AdditionalProblems2ndEd

Prévia do material em texto

Structural Geology: Problems
On these pages you will find problems related to chapters in the book. New prob-
lems will be added, and solutions will be presented separately. 
Version date: 17 March 2017
Deformation (Chapter 2)
Problem 2-1 
Calculate the extension along a) the stretched (boudinaged) 
Swiss belemnite in Figure P2.1a (from tip to tip), and b) the 
two marker horizons (top Triassic and top basement) in the 
North Sea section shown in Figure P2.1b (from A to A’). Is the 
extension evenly distributed in the two cases? For the North 
Sea section, how do the two extension estimates compare? 
How much extension is taken up by the largest 4-5 faults? 
Is there any other way that we could estimate the extension 
along the North Sea section?
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2
Structural GeoloGy/FoSSen 
Problem 2-2. 
The two pictures shown in Figure P2.2 are from the quartzite in northern Scotland called Pipe Rock. The pipes are worm bur-
rows and originally perpendicular to bedding. Where these rocks are involved in shear deformation along the Moine Thrust, 
they change orientation with respect to bedding. The upper photo shows undeformed Pipe Rock, the lower photo shows a 
sheared version. Find the angular shear and shear strain from the lower picture (the shear plane and bedding are horizontal), 
assuming that the deformation is simple shear. 
3
Structural GeoloGy/FoSSen 
Figure P2.2 Top: undeformed Pipe Rock, showing bedding-perpendicular burrows called skolitos. Bottom: 
deformation has changed the primary angular relationship. Bedding is horizontal in both images. 
Problem 2-3.
What happens to the four points in Figure P2.3 when affected by a) a simple shear with g=2, b) a pure shear where kx=2, and c) a subsimple shear where g=2 and kx=2? There is no area or volume change involved. Use the appropriate deformation matrices and graph your answers. 
Problem 2-4.
What deformation is described by this deformation matrix? Show that the deformation represented by this matrix does 
not preserve volume and find the volume change involved.
3 0 0 25
0 0 5 0
0 0 0 5
.
.
.
Problem 2-5.
Write, using 2 x 2 deformation matrices, the following sequence of deformations: compaction (vertical contraction), followed 
by simple shear, followed by pure shear (vertical contraction balanced by horizontal extension). Also write the total deforma-
tion matrix. 
x
y
2 3 4 5 6 1 
1 
3 
Figure P2.3 Points in the undeformed state. 
Figure P2.5 The three deformations and their order. 
z
x
y
z
1) Compaction 2) Simple shear 3) Pure shear
x
y
++
z
x
y
4
Structural GeoloGy/FoSSen 
2
4
4
6
6
x
y
(a)
2
4
4
6
6
x
y
(c)
2
4
4
6
6
x
y
(d)
2
4
4
6
6
x
y
(f)
2
2
4
4
6
6
x
y
(e)
2
4
4
6
6
x
y
(b)
Figure P2.6 Undeformed and deformed grid, for 6 different deformations. Connect the nodes to get the displacement vector fields. 
Problem 2-6 
Draw and describe the displacement fields based on the transformations shown in Figure P2.6. Do they involve strain, and if 
so, is the strain homogeneous? Can you find a deformation matrix that describes each of the deformations? 
5
Structural GeoloGy/FoSSen 
Problem 2-7
a) Imagine a rock with vertical foliation (strike/dip= 000/90) and vertical lineation (000/90). Sketch the rock in a coordinate 
system with the x-axis oriented along the strike direction of the foliation. 
b) The rock is exposed to simple shear along the x-axis. The shear plane is horizontal. Use the deformation matrix for simple 
shear to calculate the orientation of the lineation after shear strains of 1 and 10. 
c) Calculate the elongation of a line of unit length parallel to the lineation in the two cases. 
d) What happens to the foliation during these deformations?
