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Structural Geology: ProblemsOn these pages you will fnd problems related to chapters in the book. New prob-lems will be added, and solutions will be presented separately. Version date: 01 Feb. 2011Deformation (Chapter 2)Problem 2-1 Calculatetheextensionalonga)thestretched(boudinaged) Swiss belemnite in Figure P2.1a (from tip to tip), and b) the twomarkerhorizons(top Triassicandtopbasement)inthe North Sea section shown in Figure P2.1b (from A to A). Is the extension evenly distributed in the two cases? For the North Seasection,howdothetwoextensionestimatescompare? Howmuchextensionistakenupbythelargest4-5faults? Is there any other way that we could estimate the extension along the North Sea section?040 kmGullfaksFieldViking GrabenTop TriassicTop BasementHorda PlatformNorwayShetland PlatformUKAA20 kmFigure P2.1a Stretched belemnite in Ordovician limestone, Gross Windgllen, Uri, Switzerland. Figure P2.1b Cross-section through the northern North Sea, where post-Triassic strata have been removed. Based on deep seismic line NSDP84-1. 2 cm2STRUCTURAL GEOLOGY/FOSSENProblem 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.PROBLEMS3Figure P2.2Top: 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 fnd the volume change involved.3 0 0 250 0 5 00 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. xy234561 1 3 Figure P2.3Points in the undeformed state. Figure P2.5Te three deformations and their order. zxyz1) Compaction 2) Simple shear 3) Pure shearxy+ +zxy4STRUCTURAL GEOLOGY/FOSSEN24466xy(a)24466xy(c)24466xy(d)24466xy(f)224466xy(e)24466xy(b)Figure P2.6Undeformed and deformed grid, for 6 dierent deformations. Connect the nodes to get the displacement vector elds. Problem 2-6 Draw and describe the displacement felds based on the transformations shown in Figure P2.6. Do they involve strain, and if so, is the strain homogeneous? Can you fnd a deformation matrix that describes each of the deformations? PROBLEMS5Problem 2-7a) 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 fnd). Hint: use an equation that relates Wk, gand k. You can also use the Excel spreadsheet (sheet named Wk-based 2D-Strain) located on the web-site to fnd the solutions or to check your results. b) What is the angle a between the fow 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 unlithifed sand is 40%. After lithifcation 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?6STRUCTURAL GEOLOGY/FOSSENStress (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 acrosstheplane?UsetheMohrcircleandthencheckyour answer using Equation 4.2.Problem 4-2a) Present the following plane states of stress in the Mohr diagram and fnd 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 MPab) 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 s3are 150, 100 and 50 MPa, respectively. Illustrate the same state of stress in the Mohr diagram. What type of stress feld is this (what is it called)?c) Do the same for principal stress values of 0, 25 and 50 MPa. What type of stress feld 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 Poissons ratio of 0.3. PROBLEMS7n 1 3 s 120 MPa MPa 2 1 234561 1 3 FigureP4.1Twoplanes(1and2)representedonaMohrcircle.See problem 4.1a). 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 infu-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 confning 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 fnd 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 confning pressure of 250 MPa. Figure P7.1Listing of frictional properties (conning pressure, shear stress and normal stress) at the onset of frictional sliding on a 45 saw cut through Tennessee Sandstone.Confningpressure (MPa)25 76 1005075100181231255330125150175200287331386420410480560620130 180s(MPa)n(MPa)8STRUCTURAL GEOLOGY/FOSSEN031.635436.334133.7341.530.234529.4353.528.31926.61425.91626.435128.119.543.331.537.322.534.61838.216.537.72137.33436.018.331.64041.65348.82836.02940.15853.15750.33938.735334.81837.31929.635128.134036.033132.131137.332633.734427.735427.330043.334033.234827.331630.7301.534.830138.71936.035537.3334.819.534.823.536.7 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?PROBLEMS9Folds 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?20 cm1 cmFigureP11.1Pictureanddrawingoftwograniticveins,onethatisfoldedandonethatisnot.Tefoldedveindisplaysastylethatiscommonly described as ptygmatic. Proterozoic basement rocks, South Norway. 