CE 473/573 Groundwater
Fall 2008
Homework 5
Due Wednesday October 29
30. Investigate the accuracy of different 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., ∆x = 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?
31. Show that the centered difference formula for the second derivative is O((∆x)2 ).
*32. The confined aquifer shown from above in the figure below has fixed head on the
left and right boundaries and no flow through the upper and lower boundaries. The
thickness is 12 m. The hydraulic conductivity changes abruptly (across the slanted
line) from 1 m/d to 0.4 m/d.
a. Plot contours of piezometric head at 0.1 m intervals.
b. Explain physical reasons for the features in the contour map (e.g., spacing, alignment, etc.)
c. Compute the flow across the right boundary.
d. Repeat parts a and c with Modflow and explain any differences.
2000 m
No-flow boundary
Fixed head: h = 3 m
2000 m
Fixed head: h = 4 m
500 m
500 m
No-flow boundary
*33. Suppose an unconfined aquifer has the same configuration and conductivity as the
aquifer in problem 32. The recharge rate is 1 mm/d on a square region with a side
length of 500 m centered in the aquifer. The water table elevations from the bottom
of the aquifer are 4 m and 3 m for the left and right boundaries, respectively.
a. Plot contours of the water table elevation.
b. Plot the groundwater divide, if any, in the x-y plane.
*34. 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.
*35. 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.
36. Plot I0 (x), 10K0 (x), and K1 (x) for the range 0 ≤ x ≤ 5.