Use the Excel spreadsheet (enter sheet named Subsimple shear and set k=0.0000001) located on the website to check your 
results.
Problem 2-8 
a) Do Problem 2-7 a-c for subsimple shear with Wk=0.5 (use the same values for g, in addition there is a pure shear component 
k that you need to find). Hint: use an equation that relates Wk, g and k. You can also use the Excel spreadsheet (sheet named Wk-based 2D-Strain) located on the web-site to find the solutions or to check your results. 
b) What is the angle a between the flow apophyses in this subsimple shear deformation? What is the orientation of the long 
axis (X) of the strain ellipsoid? Use formulas in Chapter 2 and/or Figures 2.24 and 16.12. Sketch the results. 
Problem 2-9 
Assume that the porosity of unlithified sand is 40%. After lithification the sand is turned into a sandstone with a porosity of 
20%. Assume that the reduction in porosity is caused solely by physical compaction. 
a) What is the deformation matrix of this deformation?
b) How is the strain ellipsoid oriented, and what are its R-values in the three principal sections? 
c) What shape does the strain ellipsoid have and where does it plot in the Flinn diagram? 
6
Structural GeoloGy/FoSSen 
Problem 2-10 
For a unit sphere, X:Y:Z are 1:1:1:
a) Plot in the Flinn diagram the states of strain resulting from 
10%, 20%, 30%, 40%, and 50% compaction (Using a spread 
sheet is recommended).
b) Then expose each of those strain states (ellipsoids) to a 
pure shear with i) X=5, Y=1, Z=1/5, ii) X=10, Y=1, Z=1/10
Plot the data in a Flinn diagram with linear as well as log 
axes to see the difference (in Excel you can change the axes 
to logarithmic by clicking the axis values).
c) Do the same for the opposite sequence, i.e. plane strain exposed to compaction. Are the two sequences different? Prove your 
answer by means of the deformation matrix (matrix multiplication, where the first matrix represents the last deformation).
d) Draw contours for constant compaction values (1+D).
Compaction
(negative dilation)
0
0
1 0
0 1
0 0 1+Δ
7
Structural GeoloGy/FoSSen 
Strain (Chapter 3)
Problem 3-1 
Use the software EllipseFit by Fred Vollmer to estimate 2D 
strain from a deformed conglomerate. The outlines of peb-
bles have been traced and are presented in Fig. P3.1 and as 
an image file (AntilopeIsland tracing.png). Open this file in 
EllipseFit (File ->Open Image) and draw ellipses for each 
pebble. You can do this in several ways, using the tool selec-
tion in the menu (2nd option from the left, which changes 
between center point, ellipse, polygon, filledpolygon, line 
pair by clicking the icon). I used filled polygon and clicked 
inside each pebble once to define ellipses. Use the Cut tool if 
you accidentally click the wrong place. 
Once the ellipses are defined, do Fry and Rf-f analyses 
(Menu -> Analyze) and describe the strain. Read about the 
different Fry outputs in the EllipseFit Manual. Discuss pos-
sible sources of error.
Problem 3-2 
Use EllipseFit to estimate 2D strain from image Sandviksfjel-
let cgl drawing.jpg (similar to Problem 3-2). This conglom-
erate has more rounded clasts than the one in the previous 
problem
Problem 3-3 
Use the software EllipseFit by Fred Vollmer to estimate 2D 
strain from the brachiopodes shown in the attached figure. 
The answer is shown in the e-learning module for Chapter 3. 
File name: LA - Ragan 1985 F10_1a.png
Figure P3.2 Quartz and quartzite pebbles in a deformed conglomerate 
(Bergen, Norway). 
Figure P3.3 Deformed population of brachiopodes (fabricated). From 
Ragan 1985, Fig 10.
8
Structural GeoloGy/FoSSen 
Figure P3.1 Pebbles in a conglomerate (diamictite from Antilope Island, 
Utah, provided by Fred Vollmer). Note the different shapes of the clasts.
