CE 473/573 Groundwater
Fall 2010
Homework 4
Due Wednesday October 27
22. Investigate the accuracy of finite difference approximations for the first derivative.
Consider f (x) = sin(x) from 0 ≤ x ≤ π.
a. Use a grid with 11 points (i.e., Δx = π/10), and make a single plot comparing
the exact first derivative, the forward difference approximation, the backward
difference approximation, and the centered difference approximation. Which is
most accurate?
b. Repeat part a with 31 points. What is the effect of increasing the number of
points?
23. Construct a numerical model of flow in a confined aquifer with thickness 10 m, length
1500 m, and hydraulic conductivity varying linearly from 1 to 5 m/d. The upgradient
head is 35 m, and the downgradient head is 34 m.
a. Plot the variation in piezometric head.
b. Compute the flow per unit width.
c. Convince me that your result in part a is correct.
d. Compute the effective conductivity.
*24. A pump test was conducted in a 25-m thick, confined aquifer in which the initial
piezometric surface was at elevation 15 m. The well was pumped at 30 L/s, and after
one day the piezometric levels at distances of 35, 75, 110, and 150 m from the well
were 14.1, 14.4, 14.7, and 14.9 m, respectively.
a. Estimate the transmissivity and the hydraulic conductivity of the aquifer.
b. Aquifers with transmissivity less than 10 m2 /d can supply only enough water for
domestic wells and other low-yield uses (Lehr et al. 1988; Chin 2000, p. 501).
How does this aquifer’s transmissivity compare to that value?
c. Estimate the radius of influence of the well.
*25. After you start working at Realenwelt Consulting, you are asked to construct a numerical model of flow in an agricultural field. The aquifer, which is next to a stream,
is unconfined and heterogeneous; the alluvium near the stream has a conductivity of
KA = 1 m/d and a effective porosity of nA = 0.3, while the soil farther from the
stream has a conductivity of Ks = 0.3 m/d and an effective porosity of ns = 0.25.
An old tile drain was installed 20 m away from the stream many years ago, but after
several years of drought, the field (but not the buffer strip) is irrigated at a rate of
5 mm/d.
a. At this irrigation rate, does any water enter the drain?
b. Fertilizer applied to the field might cause nutrients to enter the stream. The
nutrient concentration will depend on the time the water remains in the soil. Use
your numerical solution to compute the time to travel from either the drain or
a groundwater divide to the stream. (Hint: Apply the formula we have used
between cells in your model and add the times.)
13.5 m
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5 mm/d
Corn field
0.9 m
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Drain
Buffer
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Ks
ns
1.2 m
KA
nA
Aquifer bottom
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20 m
6m
26. Set up and run the sample problem for PMWIN. Follow the instructions in chapter 2
of Processing Modflow at least up to and including step 4. For this problem you will
be most concerned with the volume of water out of the region (Table 2.2).
a. Verify that you can reproduce the results in Table 2.2 on p. 18 of the documentation.
b. Explain as many numbers in the table as you can. Try turning off the well and
verify that the flow through the constant head boundary is correct.
*27. Consider a confined aquifer of thickness 5 m, length 600 m, width 400 m, and hydraulic conductivity 25 m/d. The western boundary has a head of 10 m, and the
eastern boundary has a head of 9 m. No flow occurs across the northern and southern
boundaries. A well in the center of the aquifer pumps at 30 m3 /d.
a. Solve this problem with PMWIN (Modflow) and obtain the water balance table
as in Table 2.2 of the PMWIN documentation.
b. Solve this problem in Excel and compute the flow across the eastern boundary.
Compare to the result from Modflow.
c. Plot contours of the piezometric head and streamlines and explain your plots.
*28. A pump test was conducted in an unconfined aquifer of saturated thickness 15 m (in
the absence of pumping). The well was pumped at 8000 m3 /d, and after one day the
drawdowns at 50 m and 100 m from the well were 0.42 m and 0.27 m, respectively.
a. Show that in steady state the hydraulic conductivity can be computed from
K=
1
Q0 ln(r2 /r1 )
,
π (H − s2 )2 − (H − s1 )2
where Q0 is the pumping rate, H is the saturated thickness in the absence of
pumping, and si is the drawdown at radial distance ri .
b. Compute the hydraulic conductivity for this aquifer.