18-Env-A3 Geotechnical and Hydrogeological Engineering · May 2013
Question 6 of 6: Confined Aquifer — Flow Rate and Tracer Travel Time
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exams — May 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 slope-stability chapters; Craig & Knappett, Craig's Soil Mechanics (8th ed.) — cross-reference for the variable-head permeability test and Taylor's stability-number chart; Freeze & Cherry, Groundwater (1979) — Darcy's law, confined-aquifer flow and seepage velocity.
Question 6: Confined Aquifer — Flow Rate and Tracer Travel Time (20 marks)
Confined aquifer cross-section — flow driven from the higher-head well toward the lower-head well.
Given.
Given data
Quantity
Symbol
Value
Porosity
$n$
0.15
Aquifer thickness
$b$
30 m
Well separation
$\Delta x$
1000 m
Potentiometric head, high well
$h_2$
56.90 m
Potentiometric head, low well
$h_1$
52.35 m
Horizontal hydraulic conductivity
$k$
25 m/day
Find. (a) flow rate per unit width $q$, specific discharge (Darcy velocity) $v$, and average linear (seepage) velocity $v_s$; (b) tracer travel time between the wells, and the assumptions this requires.
Approach. Flow runs from the high-head well toward the low-head well; find the hydraulic gradient from the two heads, apply Darcy's law for specific discharge and flow per unit width, then divide by porosity to get the average LINEAR velocity that actually governs how fast a tracer particle travels.
Part (a) — hydraulic gradient and Darcy quantities.
$$i=\frac{h_2-h_1}{\Delta x}=\frac{56.90-52.35}{1000}=0.00455.$$
$$v=ki=(25)(0.00455)=\boxed{0.1138\ \text{m/day (specific discharge)}}.$$
$$q=vb=(0.1138)(30)=\boxed{3.41\ \text{m}^3/\text{day per metre of aquifer width}}.$$
Average linear (seepage) velocity. Darcy's specific discharge is a flux averaged over the FULL cross-sectional area, including solid grains; the actual pore water travels faster, through only the void fraction $n$:
$$v_s=\frac{v}{n}=\frac{0.1138}{0.15}=\boxed{0.758\ \text{m/day}}.$$
Part (b) — tracer travel time. A conservative tracer moving with the average linear velocity covers the 1000 m well spacing in
$$t=\frac{\Delta x}{v_s}=\frac{1000}{0.758}=\boxed{1319\ \text{days}\ (\approx3.61\ \text{years})}.$$
Assumptions required for this travel-time estimate: (1) the aquifer is homogeneous and isotropic between the wells, with flow strictly one-dimensional along the well-to-well line; (2) steady-state flow (heads not changing with time); (3) the tracer is conservative — no sorption, retardation, decay, or density difference from the ambient groundwater; (4) advection dominates (hydrodynamic dispersion, which would spread the tracer front around this average arrival time, is neglected); (5) Darcy's law applies (laminar, low-Reynolds-number flow, which is essentially always true in a sand/sandstone aquifer at these gradients).