Problem 3-4 
The three photos show the same magmatic rock from the 
same outcrop at three different states of strain. Try to estimate 
strain for each case, using the software EllipseFit. The feld-
spar megacrysts are our strain markers. The rock deformed 
at high temperatures 700-800 °C so that the feldspar crystals 
deformed plastically along with the rest of the rock. 
Note how the original objects start to become difficult to dis-
cern at stage C. If the strain was produced by simple shear, 
what would the shear strain be at this stage? Use the spread-
sheet or graphs in the book to find the answer.
9
Structural GeoloGy/FoSSen 
A
B
C
Stress (Chapter 4-5)
Problem 4-1 
a) Interpret the representation of planes 1 and 2 in the Mohr 
diagram shown in Figure P4.1 by drawing a three-dimension-
al sketch of the planes and s1, s2 and s3. What are the values 
of sn and ss?
b) A force of 100 N (Newtons) acts normal to a 0.1 m2 plane. 
What are the normal and shear stresses (traction) across the 
plane?
c) The plane is rotated so that it makes 45° to the force or sN 
in b). What are the normal stress and maximum shear stress 
across the plane? Use the Mohr circle and then check your 
answer using Equation 4.2.
Problem 4-2
a) Present the following plane states of stress in the Mohr diagram and find the mean stress and the deviatoric stress: 
(i) sv = 25 MPa, sh = 0 MPa
(ii) sv = 100 MPa, sh = 0 MPa
(iii) sv = 100 MPa, sh = 50 MPa
b) Consider two weak planes dipping 45 and 60°. Which of these two planes would have the largest chance of being activated 
in these three states of stress, and what would the resulting sense of slip be? 
Problem 4-3 
a) What information does the stress ellipse contain, and what information does the Mohr circle contain? 
b) Draw a stress ellipsoid for a state of stress where s1, s2 and s3 are 150, 100 and 50 MPa, respectively. Illustrate the same 
state of stress in the Mohr diagram. What type of stress field is this (what is it called)?
c) Do the same for principal stress values of 0, 25 and 50 MPa. What type of stress field is this?
Problem 4-4 
a) Draw the states of stress in the Mohr diagram in the crust at 1 km, 5 km and 10 km depth by assuming a crustal density of 
2.7 g/cm3 and a lithostatic state of stress. 
b) Do the same for the uniaxial-strain model and compare with the lithostatic model with a Poisson’s ratio of 0.3. 
Fracture (Chapter 7)
10
Structural GeoloGy/FoSSen 
σn 
σ1 σ3 
σs 
120° 
M
Pa
 
MPa 
2 
1 
2 3 4 5 6 1 
1 
3 
Figure P4.1 Two planes (1 and 2) represented on a Mohr circle. See 
problem 4.1a). 
Problem 7-1 
Cylinders of a 19 x 50 mm sandstone with a saw cut at 45° to the axis was deformed in the laboratory by one of several influ-
ential rock fracture geoscientists of the 20th century, John Handin. Jacketed in lead, the cylinders were deformed in a so-called 
triaxial rig, which is an apparatus where a confining pressure complements axial loading. The axial load was increased until 
sliding occurred on the saw-cut surface. 
Plot the critical stress data (Figure P7.1) in a Mohr diagram to find a frictional failure envelope and draw the Mohr circles. Is 
the envelope linear? If so, write the formula for the Coulomb fracture criterion and use it to predict the stress conditions under 
which the sandstone will slide at a confining pressure of 250 MPa. 
Faults (Chapter 9)
Figure P7.1 Listing of frictional properties (confining pressure, shear stress and normal stress) at the onset of frictional 
sliding on a 45° saw cut through Tennessee Sandstone.
