Problem Set #2: Force, Pressure, and Stress

Geol 542: Advanced Structural Geology Fall 2013

Problem Set #2: Force, Pressure, and Stress

1. Consider the following descriptions for states of stress:

i) isotropic and homogeneous

ii) isotropic and heterogeneous

iii) anisotropic and homogeneous

iv) anisotropic and heterogeneous

(a) Which of these terms accurately describes lithostatic stress? Explain your answer. (2)

(b) Which of these terms accurately describes hydrostatic pressure? Explain your answer. (2)

2. Work like an Egyptian. You have been commissioned to help design a great pyramid to honor your beloved king, Tutansimon. The pyramid architects tell you that for stability reasons, it is very important that the vertical stress (force per unit area) acting on any hypothetical horizontal plane at any height within the pyramid is always identical. They assume this is the most logical design because, as any pharaoh knows, the strength of materials is governed by the ratio of force to area. The area of any horizontal plane within the pyramid must gradually decrease up towards the top of the pyramid, so the architects want the force acting on these planes to decrease in the same proportion to keep the vertical stress constant.

According to the architects, the base of the pyramid will have four sides that are each 100 m in length. The pyramid will be built using limestone with a density of 2400 kg/m3. Their task for you: tell them how tall the pyramid must be to meet their needs otherwise they will chop off your head and feed your brains to bats. My questions to you:

a) Is it possible for you survive this employment opportunity with your brains intact? Explain!
(10)

Hints: (1) Draw a figure. (2) Come up with a general expression for stress as a function of height. (3) The volume of a pyramid = 1/3 x length of base x width of base x height).

b) The limestone has a compressive strength of 100 MPa. How high would the pyramid need to be in order for the structure to start collapsing under its own weight? Given this result, should we be concerned about any of the pyramids in Egypt suffering this fate? (Hint: keep track of your units during the calculation). (8)

c) Now don’t you think pyramids are cool for reasons you didn’t previously realize?!

3. Dam it! In the 1970s, the U.S. Army Corps of Engineers Dam Building Division went through a hiring strategy in which they only hired graduates of Boise State University* (*Note: some parts of this question may be fictional). In response to the great Idaho dam disaster that soon followed, the Corps is attempting a new strategy in which they qualify their dam-building specifics based on questions sent to graduate students at the University of Idaho and Washington State University. Below is the background info needed.

There is now concern about the integrity of two dams built in the 1970s. One of them is the Libby Dam in Montana, built to hold back the waters of Lake Koocanusa. The other is Dworshak Dam, built to hold back the waters of Dworshak Reservoir, near Orofino, Idaho. Locate both dams in Google Earth.

Lake Koocanusa stretches 145 km back from the 129-m-high Libby Dam, all the way into British Columbia. The volume of water in Lake Koocanusa is a whopping 7.22 km3. At the dam wall, the maximum water depth is 113 m.

Dworshak Reservoir only stretches 85 km back from the 219-m-high Dworshak Dam, the third-highest dam in the United States. The volume of water in Dworshak Reservoir is estimated at 4.28 km3, significantly less than the volume of Lake Koocanusa. At the dam wall, the maximum water depth is 217 m.

The U.S. Army Corps of Engineers is concerned that they have severely underestimated the ability of the Libby Dam to hold back Lake Koocanusa, given that it holds 169% of the volume as the Dworshak Reservoir. They are thinking of making the dam much thicker at its base just to be on the safe side. The logic being used is that the dam needs to be much stronger at Libby than at Dworshak to hold back so much water. The problem is considered particularly dire since the Dworshak Dam has repeatedly shown signs of catastrophic failure, with large cracks opening up in the dam wall soon after its construction (some up to 120 m long) and again in 1980 (a 72 m long crack).

a) Assume the dams are both made of exactly the same materials and are able to withstand an identical amount of pressure against the dam wall. Do you agree with the U.S. Army Corps of Engineers assessment that the Libby Dam should have been designed to be 169% of the strength of the Dworshak Dam and may thus start to fracture? Explain your answer quantitatively. (5)

b) Dams are actually designed to withstand a certain amount of force across their surface areas. If the strength of the dam concrete is only a function of the force exerted by the water, does it matter what the concavity of the dam wall is, or if the dam is completely linear, for the concrete integrity? (2)

c) Given your answer to part (b), why then are dams commonly built in such a way
that they are concave downstream? After all, it would use less concrete to just
make it linear across the valley. Hint: your answer should consider the resolution
of a force vector into components within the appropriate coordinate system
designed for this problem. (5)

4. Consider a region of the Earth’s crust where the rock density is a depth-independent 2400 kg m-3.

a) What is the vertical stress (lithostatic stress or overburden) at a depth of 5 km? (3)

b) In the absence of any tectonic stresses, what are the magnitudes of principal stresses in the horizontal plane? (2)

c) If the rocks are water saturated, and the water table is at a depth of 500m, what is the magnitude of the hydrostatic pressure at this 5 km depth? Assume the density of water is 1000 kg/m3. (4)

d) What is the magnitude of the effective stress at 5 km depth given this water pressure? (2)

4. Now consider a region of the Earth’s crust where a borehole is being drilled into a hydrocarbon reservoir. The stratigraphy consists of the following rock types: a 1 km thick shale layer (density 2100 kg/m3), a 1.5 km thick sandstone layer (density 2400 kg/m3) and a limestone unit (density 2600 kg/m3) of unknown thickness but which extends deeper than the maximum depth of the borehole.

a) Calculate the lithostatic stress that exists along the walls of the borehole at depths of
(i) 1 km, (ii) 2 km and (iii) 3 km (the bottom of the borehole). (15)

b) Now calculate the effective stress at each of these depths, assuming the rocks are saturated up to the Earth’s surface. (9)

c) Assuming the borehole is cased down to the depth of hydrocarbon recovery 3 km down, what range of density of drilling mud would the borehole operators need to use to ensure that the borehole neither collapses (in response to the formation pressure) nor explodes (by exceeding the confining pressure) at a depth of 3 km? (9)

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