Homework #3 GEOL 4880 Humphrey Fall 2020

Due Tuesday Sept 22nd. 

Show your work. And please be neat!  (You may submit via PDF/email if you want)

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 (above), that may help.

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 will have to come up with several reasonable estimates of: debris density, debris heat capacity etc.

a) Calculate the total energy lost by a cubic meter of rock, falling from the mountain to the town.

b) Calculate how hot the rock could have become from frictional heating.

c) Calculate how much volume of ice/snow could have been melted by each cubic meter of rock.

d) Calculate the approx. critical shear stress of the debris flow.

e) Calculate the approx. velocity of the flow in the Llanganuco valley before it crested the ridge to Yungay.

f) Calculate the approx. thickness of the viscous part of the flow.

g) Calculate the effective viscosity of the flow.