Con�ning
pressure 
(MPa)
25 76 100
50
75
100
181
231
255
330
125
150
175
200
287
331
386
420
410
480
560
620
130 180
σs
(MPa)
σn
(MPa)
11
Structural GeoloGy/FoSSen 
341.5 30.2
345 29.4
353.5 28.3
19 26.6
14 25.9
16 26.4
351 28.1
19.5 43.3
31.5 37.3
22.5 34.6
18 38.2
16.5 37.7
21 37.3
34 36.0
18.3 31.6
40 41.6
53 48.8
28 36.0
29 40.1
58 53.1
57 50.3
39 38.7
353 34.8
18 37.3
19 29.6
351 28.1
340 36.0
331 32.1
311 37.3
326 33.7
344 27.7
354 27.3
300 43.3
340 33.2
348 27.3
316 30.7
301.5 34.8
301 38.7
19 36.0
355 37.3
3 34.8
19.5 34.8
23.5 36.7 
Folds and folding (Chapter 12)
Problem 9-1 
A list of fault surface orientations (strikes and dips, right hand rule) from a fault in the North Sea Gullfaks Field is given below. 
The fault orientations have been calculated along a surface that was interpreted on seismic data and thereafter depth converted. 
They are listed the way they were measured, from south to north. Plot the data as poles in a stereonet. What does it tell us about 
the geometry of the fault? Describe and make a sketch. What can be inferred about the extension direction?
0 31.6
354 36.3
341 33.7
12
Structural GeoloGy/FoSSen 
Problem 9-2
Faults are often interpreted from reflection seismic data. Even without much knowledge about seismic data it is useful to try 
to use such data for structural interpretation and analysis. An important thing to remember is that the vertical scale of most 
seismic sections is in time (seconds) rather than depth (meters). Hence the geometries that we see are not 1:1 unless we do a 
depth conversion, which requires a velocity model. Also note that seismic data always contain noise. This means that we have 
to use common sense and our knowledge of structural geology and stratigraphy to make sensible interpretations. 
a) The seismic line transects the northern North Sea Rift. What kind of faults would we expect to see in a section through a 
rift? 
b) Make a crude interpretation of seismic line and the more detailed image of its right-hand (ESE) part. Use the stratigraphic 
information is annotated on the line. Top Basement in this context is the top of crystalline (metamorphic and magmatic) rocks. 
c) Define major faults and fault blocks. What is the characteristic 1st order features of the rift in this section?
d) Describe the structures. What is the general fault geometry like? Are the faults planar? Note that the faults look steeper than 
they would be in a 1:1 depth-converted section. 
e) Look for evidence of fault growth in the seismic reflectors. Parallel reflectorsindicate that layers were deposited before or 
after rifting. Wedge-shaped sequences form in the hanging walls of active normal faults during the time of faulting.
f) Estimate the amount of horizontal extension for top basement in the lower (detailed) section and compare to shallower level 
extension for the same section. 
g) Can we explain the deformation (faulting) by a single phase of rifting or is there evidence for more than one phase?
13
Structural GeoloGy/FoSSen 
20 km
20 km
ESE
W
NW
12345678
TW
T 
(s)12345678
TW
T 
(s)
Base Q
uarternary
Base Cretaceous
Top Shetland
Jurassic
Cretaceous
Cenozoic
Triassic
Basem
ent
Basem
ent
Jurassic
Cretaceous
Cenozoic
Triassic
Top basem
ent
Top basem
ent
20 km
14
Structural GeoloGy/FoSSen 
Problem 12-1 
Figure P12.1 shows two cm-thick granitic veins (ptygmatic veins) in a magmatic rock that have been exposed to deformation. 
Note the cross-cutting (relative age) relationship between the two. 
a) Draw the approximate orientation and magnitude of the strain ellipse. Discuss the assumptions that need to be made. 
b) Consider the folded granitic layer. What fold class are we dealing with (Class 1A-C, 2 or 3)? 
c) What is its dominant wavelength Ld? What can we say about its viscosity at the time of deformation, using Equation 12.2 in the textbook?
