Structural Geology: Problems

On these pages you will find problems related to chapters in the book. New problems will be added, and solutions will be presented separately.

Version date: 01 April 2011 Structural Geology/Fossen Deformation (Chapter 2)

Problem 2-1 Norway Calculate the extension along a) the stretched (boudinaged)

Swiss belemnite in Figure P2.1a (from tip to tip), and b) the A’ 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 Horda Platform along the North Sea section? Top Triassic Top Basement Viking Graben Field Gullfaks Shetland Platform 2 cm 20 km A Cross-section through the northern North Sea, where post-Triassic strata have been removed. Based on deep seismic line NSDP84-1. Based been removed. strata have post-Triassic where Sea, the northern through North P2.1b Cross-section Figure UK Stretched belemnite in Ordovician limestone, Gross Windgällen, Uri, Switzerland. Switzerland. Uri, Windgällen, limestone, Gross belemnite in Ordovician P2.1a Stretched Figure 0 km 40

2 Problems

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 burrows 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.

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.

3 Structural Geology/Fossen

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.

x 1 2 3 4 5 6 Figure P2.3 Points in the undeformed state.

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.

 3002. 5     00.50  00 05. 

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 deformation matrix.

y y y

x + x + x z z z 1) Compaction 2) Simple shear 3) Pure shear

Figure P2.5 The three deformations and their order. 4 Problems

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?

y y

6 6

4 4

(b) (a) x x 2 4 6 2 4 6

y y

6 6

4 4

y y

6 6

4 4

(e) x (f) x 2 4 6 2 4 6

Figure P2.6 Undeformed and deformed grid, for 6 different deformations. Connect the nodes to get the displacement vector fields. 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 kW , 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 15.12. 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 Problems

σs

3 Stress (Chapter 4-5)

MPa 1 Problem 4-1 a) Interpret the representation of planes 1 and 2 in the Mohr 1 diagram shown in Figure P4.1 by drawing a three-dimension- 120° σ3 σ1 al sketch of the planes and s1, s2 and s3. What are the values σn 1 2 3 4 5 6 of sn and ss? MPa 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? 2 c) The plane is rotated so that it makes 45° to the force or sN Figure P4.1 Two planes (1 and 2) represented on a Mohr circle. See in b). What are the normal stress and maximum shear stress problem 4.1a). 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.

7 Structural Geology/Fossen Fracture (Chapter 7)

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 7.1) 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.

Con ning σs σn pressure (MPa) (MPa) (MPa) 25 76 100 50 130 180 75 181 255 100 231 330 125 287 410 150 331 480 175 386 560 200 420 620

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.

8 Problems Faults (Chapter 8)

Problem 8-1 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 39 38.7 354 36.3 353 34.8 341 33.7 18 37.3 341.5 30.2 19 29.6 345 29.4 351 28.1 353.5 28.3 340 36.0 19 26.6 331 32.1 14 25.9 311 37.3 16 26.4 326 33.7 351 28.1 344 27.7 19.5 43.3 354 27.3 31.5 37.3 300 43.3 22.5 34.6 340 33.2 18 38.2 348 27.3 16.5 37.7 316 30.7 21 37.3 301.5 34.8 34 36.0 301 38.7 18.3 31.6 19 36.0 40 41.6 355 37.3 53 48.8 3 34.8 28 36.0 19.5 34.8 29 40.1 23.5 36.7 58 53.1 57 50.3

9 Structural Geology/Fossen Folds and folding (Chapter 11)

Problem 11-1 Figure P11.1 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 11.2 in the textbook?

1 cm 20 cm

Figure P11.1 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.

10 Problems

Problem 11-2 Figure P11.2 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.

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?

1.5 mm

1 cm

Figure P11.2 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. 11 Structural Geology/Fossen

Problem 11-3 What class(es) of folds are portrayed in Figure P11.3? 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 11.1 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 properties of the layers during folding?

Figure P11.3 Folds in strongly deformed quartzite, South Norwegian Caledonides. The height of the picture is about one meter. 12 Problems

Problem 11-4 This example is of multilayered rocks that shortened by folding. It appears that the layers have different properties and different 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?

