Calculation and Fill in the Blank: 04-BS-14, May 2015, Q3

Question 3: Calculation and Fill in the Blank (20 marks)

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.

1) & 2) Advective travel time and piezometer pressure

Given.

Quantity	Value
Plan distance landfill–stream, L	1000 m
Water-table elevation under landfill	210 m
Water-table elevation at stream	203 m
Hydraulic conductivity, K	3×10^-5 m/s
Porosity, n	0.27
Piezometer intake elevation	205 m

Find. (1) The advective travel time for a solute to move with the groundwater from beneath the landfill to the stream. (2) The pressure read at the bottom of the standpipe piezometer.

Approach. Compute the hydraulic gradient from the two water-table elevations, use Darcy's law for the specific discharge, convert to the seepage (linear, pore) velocity that actually carries a dissolved solute under pure advection, then divide distance by that velocity; the piezometer pressure follows directly from the height of water standing above the intake.

Hydraulic gradient. The head drops from 210 m to 203 m over the 1000 m flow path (perpendicular to the stream, as stated): $$ i = \frac{\partial h}{L} = \frac{210 - 203}{1000} = 0.007 $$
Darcy (specific discharge) velocity. $$ v = -Ki = (3\times10^{-5}\ \text{m/s})(0.007) = 2.1\times10^{-7}\ \text{m/s} $$ This is the bulk (Darcy) flux through the full cross-section, not the actual speed of water in the pores.
Seepage velocity. Under pure advection the solute travels at the average linear (seepage) velocity of the water in the pores, not the Darcy velocity: $$ v_s = \frac{v}{n} = \frac{2.1\times10^{-7}}{0.27} = 7.78\times10^{-7}\ \text{m/s} $$
Travel time. $$ t = \frac{L}{v_s} = \frac{1000\ \text{m}}{7.78\times10^{-7}\ \text{m/s}} = 1.286\times10^{9}\ \text{s} $$ Converting to years (1 yr ≈ 3.156×10⁷ s): $$ \boxed{t \approx 40.7\ \text{years}} $$
Piezometer pressure. The standpipe piezometer's water level rises to the local piezometric (water-table) elevation of 210 m at the landfill; its intake sits at 205 m, so the pressure head is the difference between these two elevations: $$ h_p = z_{water\,table} - z_{intake} = 210 - 205 = 5\ \text{m} $$ Converting to pressure with the unit weight of water γ_w = 9.81 kN/m³: $$ p = \gamma_w h_p = (9.81\ \text{kN/m}^3)(5\ \text{m}) = \boxed{49.05\ \text{kPa}} $$

Result	Value
Hydraulic gradient, i	0.007
Darcy velocity, v	2.1×10^-7 m/s
Seepage velocity, v_s	7.78×10^-7 m/s
Advective travel time	≈ 40.7 years
Piezometer pressure at 205 m	49.05 kPa

3) Fill in the blanks – Figure Q3-1, Water Related Features

[Figure not reproduced: Water related features block diagram. See the official exam paper or the cited reference text.]

Figure Q3-1 from the source exam: hillslope block diagram with two unlabelled boxes on the hillslope (well/pipe callouts) and two unlabelled "(Zone)" boxes at the right, plus the labelled "Unsuccessful well".

Reading the diagram left to right / top to bottom, the six blank boxes correspond to the standard vadose-to-saturated-zone terminology illustrated by the hillside well cross-section:

Box (left to right, top to bottom)	Label
Upper hillslope box (near the recharge area / house)	Recharge area (infiltration zone)
Lower-left hillslope box	Water table (unconfined)
Box near the base, right of the successful well	Zone of saturation (saturated zone)
Bottom box, base of the block	Impermeable layer (aquitard/bedrock)
Upper right "(Zone)" callout (above the unsuccessful well tip)	Zone of aeration (unsaturated/vadose zone)
Lower right "(Zone)" callout (below the water table, at the successful well)	Zone of saturation (perennial water-bearing zone)

The "Unsuccessful well" shown in the figure was drilled too shallow, terminating in the zone of aeration above the water table (it never reaches saturated, water-yielding material), while a well penetrating into the zone of saturation below the water table will yield water reliably.

Check: the piezometric level at the landfill is taken as the stated water-table elevation of 210 m (the piezometer is described as "installed through the landfill", i.e. at the landfill location).

Topic: Darcy flow, seepage velocity and advective solute transport; standpipe piezometer pressure; vadose/saturated zone terminology.

Key relations: Darcy's law v = -Ki; seepage velocity v_s = v/n; travel time t = L/v_s; piezometer pressure p = γ_w(z_{water table} - z_intake); zone of aeration (above the water table) vs. zone of saturation (below it).

Why this works: the Darcy (specific discharge) velocity describes bulk flow through the full soil cross-section including grains, but a dissolved solute physically travels only through the interconnected pore space, so its true average speed is the Darcy velocity divided by porosity (n < 1, so v_s > v always); the piezometer reads the static pressure of the water column standing above its intake at the local piezometric surface.

Common pitfall: using the Darcy velocity directly for solute travel time (over-predicts travel time by a factor of 1/n ≈ 3.7× here); forgetting that porosity, not the void ratio, is the correct divisor for seepage velocity.

Source: Freeze & Cherry, Groundwater, Ch. 2 (Darcy's law, seepage velocity) and Ch. 6 (contaminant transport, advection).

Question 3 of 7: Calculation and Fill in the Blank

Question 3: Calculation and Fill in the Blank (20 marks)

1) & 2) Advective travel time and piezometer pressure

3) Fill in the blanks – Figure Q3-1, Water Related Features