Problem 12-2 
Figure P12.2 shows five ptygmatic veins with a variation in thickness. The vein material is the same, as is the matrix. This 
means that we can consider the viscosity contrast to be constant as we compare the folded granitic layers. 
20 cm
1 cm
Figure P12.1 Picture and drawing of two granitic veins, one that is folded and one that is not. The folded vein displays a style that is commonly 
described as ptygmatic. Proterozoic basement rocks, South Norway. 
15
Structural GeoloGy/FoSSen 
a) Measure the dominant wavelength (Ld) and plot it against layer thickness (h). 
b) Estimate the amount of shortening expressed by the folding. Do the layers all indicate the same amount of shortening?
Problem 12-3 
What class(es) of folds are portrayed in Figure P12.3? These folds are found in mylonitic quartzite in a major shear zone. Add 
dip isogons to the drawing. Plot some of the folded layers in the diagram shown in Figure 12.11 in the textbook. The fact that 
the axial traces are not linear and parallel introduces an error. Do the results give us information about the mechanical proper-
ties of the layers during folding? 
1.5 mm
1 cm
A
B
C
D
E
Figure P12.2 Picture and drawing of folded granitic veins. The folded veins have different thicknesses and appear to have different wave lengths too. 
Proterozoic deformation within the Caledonian Jotun Nappe, South Norway. 
16
Structural GeoloGy/FoSSen 
Figure P12.3 Folds in strongly deformed quartzite, South Norwegian Caledonides. The height of the picture is about one meter. 
Problem 12-4 
This example is of multilayered rocks that shortened by folding. It appears that the layers have different properties and differ-
ent fold geometries. Analyze the folds geometrically like we did in the previous question. What classes of folds do we have? 
Which layers are more competent? 
17
Structural GeoloGy/FoSSen 
Figure P12.4 Multilayer-folding of late Proterozoic sedimentary rocks in Finnmark, northernmost Norway. 
Shear zones and mylonites (Chapter 16)
A
C
F H
B D
E
G
I
18
Structural GeoloGy/FoSSen 
Problem 16-1 
a) Make two shear strain profiles across the shear zone shown in Figure P16.1, which formed in a magmatic rock. b) Cal-
culate the offset across the zone. Also estimate offset by finding y (=tang) at various locations across the profiles. Assume 
simple shear. c) What is the maximum strain value R in the shear zone? 
Problem 16-2 
Assume that the shear zone shown in Figure P16.2 is deformed by simple shear. This shear zone is affecting a pre-existing 
foliation, marked as “layering” (orange dashed line). The approximate orientation of the shear zone is indicated. 
a) Make a shear strain profile (graph) across the zone (perpendicular to the margins of the zone), for instance along the black 
& white “ruler” from A to B. 
b) Estimate the offset along the zone from the strain profile. How does this compare with the offset of markers seen in the 
picture? 
c) Is there anything about this zone that suggests a deviation from the ideal simple shear zone model? 
Strike-slip, transpression and transtension (Chapter 19)
Figure P16.1 Small-scale shear zone in Proterozoic magmatic rock, Sognefjellet, South Norway. 
1 cm
A B
19
Structural GeoloGy/FoSSen 
10 cm
A
B
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20
Structural GeoloGy/FoSSen 
Problem 19-1 
A N-S striking vertical shear zone dominated by brittle structures is illustrated in Figure P19.1, and a set of orientation data 
are listed below. 
a) Plot the data using a stereo net. 
b) What is the kinematics (sense of shear) and type of deformation (simple shear, pure shear or something else) based on the 
structures and their orientations? 
c) Draw the ISA (Instantaneous Stretching Axes) onto Figure P19.1, assuming that the deformation is simple shear.
d) Make an illustration similar to Figure P19.1 that shows the type and orientation of small-scale structures that can be ex-
pected on the cm and dm scale. 
