Homework #3 GEOL 4880 Humphrey Fall 2022
Due Tuesday Sept 22nd.
Show your work. And please be neat! If you submit via email, please reduce your submission below 5MB (it clogs up my email)
You will have to be very careful of units etc. to come up with the correct answers. For this homework I will actually expect at least 2 correct digits in your answers to #1. Question 2 will require more work and your answers will only be approximate. Note the course web page has a page of equations if you want to check if your notes are correct.
1 Let us look at the hydrology of the shallow soiled hillslope (the slope failure from homework #2) near Centennial, with a soil depth of 2m (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 25 degrees, and the internal angle of friction of the soil is about 30 degrees. The soil is homogeneous and the bedrock has very low conductivity. The soil has a hydraulic conductivity of 10-6m/s. It is raining very hard and the soil is saturated from the bedrock up to 0.5m below the surface (the tilted water table is 0.5m below the surface [ignore any stability problems]). The soil density is about 1800kg/m^3 and has a porosity of about 30%. Calculate some of the water flow values using Darcys law, and our ‘shallow soiled hillslope flow’ approximation.
a) If we consider only 1 meter width (across the slope, not down), what is the cross-sectional area through which water will flow?
b) What is the Head gradient (remember it is dimensionless) in the direction of water flow.
c) Calculate the flux of water, per meter width of hillslope (use Darcy’s law). Answer in cubic meters of water per day.
d) and for good luck, what is the specific flux (q). Specific volume flux is the volume flow per square meter cross-section of the flow path. Note the answer is in meters per day (volume divided by area).
e) and if the hillslope is 100 meters wide, what is the total flux (Q) per day?
f)
And a little geometric problem: What is the vertical (not slope perpendicular!)
soil depth?
2
I quickly sketched some of the calculations we could perform on the Huascaran/Yungay
slide/debris flow, so I am going to ask you to do them more carefully. Note that there is a page of debris flow equations on the web page, that may help. As usual this early in the course I am
partially asking to make sure you can do order-of-magnitude questions without
losing or gaining factors of 10.
The
slide started as a rock-fall on the face of the mountain, at an elevation of about
21,000ft. Within the first horizontal
mile, the mass had fallen nearly 1000m onto the steep glacier under the face,
and then another 2000m into the Llanganuco valley/river. It then flowed down
the valley slope (~4 degrees) for about 6 miles as a ~5m deep debris flow. Part of the flow failed to flow past a sharp
bend in the valley, and climbed directly up a ~100m high ridge, which separated
the main valley from an older outwash fan.
Unfortunately, the town of Yungay was located on this old flat, at an
elevation of 8000ft. The debris flow
stopped on the 2 degree slope of the town, completely burying most of the town
in a 5m deep pile of mud and rock. You can use a density of about 2000 kg/m^3
for the debris after it has incorporated water from the glacier and the valley. The first 3 questions are related to the
energy cascade, the latter 4 are debris flow questions.
a)
Calculate
the total energy lost by a cubic meter of rock (assuming it stayed intact),
falling from the mountain to the town.
b)
Calculate
how hot the rock potentially could have become from frictional heating from the
mountain to the town.
c)
Calculate
how much volume of a cubic meter of ice/snow could have been melted by such a
cubic meter of rock.
d)
Calculate
the approx. critical shear stress of the debris flow (hint: use the stopped
depth and slope to find the stress at which the flow stopped).
e)
Calculate
the approx. surface velocity of the flow in the Llanganuco valley before it
crested the ridge to Yungay (hint this
is a little tricky, but a good approach is to use an energy cascade argument: to
be able to overtop the 100m high ridge the flow had to convert kinetic energy
into potential energy. So you can answer
the question by asking what was the velocity needed for a cubic meter of the
debris to have kinetic energy equal to 100m of potential energy)
f)
Calculate
the approx. thickness of the viscous part of the flow, while in the Llanganuco
valley. (hint these last 2 questions are just plug and chug from the debris
flow equations)
g)
Calculate
the effective viscosity of the flow.
This will be in Pascal-sec, and you can compare to water which is 10^-3.