Earth Surface
Processes, GEOL 4880 Humphrey, fall 2013, Homework #2
Note question 4 requires a field trip!
Leave yourself enough time to do the field work!
Please be careful… the homework gets posted just
before it is assigned. Links to homework
further on in the course lead to old homework.
I change some questions, and also change some of the data each year, so
you will be doing the wrong homework (the due date has to be this years for the
homework to be valid). The last part of
question 5 is hard.
Many of you are making simple mistakes, both algebraic and conceptual. Here is a summary table of most of the equations used so far, please use it and compare with your notes.
1
A soil with a porosity of 40% and a dry bulk density of 1600kg/cubic meter lies
2 meters deep over solid bedrock. Assume
the slope and the soil are horizontal and that
ground water motion is zero. After a
heavy rain the soil is saturated to the
surface.
a)
Calculate the wet bulk density of the soil
b)
Calculate the water pressure at the soil/bedrock interface
c) Calculate
the effective normal stress across the soil/bedrock interface
d)
What is the Hydraulic Gradient from the surface to the soil/bedrock interface
(a fancy way of asking the Total Head difference from surface to bedrock)
e)
Now change the problem a little, assume the water table is 0.5m below the
surface, and has a very slight gradient or slope of 0.5degrees. Use the Dupiut
assumption, and a saturated hydraulic conductivity of 5x10-3m/sec to
calculate the flux of water, per meter width in the flow direction (use Darcy’s
law, and to be precise, calculate the flux per meter width over the entire
depth of flow).
2
Let us look at a shallow soiled hillslope, with a
soil depth of 1m (slope perpendicular depth)
over solid bedrock. The slope is uniform, and the soil is essentially a slab
lying on a uniform tilted slope of bedrock. The slope is 30 degrees, and the
internal angle of friction of the soil is about 33 degrees. The soil is
homogeneous and is similar to the previous problem’s soil. The soil has a hydraulic conductivity of
0.5cm/sec, and the soil has no cohesion. It is
raining very hard and the soil is saturated to a depth of 1m (water table at
surface). Calculate:
a) the water pressure at the soil/bedrock interface [remember
that the water is now moving, since it is on a slope]
b) the driving stress (shear stress) at the soil/bedrock
interface (you need to add the water weight to the soil weight),
c) the resisting stresses from friction (note you will have to use the water
pressure from part ‘a’)
d) calculate the factor of safety for a failure along the soil/bedrock
interface
e) compare to the DRY factor of safety.
f)
Calculate the flux of water, per meter width of hillslope
(use Darcy’s law).
g)
What is the vertical soil depth?
3 Turbidity, or the opacity of water is often used as a rough
estimate of the amount of sediment in rivers. If the sediment size is fairly
constant with time this works well, with some initial calibration. However the
turbidity depends heavily on grain size and if the grain size varies, estimates
of sediment transport based on turbidity can have large errors. Investigate this problem by looking at the
turbidity created by two different crushed rock samples mixed into water.
a)
Calculate the amount (volume) of crushed rock that is needs to be suspended in
water to make a 1 meter cube of water 10% opaque, (in other words: so that 10%
of the light hits a particle while traveling through the 1 meter cube of
water). Use two different grain sizes, fine silt, and coarse sand.
(hint: it is x-section area of each grain that blocks
light, but the mass is due to the volume.
Make the simplifying assumption that no rock particles hide behind each
other.).
4 Time for a field trip:
a)
You
need to go down to the Laramie river (the Greenbelt is a nice place)
b)
Estimate
the water discharge. Hint: a reasonable estimate is just to multiply width x
depth x velocity. Pages 391+ in your
book (A&A) suggest
several methods, however your best bet is throwing several sticks
in the water to get an average velocity.
Knowing the length of your pace is useful to measure the width. You can use triangulation, or use a bridge to
get width. Depth may be the most
difficult, but you can actually see the bed in most places, so at the very
least, estimate.
c)
Estimate the discharge at another location, or compare your answer
to another student’s who did a completely independent estimate.
d)
Why
are the 2 answers different? Should they
be?
e)
Try
to list the 3 largest potential errors in your estimate, in order of size.
f)
The
USGS no longer monitors water discharge at Laramie, as far as I can tell. There is data from Bosler
in a Water Report to the state for data up till 2001 (on the web). There are (or at least were) gauging sites at
Woods Landing and at Bosler. See if you can locate any data, either
historical or current, for the day you went to the river (or for the time of
year that you went), and compare to your estimate of discharge. If you find current data, please record the
source. [note that
5 (part d is very difficult to do
correctly, but everybody should try part a,b,c:
HINT, it does not need any factor of
safety or other complex calculation(!), clear thinking is required, but very
little work)
a)
How
much does the block lengthen each day (and contract each night)?
b)
Calculate
the rate of motion, in meters per year of the slab, resulting from the diurnal
thermal cycle.
c)
Does
the angle of the slope affect the motion?
d)
(VERY
hard) Assume the slab touches the underlying rock along its entire length, not
just at the ends, now how fast does it move? A complete answer would give
a definite speed, however, a description of the details of motion is probably
sufficient, but such a description should include a comment on the speed
dependence on the slope angle, which is different from the answer to part c.