Analytical Paper 5
Explore the processes of groundwater flow in a shallow, unconfined aquifer with uniform porosity
and permeability.
Format: As usual, you should frame your answers to the questions listed below in a single, coherent
essay, with an introduction paragraph and a concluding paragraph. All symbols should be defined
and units or dimensions given for each.
1. Darcy’s Law. Present and describe Darcy’s law. Your description should:
• As always, define all mathematical symbols, along with their dimensions or units.
• Define hydraulic conductivity and permeability, and give the formal mathematical definition of conductivity (i.e., show how conductivity and permeability relate to one another).
• Summarize the factors that control permeability and conductivity.
• Define hydraulic head, elevation head, and pressure head.
• Give an example of a type of geological material that tends to have high permeability,
and one that tends to have low permeability.
2. Mass balance for an unconfined, shallow aquifer. Imagine that you have a shallow, unconfined aquifer in a permeable sandstone formation (labelled “sand layer” in the diagram
below). The sandstone sits on top of a horizontal, impermeable shale bed (labelled “impermeable layer”). Use the principle of mass conservation to derive an equation for the time
rate of change of water table height at a particular point in space. Assume that all flow occurs along the x axis, which runs from left to right, and that the control volume shown in
the sketch is W meters wide in the y direction (in and out of the page). The symbols in the
diagram are defined as follows, with dimensions given in square brackets:
H = height of water table in the cell [L]
@H
@t = rate at which water table height varies with time [L/T]
q = uh = volumetric flow rate of groundwater into or out of the control volume, per unit
width [L2/T]
u = discharge of groundwater per unit cross-sectional area (i.e., Darcian velocity) [L/T]
R = recharge rate from infiltrating rain and snowmelt [L/T]
ρ = density of water [M/L3]
φ = porosity of the sand layer [dimensionless]
Use this framework and the continuity of mass principle to derive an equation for @H=@t as
a function of q and R. Assume R, φ, and ρ are constant. (Hint: follow the general steps in a
2 Handout 5: Analytical Paper 5
mass balance that we went over in class. The pattern should be very similar to our hillslope
evolution and chemical diffusion case studies.)
IMPERMEABLE LAYER
SAND
LAYER
water table
H
∆x
q(x) q(x+∆x)
ground surface
R
recharge
3. Now combine your equation from the question above with Darcy’s law. For this problem,
you can use a simple version of Darcy’s law that is based on the Dupuit approximation. The
Dupuit approximation simply says that if you have a broad, shallow unconfined aquifer like
the one sketched below, and the slope of the water table is relatively small, then nearly all
the groundwater motion will be horizontal. Furthermore, the rate of horizontal groundwater
flow will depend on the slope of the water table, as follows:
| u = –K @x |
(1) |
@H
where u is as defined above, and K is hydraulic conductivity, which is constant in our sandstone aquifer.
Substitute this version of Darcy’s law (equation 1) and q = uH into the equation you derived
in the previous question, and show the resulting equation. (Note: you don’t need to solve
the equation for H; just find an equation for @H=@t, the rate of water table rise or fall, and
briefly describe it in words).
4. You have been given a MATLAB program that simulates the water table in our sandstone
aquifer. The boundary conditions are a seepage face on the right side (water flows out in a
spring at the base of a cliff), a groundwater “ridge” on the left (no water flows in or out on
that side), and infiltrating precipitation (a.k.a., recharge) from above. Use the program to
explore how the system works:
• Run the model until it reaches a steady state (water table stops changing over time).
Show a plot of (a) Darcian velocity versus distance, (b) unit discharge versus distance,
Handout 5: Analytical Paper 5 3
and (c) water table height (=aquifer thickness) versus distance. Note the shape of the
water table. Does it slope up or down to the right, and why? Is it planar, convex-up, or
concave-up, and why?
• Experiment with increasing and decreasing the hydraulic conductivity by changing the
K parameter in the code and running it to steady state again. Show a plot with higher
K, and one with lower K. How and why do changes in K influence the thickness of
the aquifer and the slope of the water table?
• Experiment with increasing and decreasing the recharge by changing the corresponding
parameter in the code and running it to steady state again. Show plots with higher and
lower recharge. How and why do changes in recharge influence the thickness of the
aquifer and the slope of the water table?
