Earth Surface Processes, GEOL 4880 Humphrey, fall 2016, Homework #2
Here are 5 questions to work on. It is shorter than the last homework, but requires more careful thinking. You will not be able to do all of question 1 until after Thursday.
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.
Question 1 you need to be able to do. Question 2 makes you think about surfaces and volumes. Question 3 is based on our geomorphic isostasy discussion. Question 4 requires some fairly careful thinking (if nobody gets it, I may ask it again). The last part of question 5 is hard (I don’t think anybody has ever answered it). Questions 1,2 and 3 require short calculations, question 4 and 5 only require careful thinking.
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 water pressure at the soil/bedrock interface
b) Calculate the effective normal stress across the soil/bedrock interface
c) Calculate the shear stress on the 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)
2 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.
Calculate the amount (volume) of crushed rock that is needed 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.
3 This is a simplified geomorphic Isostasy problem. The North Platte River, running through Casper, used to flow across a broad plain that was at a similar elevation to the Seminoe reservoir, but the river has since down cut north of the Seminoe. As a result the entire region around Casper has been eroded down to a new surface that currently stands about 500ft below its old level (you can see the steep drop-off if you look on Google Earth at the north end of the Shirley Basin north of Medicine Bow). Calculate how much erosion had to occur for this elevation difference to persist after erosion? Assume the area was isostatically compensated, as we discussed in class when talking about how erosion can drive uplift. You can use these gross approximations: crustal rock density in that area 2700kg/m^3, excess depth of crustal rocks 10Km (thickness difference of total crust in Wyoming compared to surrounding regions), and density of fluid mantle/lower crust 3300kg/m^3. Note this region is in reality too small to ignore the crustal flexure strength, plus it turns out that one of the puzzles of this region we live in is that it is not very well isostatically compensated. Thus this calculation yields much too large a number. However, there is strong evidence of many tens of meters of isostatic uplift around the Casper basin, so although unrealistic in detail, this rebound process appears to have occurred in the area.
4 (This may be hard for some of you) In class I showed that for a dry slope failure on a slope at the angle of repose (Φ), the H/L ratio for a sliding block is the same as the tangent of the angle of repose. Use a little algebra, and show that for any constant slope (α) dumping onto a flat runout plane, that a sliding block with have a similar H/L ratio. Remember, H is the vertical height of the block above the plane, and L is the horizontal distance from directly below the block, to where it stops. This is true, independent of the complexity of the path taken by the block... curved slide paths also end up with an H/L ratio of tangent of Φ, although this is much harder to prove for a curved path. If you can’t do this as a general result, chose two angles for α, and show that both give the same runout distance: choose α = Φ for one of the slopes and α steeper than Φ for the other (e.g. use 30 degrees for Φ and the first α and for the second α use something like 35 degrees). Now: Use this result to solve question 1d in the previous homework, without doing any work, except a simple trigonometric calculation.
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) At Vedauwoo, a sheet of exfoliating granite 1 meter in downslope length lies on an exposed granite slope, with slope less than the friction angle (let’s say a slope angle 20 degrees, friction angle 30 degrees). The granite slab is only a few centimeters thick and warms up and cools down with the diurnal (daily) cycle. The underlying granite is such a large thermal mass that it hardly heats or cools at all. Assume the granite slab goes through a steady 10 degree C peak to peak sinusoidal diurnal cycle. Also assume the slab is basically a rectangular block that is touching the underlying rock only at its upper and lower ends. Coefficient of thermal expansion of granite is 0.00001 per degree C (ie. dimensionally a strain per degree)
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. Although this example of thermal creep is extreme, in fact, most loose materials (soils, rock debris etc.) creep downhill under the action of a large variety of thermal, water, freeze-thaw, biological disturbances etc. This slow downslope motion is important and we will examine in detail later.
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.