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: 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 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 Cross-section 2 cm 20 km Stretched belemnite in Ordovician limestone, Gross Windgällen, Uri, Switzerland. Switzerland. Uri, Windgällen, limestone, Gross belemnite in Ordovician Stretched A Figure P2.1b Figure UK Figure P2.1a 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 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. 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. y 3 1 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 deforma- tion 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 (c) x (d) x 2 4 6 2 4 6 y y 6 6 4 4 2 (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. Conning 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.