CE 473/573 Groundwater
Fall 2011
Comments on homework 6
39. The quick way to do part c is to recall the diffusion estimate for the time scale:
t ∼ length2 /diffusivity. In this case, the “diffusivity” is T /S. Therefore, t1 /t2 =
(x1 /x2 )2 = 1/4.
40. All groups approached this problem correctly.
41. All but one group assumed the peak in time to occur at t = ta = x/v . The peak
in space occurs at x = v t, but the peak in time occurs before t = ta because of
dispersion. I plotted C vs. t and found the peak, but one could also solve ∂C/∂t = 0
for time. The result is
1
x −2 2
−1
1+P
tp =
−P
,
v
is the Péclet number.
where P = v x/DL
43. Most groups had most of the details computed correctly. A few groups had trouble—
some more substantial than others–with the ERFC function in Excel. Remember
some key values: erfc(−∞) = 2, erfc(0) = 1, and erfc(∞) = 0. As time becomes
large, the second term in the sum of our full solution should approach zero, while
the argument of the erfc in the first term becomes large and negative, thus making
that erfc approach 2. While I estimated the dispersivity using Lpa , the steady plume
length ignoring dispersion, Tim and Ryan used an iterative approach to recompute
the dispersivity as the estimate of the plume length changed. My answers are shown
below; plume lengths are in meters, and all values are steady after 50 years.
Compound
Benzene
Toluene
Ethylbenzene
p-xylene
Lpa (m)
1392.2
16.4
44.4
23.1
Lpd (m)
1498.7
20.0
39.8
24.9
1 year
342.4
20.0
53.2
22.2
5 years
1000.2
20.0
53.2
24.9
10 years
1446.7
20.0
53.2
24.9
50 years
1498.7
20.0
53.2
24.9