Orientation data (right-hand rule):
Axial planes, gentle to open folds:
034/88
215/89
Axial planes, tight folds:
025/87
027/90
Small normal faults:
124/60
125/58
305/60
Large normal faults: 
118/61
114/58
292/62
Small thrust faults:
035/30
033/34
034/50
214/28
213/35
Large thrust faults:
022/15
028/33
200/26
Sh
ea
r z
on
e
Normal 
fault
Reverse
fault
Fold
axial 
trace
N
Figure P19.1 Structures in a fictive strike-slip shear zone (map scale).
21
Structural GeoloGy/FoSSen 
Balancing and restoration (Chapter 21)
Problem 21-1 
The effect of choice of shear angle, exemplified by a hanging-wall block extended above a listric fault. Construct the hang-
ing wall roll-over if the hanging wall deforms by (ductile) vertical shear and antithetic (45°) shear. Describe the differences 
between the two cases. 
hh
hh
Vertical shear
Antithetic shear (45°)
Figure P21.1 Deformation above a listric fault. Extension of the hanging wall is indicated by a vector h (the heave). The collapse of the hanging wall 
onto the fault is to be constructed. 
22
Structural GeoloGy/FoSSen 
Problem 21-2Restore the section across the North Sea Gullfaks Field for 
the Jurassic top Statfjord Formation level and for the Triassic 
reflector called Upper Teist Formation. Exclude the rightmost 
(eastern) downfaulted block. Do this by performing a rigid 
block reconstruction (make a copy of the line, cut the fault 
blocks using a pair of scissors and glue the blocks up on a 
sheet of paper). 
a) What is the extension at each level? 
b) Is there evidence of early fault activity and stratigraphic 
thickness variations? 
c) Is the section balanced (is the restored version sound)? 
d) What was the initial dip of the faults according to the re-
construction? Is this a likely initial dip? 
e) Any indication of ductile or “soft” deformation?
50
00
m
bs
l
12
00
13
60
10
40
CD
P 
40
0
11
20
96
0
12
80
40
00
30
00
10
00
20
00
Li
ne
 7
36
1 
km
St
at
fj. 
Fm Lu
nd
e 
Fm
Up
pe
r T
eis
t F
m
Br
en
t G
r.
23
Structural GeoloGy/FoSSen 
0 1km
Gullfaks Field, 
Deformed State
N
Problem 21-3 
Reconstruct this map of the top Statfjord Formation of the Gullfaks Field. Do this by cutting out each important fault block 
using a pair of scissors and placing them together so that overlaps/open gaps are minimized. What is the extension direction 
and how much extension is there? Do we have plane strain or non-plane strain? What orientation would you chose for section 
balancing based on this exercise? How could we map the displacement field? Is the map restoration acceptable, or must the 
interpretation be refined?
Figure P21.3 Map of faults at the top Statfjord Formation stratigraphic level, some 3 km below the North Sea. Contour lines have been omitted. 
24
Structural GeoloGy/FoSSen 
Problem 21-4.
a) Construct an interpretation of the cross-section (Figure 
P21.4) based in the three wells and surface dip data. 
The section is from a fold-and-thrust belt, and contains 
thrusted sedimentary rocks (mostly) and fault-bend folds. 
Note stratigraphic repetition in two of the wells. Assume 
that stratigraphic thicknesses are constant across the 
section and that the folds have kink-like geometries. The 
stratigraphic units are Precambrian, Cambrian, Silurian, 
Devonian and Carboniferous. Name the structures. Dip as 
observed at the surface is indicated, and dip domains are 
separated by dashed lines.
b) Does your interpretation balance? To find out, try to restore 
the section, assuming constant bed length. Pin your cross-
section in the right-hand end. 
c) How much shortening has taken place? 
PcCOOSDC C
OSDC Pc
PcCOS SDC
2 
km
C
Fi
gu
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25
Structural GeoloGy/FoSSen