← Module 3 · OpenHydroQual Exercise 2

Darcy's column in a computer

About 1.5 hours · submit a short report (PDF) to Brightspace → Assignments

What you will do

You will run a numerical version of Darcy's experiment: a horizontal sand column cut into ten cells, between two reservoirs held at fixed water levels. You will check that the model reproduces Darcy's law, see which inputs change the flow and which change the heads, and then put a low-conductivity layer in the column and test the series formula for \(K_v\). The small disagreement you will find there is a first lesson in how numerical models work.

The model:
  • Cell1 … Cell10 (Groundwater cell blocks): a column 1.0 m long, 0.20 m wide and 0.25 m high (cross-section \(A = 0.05\) m²), cut into ten cells 0.10 m long. Porosity 0.35. All cells are saturated.
  • Upstream and Downstream (fixed_head blocks): reservoirs held at heads of 1.50 m and 1.00 m. Upstream is given a very large storage so it never runs dry; Downstream's storage starts at zero and accumulates the water that passes through.
  • Links: groundwater_link between neighbouring cells (length 0.10 m, area 0.05 m²) and groundwater_to_fixedhead at the two ends (length 0.05 m: from the centre of the end cell to the column face). Each link computes a Darcy flow, \(Q = K A\,\Delta h / L\), from the heads of the two objects it joins.

The model runs for one day, long enough to reach steady state (the flow becomes constant).

Step 0 · Download

darcy_column.ohq darcy_column_layered.ohq

Part 1 · Predict with Darcy's law (before running)

  1. Open darcy_column.ohq (File → Open). Click the cells, links and reservoirs and record: K of the cells, cell length, link lengths, cross-sectional area, and the two reservoir heads.
  2. By hand, compute: (a) the hydraulic gradient, (b) the steady discharge \(Q\) (m³/day), (c) the Darcy flux \(q\), (d) the average pore velocity \(v\) and the time for water to cross the column.
  3. Predict the head at the centre of each cell (\(x = 0.05, 0.15, \dots, 0.95\) m). Hint: in a homogeneous column the head varies linearly between the two faces.

Part 2 · Run and compare

  1. Click Run. Right-click a few cells and choose Results → head; right-click the Outlet link and choose Results → flow. Use Copy Curve to bring the numbers into Excel.
  2. Make a table of the ten steady heads and plot head against distance, together with your Part 1 prediction.
  3. Compare the model's steady outlet flow with your hand-calculated \(Q\). Then look at the Inlet flow: why does OpenHydroQual report it with a negative sign? (Look at which way the link is drawn: from Cell1 to Upstream.)
  4. How long does the model take to reach steady state? Look at the early part of the outlet flow curve.

Part 3 · What controls Q, and what controls h?

  1. Predict first, then test each change on its own (change it back afterwards):
    1. Raise the Upstream head from 1.50 to 2.00 m.
    2. Halve the K of all ten cells (10 → 5 m/day).
    For each, record the new steady \(Q\) and the head profile.
  2. One of the two changes leaves the head profile exactly the same while changing \(Q\). Which one, and why? Write Darcy's law for two neighbouring links to explain it.

Part 4 · A layered column

  1. Open darcy_column_layered.ohq. Cells 1–5 have \(K = 10\) m/day, cells 6–10 have \(K = 1\) m/day. Everything else is the same.
  2. By hand, with the series formula: compute the equivalent \(K_v\) of the column (two layers, 0.5 m each), the steady \(Q\), and the head at the interface between the two layers (\(x = 0.5\) m).
  3. Run the model. Plot the head profile. Where does most of the head drop happen? Compare your interface head with the model.
  4. Compare the model \(Q\) with your hand calculation. You should find the model about 7% higher.

Part 5 · Why is the model 7% off?

Each link computes \(Q = K_{link} A\,\Delta h / L\). A link joining two cells with different K needs one value for \(K_{link}\). OpenHydroQual's groundwater_link uses the arithmetic mean of the two cells' K values.

  1. Think of each link as a resistance \(R = L/(K_{link}A)\) in series, so \(Q = \Delta h_{total}/\sum R\). List the 11 links (inlet, nine internal, outlet) with their \(L\) and \(K_{link}\). The fixed-head links use the K of the cell they start from. Compute \(\sum R\) and \(Q\). Do you get the model's value?
  2. Repeat with the harmonic mean, \(K_{link} = 2K_1K_2/(K_1+K_2)\), for the one link that crosses the interface. Which averaging reproduces the series formula exactly? Why is it the physically correct choice for flow across a boundary? (Think about Animation 2 in the module.)
  3. If the column were divided into 100 cells instead of 10, would the error grow or shrink? Explain without running anything.
  4. Optional: swap the layers (cells 1–5 at 1 m/day, cells 6–10 at 10 m/day). Does \(Q\) change? Does the head profile?

What to submit

  • Your Part 1 and Part 4 hand calculations.
  • One plot with the head profiles of the base case, the two Part 3 cases, and the layered case, plus your predictions.
  • A table of steady \(Q\): hand calculation vs. model, for every case.
  • Answers to Parts 2–5 in a few sentences each, with your resistance table from Part 5.
  • At most 3 pages, as a single PDF, uploaded to Brightspace → Assignments → OHQ Exercise 2.