Homework #3 GEOL 4880 Humphrey Fall 2015

Due Tues 29^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 Let us look at a shallow soiled hillslope, 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 20 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, and the soil has no cohesion. It is raining very hard and the soil is saturated with a water table 0.5m below surface. Calculate:

a) the water pressure at the soil/bedrock interface [remember that the water is now moving, since it is on a slope]

b) the driving stress (shear stress) at the soil/bedrock interface (you need to add the water weight to the soil weight),

c) the effective normal stress across the soil/bedrock surface

d) What is the vertical (not slope perpendicular) soil depth?

e) Will the slope fail ( factor of safety )?.

f) Calculate the flux of water, per meter width of hillslope (use Darcy’s law).

g) and for good luck, what is the specific flux (q)

h) and if the hillslope is 100 meters wide, what is the total flux?

3 Two questions on stress and strain

a) 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]

b) In class I drew a plot on the board showing the behavior of a perfect fluid. I plotted shear stress on the x axis, and shear strain rate on the vertical. What is the relationship between the slope of the plot, and the value of viscosity for the fluid?

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 if (despite the different magnifications) 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]