F H

E I BDC G

Figure P11.4 Multilayer-folding of late Proterozoic sedimentary rocks in Finnmark, northernmost Norway. 13 Structural Geology/Fossen Shear zones and mylonites (Chapter 15)

Problem 15-1 a) Make two shear strain profiles across the shear zone shown in Figure P15.1, which formed in a magmatic rock. b) Calculate 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?

A B

1 cm

Figure P15.1 Small-scale shear zone in Proterozoic magmatic rock, Sognefjellet, South Norway.

Problem 15-2 Assume that the shear zone shown in Figure P15.2 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?

14 Problems B

Shear zone orientation A layering 10 cm Caledonian shear zone in Proterozoic granulite rock, Holsnøy, South Norway, affecting a Proterozoic granulite-facies foliation. Proterozoic affecting a Norway, South Holsnøy, granulite rock, in Proterozoic P15.2 Caledonian shear zone Figure 15 Structural Geology/Fossen Strike-slip, transpression and transtension (Chapter 18)

Problem 18-1 A N-S striking vertical shear zone dominated by brittle structures is illustrated in Figure P18.1, 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 P18.1, assuming that the deformation is simple shear. d) Make an illustration similar to Figure P18.1 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): Small thrust faults: Axial planes, gentle to open folds: 035/30 034/88 033/34 215/89 034/50 214/28 Axial planes, tight folds: 213/35 025/87 027/90 Large thrust faults: 022/15 Small normal faults: 028/33 124/60 200/26 125/58 305/60 Strike-slip slip surfaces (not shown on illustration): 002/88 Large normal faults: 240/89 118/61 355/87 114/58 000/90 292/62 358/89 Shear zone Normal Reverse Fold

fault fault axial N trace

Figure P18.1 Structures in a fictive strike-slip shear zone (map scale).

16 Problems Balancing and restoration (Chapter 20)

Problem 20-1 The effect of choice of shear angle, exemplified by a hanging-wall block extended above a listric fault. Construct the hanging wall roll-over if the hanging wall deforms by (ductile) vertical shear and antithetic (45°) shear. Describe the differences between the two cases.

h h

Vertical shear

h h

Antithetic shear (45°)

Figure P20.1 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.

17 Structural Geology/Fossen

Problem 20-2 Restore the section across the North Sea Gullfaks Field for 2000 4000 3000 1000 the Jurassic top Statfjord Formation level and for the Triassic 5000 mbsl 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 1360 sheet of paper). a) What is the extension at each level? 1280 b) Is there evidence of early fault activity and stratigraphic thickness variations? Line 736 1200 c) Is the section balanced (is the restored version sound)? d) What was the initial dip of the faults according to the reconstruction? Is this a likely initial dip? 1120 e) Any indication of ductile or “soft” deformation? 1040 960

400 Upper Teist Fm Teist Upper

CDP CDP

Lunde Fm Lunde

Statfj. Fm Statfj. Brent Gr. Brent 1 km

18 Problems

Problem 20-3 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?

Gullfaks Field, Restored State Area difference = 42% Deformed State of undeformed state

0 1km

Figure P20.3 Map of faults at the top Statfjord Formation stratigraphic level, some 3 km below the North Sea. Contour lines have been omitted.

19 Structural Geology/Fossen

Problem 20-4. a) Construct an interpretation of the cross-section (Figure P20.4) based in the three wells and surface dip data. 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 Carbon- iferous. Name the structures. 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 D S S O C Pc c) How much shortening has taken place? C D S O C O C Pc 2 km C D S O Pc C Cross-section from a foreland fold-and-thrust belt. Based on section shown in Marshak & Woodward (1988, in Marshak & Mitra, Basic methods of structural geology. Prentice Hall, Hall, Prentice methods of structural geology. Basic & Mitra, (1988, in Marshak Woodward & in Marshak on section shown fold-and-thrust belt. Based a foreland from P20.4 Cross-section Figure 303-332). p.