Question 6 of 6: Discharge and Tracer Travel Time for a Well in an Unconfined Aquifer
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper format. 04-Agric-A2 Soil Physics & Mechanics,
National Exams May 2017 — a three-hour open-book examination;
any non-communicating calculator is permitted. The cover page states that five (5)
questions constitute a complete exam paper and that only the first five as they
appear in the answer book are marked, that each question is of equal value, and that
some questions require a written answer whose clarity and organization matter for
marks. All six printed questions are worked here, because the set is a study resource
rather than a timed attempt; on exam day a candidate submits only the first five, in
order.
Reference texts. B.M. Das, Principles of Geotechnical
Engineering, 9th ed. (weight-volume relationships, permeability, grain-size
analysis, USCS classification, compaction, slope stability, well hydraulics); R.F.
Craig, Craig's Soil Mechanics, 9th ed. (effective stress, seepage and flow
nets, shear strength); USDA NRCS National Engineering Handbook (compaction
and earthwork field practice).
Question 6: Discharge and Tracer Travel Time for a Well in an Unconfined Aquifer (20 marks)
Find. The maximum discharge for a 2 m maximum drawdown at the
well (a); the tracer travel time from r1 = 10 m to the pumping well at a
steady Q = 500 m³/d (b).
Figure 4 (schematic): the pumping well, two observation wells,
and the cone of depression in the unconfined aquifer.
Direct inspection of the printed diagram
(Fig. 4) shows this note was itself mistaken: both arrows originate at the pumping
well, and the shorter r1 arrow terminates at the closer well (head
h1) while the longer r2 arrow reaches the farther well (head
h2) — consistent with the problem text. r1 = 10 m
(closer) and r2 = 100 m (farther) are used throughout below.
Approach. Use the Dupuit–Thiem equation for steady radial
flow to a well fully penetrating an unconfined aquifer. For (a), apply it between the
well itself and the radius of influence (where drawdown vanishes). For (b), use the
same governing relation to find the saturated thickness h(r) at the FIXED pumping
rate of 500 m³/d, then integrate the radial seepage velocity from
r1 down to the well to get the travel time.
Static and pumped saturated thickness.
$$h_0=20-2=18\ \text{m (static)}, \qquad h_w=h_0-2=16\ \text{m (2 m drawdown at the well)}$$
a) Maximum discharge. Applying Dupuit–Thiem between the
well (rw, hw) and the radius of influence (R, h0):
$$Q_{\max}=\frac{\pi K\left(h_0^2-h_w^2\right)}{\ln(R/r_w)}
=\frac{\pi(20)(18^2-16^2)}{\ln(500/0.15)}=\frac{\pi(20)(68)}{8.111}
=\boxed{527\ \text{m}^3/\text{d}}$$
b) Saturated thickness at r1 and at the well, for Q = 500 m³/d.
Re-arranging the same Dupuit relation with h0 at R as the reference,
$$h(r)^2=h_0^2-\frac{Q}{\pi K}\ln\frac{R}{r}$$
$$h(r_1=10\,\text{m})=17.11\ \text{m}, \qquad h(r_w=0.15\,\text{m})=16.11\ \text{m}$$
The saturated thickness barely changes (18 → 16.1 m) across the whole flow
path, so the well's cone of depression is shallow relative to the aquifer here.
b) Travel time. The radial (seepage) velocity of a conservative
tracer at radius r is the Darcy flux divided by porosity,
vs(r) = Q/(2πr h(r) n); integrating dr/vs(r) from
the well out to r1 (evaluated numerically, since h(r) is not constant):
$$t=\int_{r_w}^{r_1}\frac{2\pi n\,r\,h(r)}{Q}\,dr=\boxed{3.74\ \text{days}}$$
