Question 9 of 21: Question 3, Part 3: Hydraulic Gradient and Darcy Flow Calculations

Question 3, Part 3: Hydraulic Gradient and Darcy Flow Calculations (8 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.

Given.

Find. (a) The hydraulic gradient between Points 1 and 2; (b) the volume of water discharged from the sample in 5 minutes.

Approach. (a) Gradient is simply elevation difference over map distance. (b) Use Darcy's law $Q = KiA$ with the sample's full cross-sectional area (the specific discharge $v=Ki$ is a bulk, apparent velocity already referenced to the total area, not just the pore area), then multiply by elapsed time for the volume.

1. (a) Hydraulic gradient. $i = \dfrac{\Delta h}{L} = \dfrac{270 - 255}{1500} = \boxed{0.01}$ (dimensionless) – the water table (or piezometric surface) rises 0.01 m for every 1 m travelled horizontally between the two points.
2. (b) Cross-sectional area of the sample. $A = \pi (d/2)^2 = \pi(0.05\text{ m})^2 = \boxed{7.854\times10^{-3}\text{ m}^2}$.
3. (b) Darcy (specific-discharge) velocity. $v = Ki = (5\times10^{-5})(0.05) = \boxed{2.5\times10^{-6}\text{ m/s}}$.
4. (b) Volumetric flow rate and total volume. $Q = vA = (2.5\times10^{-6}) (7.854\times10^{-3}) = 1.9635\times10^{-8}\text{ m}^3/\text{s}$. Over $t=300$ s: $V = Qt = (1.9635\times10^{-8})(300) = 5.89\times10^{-6}\text{ m}^3 = \boxed{5.89\text{ mL}}$.

The porosity ($n=0.45$) given in the problem is a distractor for THIS particular question: Darcy's law already expresses discharge through the full (gross) cross-sectional area of the sample, so the total volumetric outflow does not require porosity at all. Porosity would only be needed to convert the Darcy (specific-discharge) velocity into the true, faster seepage (average linear pore) velocity $v_s = v/n$ – relevant for travel-time questions, not for the bulk volumetric flow asked for here.