Homework #3 GEOL 4880 Humphrey Fall 2014

Due Thursday 9^th. Give yourself at least a couple of hours. Show your work. And please be neat!

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 0.1 cm 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?

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 0.25 meter depth (down from the surface) in b)?

e) (a little hard, and only for those that have had strain in another course) 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 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. 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]).

3 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 directly 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]

4 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]

5 A hillsope has an angle of 20 degrees. 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. The soil has a porosity of 40% and a dry bulk density of 1600kg/cubic meter, the angle of repose is 30 degrees and the cohesion is very small at 1000Pa. Calculate (using the shallow hillslope approximation with flow directly parallel to slope):

a) the flux of water per unit width.

b) the water pressure on the granite surface.

c) Will the slope fail? (calculate the factor of safety)