Homework #3 GEOL 4880 Humphrey Fall 2013

Due Tuesday 24^th. Give yourself at least a couple of hours. Show your work. And please be neat! Question goes back to isostacy in a geomorphic context, and although the full question is hard, you should all be able to see if the basic idea is somewhat reasonable or just not even worth pursuing. I advise using Google earth to explore this part of Wyoming and using that program to get elevations and distances.

1a) In the central valley of Calif., the extraction of groundwater is causing the ground to subside. In one location the aquifer is about 400m thick and the ground has dropped about 10m. What is the approximate vertical strain in the aquifer?

b) Soil on slopes creeps down slope, slowly. In places in the Laramie Range, the soil surface moves down slope over 0.05 inches per year, while the soil near the bedrock does not move at all. Assume the soil is 0.5m thick over the rock, what is the depth averaged shear strain in one month? (be a little careful with this one, lots of conversions)

c) What is the strain rate (per sec) in b)?

d) If the speed of the creeping soil decreases linearly with depth, from the max at the surface, to zero at the soil/rock interface, what is the shear strain rate at 0.4 meter depth (down from the surface) in b)?

e) (a little hard) What is the shear strain on a plane that is at 45 degrees to the slope (that is, not parallel to the bedrock-soil interface, but dipping at a 45 degree angle to the slope angle? The strike is parallel to slope strike.)

2 a) What is the normal and shear stress (relative to the slope surface) under a rock block or slab that is 10 meters by 10 meters by 2 meter thick (thickness normal to the slope), resting loosely on a 20 degree slope of soil? (the rock has a density of ~2700kg m^-3)

b) If the soil trapped under the block is saturated and the water pressure under the rock is measured to be 10⁴Pa, what is the pressure head of the water?

c) What is the effective normal stress under the block?

d) Assume that granite is incompressible (not a bad assumption for small strains). You stand on a perfect block of granite which results in a vertical compressional strain of 10^-8. What is the approx value of the two components of horizontal strain. [hint you have been given strain in z direction, what must the strain be in x and y to conserve mass or volume]

3 Many erosion processes reduce the size of rock particles. Physical (as in grinding, abrasion, mechanical weathering etc.) geologic process can not reduce the size of the particles to much less than 1 micron in size. At this small size, the surface area to volume ratio of particles increases to the point where surface effects dominate (the strength of surface effects, such as surface chemical bonding to surrounding materials become as large as the physical stresses). As a result smaller rock dust is rare and is usually associated with some chemical process (the exception to this rule is glacial flour which is what makes mountain lakes that beautiful postcard green-blue, [and which is a grinding process]).

a) What is the starting, and the ending surface area of a 1m cube of rock that is crushed to 1 micron sized chunks? (It is ok to assume the chunks are cubical). Express the surface area in square meters.

b) If you take two photographs of part of an object at totally different magnifications and (despite the different magnifications) if the pictures look basically the same, then the object is referred to as self similar, scale invariant or fractal. In geology, many things are fractal, hence we are always putting lens caps, people etc. in pictures for scale.

Imagine making a mixture of rock particles of all sizes. I want you to figure out how you would choose the numbers of the particles of the various sizes so that the mixture will look fractal. In other words, if you look at your mixture at (let's say) triple the magnification, then you will see a similar mixture of small and large particles as at no magnification. It is easiest to do this problem thinking of only discrete particle sizes, say 0.1, .01, .001cm etc., and saying how many particles are needed in each size, to make the resulting pile look fractally distributed.

Illustrate your explanation by listing two (different) scale invariant distributions of particle sizes. [make a table with 3 columns, the first column will have particle size ‘bins’, such as 0.1, .01, .001cm etc (you should have about 5 bins), and you fill in the 2^nd and 3^rd columns with the number of particles needed in each size fraction to make the sample look fractal]

4) A hillsope has an angle of 15degrees. The soil is 2m deep (perpendicular to slope) over good solid impermeable granite. The soil has a saturated hydraulic conductivity of 10^-6m/s. The slope is 100m wide. After a heavy rain the soil is saturated to the surface. Calculate (using the shallow hillslope approximation with flow directly parallel to slope):

a) the total flux of water down the entire slope (Q)

b) the water flux (q)

c) the flux of water per unit width (unfortunately also usually designated q)

4 (Hard) Large enclosed and internally drained basins are unusual, since usually any rain input has to escape somewhere as an outlet river (Great Salt Lake is an example of an enclosed drainage basin, and is the largest in North America). Most basins are formed by tectonic upheaval. However, some may be formed by geomorphic processes. The south central region of Wyoming encompasses the 2nd largest enclosed basin in North America. The Red Dessert or Great Divide Basin stretches from Rawlins to Rock Springs and is completely internally drained, without any current outlet streams. The lowest part of the rim of the basin is SW of the Pathfinder reservoir (south of Casper), but even this low potential ‘spill over’ or outlet point is about 50m elevation above the lowest point inside the basin [north east of Rawlins and South east of Muddy Gap], and therefore it appears that something has tilted the basin or at least lifted the rock near the NE corner of the basin.

The problem is that there does not appear to have been any recent active tectonics in this region. However; there has been considerable erosion of rock recently from the region of the Pathfinder reservoir and the north western Shirley basin. The Platte river in this region appears to have been incised from the elevation of the Shirley or Hanna Basins down to the elevation of the Alcova Reservior. It has been suggested that the erosion of this valley system might have caused enough isostatic uplift of the rim of the divide basin to force the closure of the basin.

Comment on this idea. You should probably do enough of an isotacy calculation to see if the orders of magnitudes of this idea work. You can also comment on likely problems, even if the order of magnitude says that the idea is worth pursuing.