10STRUCTURAL GEOLOGY/FOSSENProblem 11-2 Figure P11.2 shows fve 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 mm1 cmABCDEFigure P11.2Picture and drawing of folded granitic veins. Te folded veins have dierent thicknesses and appear to have dierent wave lengths too. Proterozoic deformation within the Caledonian Jotun Nappe, South Norway. PROBLEMS11Figure P11.3Folds in strongly deformed quartzite, South Norwegian Caledonides. Te height of the picture is about one meter. 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 proper-ties of the layers during folding? 12STRUCTURAL GEOLOGY/FOSSENFigure P11.4Multilayer-folding of late Proterozoic sedimentary rocks in Finnmark, northernmost Norway. Problem 11-4 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? ACF HB DEGIPROBLEMS13Shear zones and mylonites (Chapter 15)Problem 15-1 a) Make two shear strain profles 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 fnding y (=tang) at various locations across the profles. Assume simple shear. c) What is the maximum strain value R in the shear zone? 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 profle (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 profle. 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?Figure P15.1Small-scale shear zone in Proterozoic magmatic rock, Sognefellet, South Norway.1 cmAB14STRUCTURAL GEOLOGY/FOSSEN10 cmABShear zone orientationShear zone orientationlayeringlayeringFigure P15.2 Caledonian shear zone in Proterozoic granulite rock, Holsny, South Norway, aecting a Proterozoic granulite-facies foliation. PROBLEMS15Strike-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):Axial planes, gentle to open folds:034/88215/89Axial planes, tight folds:025/87027/90Small normal faults:124/60125/58305/60Large normal faults: 118/61114/58292/62Small thrust faults:035/30033/34034/50214/28213/35Large thrust faults:022/15028/33200/26Strike-slip slip surfaces (not shown on illustration):002/88240/89355/87000/90358/89Shear zoneNormal faultReversefaultFoldaxial traceNFigure P18.1Structures in a ctive strike-slip shear zone (map scale).16STRUCTURAL GEOLOGY/FOSSENBalancing and restoration (Chapter 20)Problem 20-1 The effect of choice of shear angle, exemplifed 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.h hh hVertical shearAntithetic shear (45)Figure P20.1Deformation above a listric fault. Extension of the hanging wall is indicated by a vector h (the heave). Te collapse of the hanging wall onto the fault is to be constructed. PROBLEMS17Problem 20-2 RestorethesectionacrosstheNorthSeaGullfaksFieldfor the Jurassic top Statfjord Formation level and for the Triassic refector called Upper Teist Formation. Exclude the rightmost (eastern)downfaultedblock.Dothisbyperformingarigid blockreconstruction(makeacopyoftheline,cutthefault blocksusingapairofscissorsandgluetheblocksupona sheet of paper). a) What is the extension at each level? b)Isthereevidenceofearlyfaultactivityandstratigraphic 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?5000mbsl120013601040CDP 400112096012804000300010002000Line 7361 kmStatfj. FmLunde FmUpper Teist FmBrent Gr.18STRUCTURAL GEOLOGY/FOSSEN0 1kmGullfaks Field, Deformed StateRestored StateArea difference = 42% of undeformed stateNProblem 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 feld? Is the map restoration acceptable, or must the interpretation be refned?Figure P20.3Map of faults at the top Statford Formation stratigraphic level, some 3 km below the North Sea. Contour lines have been omitted.PROBLEMS19Problem 20-4.a)Constructaninterpretationofthecross-section(Figure P20.4)basedinthethreewellsandsurfacedipdata.Note stratigraphic repetition in two of the wells. Assume that strati-graphicthicknessesareconstantacrossthesectionandthat thefoldshavekink-likegeometries.Thestratigraphicunits are Precambrian, Cambrian, Silurian, Devonian and Carbon-iferous. Name the structures. b) Does your interpretation balance? To fnd out, try to restore thesection,assumingconstantbedlength.Pinyourcross-section in the right-hand end. c) How much shortening has taken place? PcC O O S D CCO S D CPcPcC O SS D C2 kmCFigure P20.4 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, p. 303-332). 20STRUCTURAL GEOLOGY/FOSSEN