at the outer well — unlike straight-line flow, the travel time can NOT be found from a single average velocity. A tracer particle satisfies $dr/dt=-v(r)$ (moving inward, toward the pump), so
$$t=\int_{r_1}^{r_2}\frac{dr}{v(r)}=\int_{r_1}^{r_2}\frac{2\pi b\,n\,r}{Q_w}\,dr=\frac{\pi b\,n\left(r_2^2-r_1^2\right)}{Q_w}.$$
Check: assumes a fully confined, homogeneous, isotropic aquifer of constant thickness under Dupuit-Thiem steady-state radial flow (no significant recharge/boundary effects between the wells), and that the effective (transport) porosity for the conservative tracer equals the stated total porosity $n=0.15$.
