Quantification and Modeling of Deforamtion Processes ... - heim.ifi.uio.no

Quantification and Modeling of Deformation Processes; Motivated by Observations from the Contact to the Hornelen Basin, Bremangerlandet Thesis Submitted in Fulfillment of the Requirements for the Degree of M. Sc. in Natural Science Physics of Geological Processes Department of Physics University of Oslo

Torbjørn Ersfjord Bjørk 2006

for GB

Acknowledgements Thanks are first due to my immensely wonderful supervisors Håkon Austrheim, Dani Schmid and, Karen Mair; without whom it is most unlikely that this work would ever have seen the light of day. Thank you for your perceptive comments, suggestions and encouragement along the way. I have benefited tremendously from having you as my supervisors. I would also like to thank everyone at PGP for making this a such exiting place. I would especially like to thank the “the Nice Master Students” in 437 for making these two years very enjoyable. The culprits are in alphabetical order: Anders Nermoen, Berit Mattson, Grunde Waag, Hilde Henriksen, Kirsten Haaberg, and Solveig Rønjom. Special thanks to Anders for checking my “equation gymnastics”. I would also like to thank my friends, both from back home and in Oslo for their continued encouragement, although the general consensus have been that geology is for people with special interests. And above all, I would like to thank my mother and my little sister and the rest of my family for their infinite patience and great love.

Contents Page Preface

1

1 Geological Setting

3

1.1

Regional Geology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.1.1

The Kalvåg Mélange . . . . . . . . . . . . . . . . . . . . . . . .

4

1.1.2

The Bremanger Granitoid Complex . . . . . . . . . . . . . . . .

5

1.1.3

Hornelen basin . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2 Fieldwork

7

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.2

Observed deformation features . . . . . . . . . . . . . . . . . . . . . .

7

2.2.1

Deformation features observed in the Granodiorite (BGC) . . .

8

Shear zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

Breccias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

Faults, fractures with granulated material and ultra cataclastic

2.2.2

veins . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

Quartz veins and quartz net-veining . . . . . . . . . . . . . . .

16

Deformation features in the granodiorite near the basin margin

23

Features and deformation observed in conglomerate near the basin margin . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

2.2.3

Deformation features observed in the mélange . . . . . . . . . .

29

2.2.4

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

3 Methods

33

3.1

Field work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

3.2

Image analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

3.2.1

34

Particle size and shape determination . . . . . . . . . . . . . . i

3.2.2 3.3

Critical evaluation of image analysis technique . . . . . . . . .

40

Numerical modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

3.3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

3.3.2

Two dimensional stress and strain (rates) . . . . . . . . . . . .

45

Stress tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

The symmetry of the stress tensor . . . . . . . . . . . . . . . .

48

Strain (rate) tensor . . . . . . . . . . . . . . . . . . . . . . . . .

50

Normal strain . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

Shear strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . .

55

Conservation of mass - The continuity equation . . . . . . . . .

57

Conservation of momentum - The equations of equilibrium . . .

59

Viscous rheology - The constitutive equation . . . . . . . . . .

60

Newtonian fluid . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

Shear stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

Normal stress . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

3.3.5

Two-dimensional iso-parametric elements

. . . . . . . . . . . .

69

3.3.6

Finite element formulation . . . . . . . . . . . . . . . . . . . . .

72

3.3.7

Numerical integration . . . . . . . . . . . . . . . . . . . . . . .

77

3.3.8

Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . .

79

3.3.9

Mesh generation . . . . . . . . . . . . . . . . . . . . . . . . . .

80

3.3.10 FEM code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

3.3.3

3.3.4

4 Petrography

81

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

4.2

Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

4.2.1

Back scatter electron imaging . . . . . . . . . . . . . . . . . . .

82

Petrography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

4.3.1

Particle size . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

Particle size distribution . . . . . . . . . . . . . . . . . . . . . .

83

2D versus 3D measurements . . . . . . . . . . . . . . . . . . . .

85

PSD and gouge comminution models . . . . . . . . . . . . . . .

86

Shape descriptors . . . . . . . . . . . . . . . . . . . . . . . . . .

90

Elongation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

Circularity

91

4.3

4.3.2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4

Statistical tools . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

Fault gouge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

4.4.1

Locality BRE60-05 . . . . . . . . . . . . . . . . . . . . . . . . .

94

Sample BRE1505-2 . . . . . . . . . . . . . . . . . . . . . . . . .

95

Wall-rock . . . . . . . . . . . . . . . . . . . . . . . . . .

95

Fault gouge . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.4.2

Image analysis results sample BRE1505-2 . . . . . . . . . . . . 108 Particle size distribution . . . . . . . . . . . . . . . . . . . . . . 108 Particle shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Epidote . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 K-feldspar . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Plagioclase & quartz . . . . . . . . . . . . . . . . . . . . 122 Particle shape distributions . . . . . . . . . . . . . . . . . . . . 127 Elongation . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Circularity . . . . . . . . . . . . . . . . . . . . . . . . . . 133

4.5

Sandstone dike . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.5.1

Locality BRE59-05 . . . . . . . . . . . . . . . . . . . . . . . . . 136 Sample BRE1305A-1 . . . . . . . . . . . . . . . . . . . . . . . . 140 Wall-rock . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Sandstone dike filling . . . . . . . . . . . . . . . . . . . . 147

4.5.2

Image analysis results sample BRE1305A1 . . . . . . . . . . . . 148 Particle size distribution . . . . . . . . . . . . . . . . . . . . . . 148 Particle shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Epidote . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 K-feldspar . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Plagioclase & quartz . . . . . . . . . . . . . . . . . . . . 163 Particle shape distributions . . . . . . . . . . . . . . . . . . . . 168 Elongation . . . . . . . . . . . . . . . . . . . . . . . . . . 168 Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Circularity . . . . . . . . . . . . . . . . . . . . . . . . . . 174

4.6

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 4.6.1

Particle size analysis . . . . . . . . . . . . . . . . . . . . . . . . 177 fault gouge sample BRE1505-2 . . . . . . . . . . . . . . . . . . 177 Sandstone dike sample BRE1305A1 . . . . . . . . . . . . . . . . 179

4.6.2

Comparison between fault gouge and sandstone dike . . . . . . 181

Combined shape and size analysis . . . . . . . . . . . . . . . . . 181 Epidote shape and size . . . . . . . . . . . . . . . . . . . 181 Epidote as comminution and fluid flow signature . . . . 181 K-feldspar shape and size . . . . . . . . . . . . . . . . . 183 Plagioclase & quartz shape and size . . . . . . . . . . . . 185 Caveats to the extension of measured 2D values to 3D D-values . . . . . . . . . . . . . . . . . . . . 187 Coupled reaction-deformation processes . . . . . . . . . 188 4.7

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

5 Intrusion between rigid plates

205

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

5.2

Model setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

5.3

5.2.1

Incompressible flow in two dimensions . . . . . . . . . . . . . . 206

5.2.2

Intrusion between rigid plates . . . . . . . . . . . . . . . . . . . 208

5.2.3

Characterization of analytical solution for spreading plates . . . 210

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 5.3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

5.3.2

Pressure and pressure deviation in gap material . . . . . . . . . 217 Vertical centerline profile a − a0 . . . . . . . . . . . . . . . . . . 217 Pressure deviation contours of gap material . . . . . . . . . . . 221

5.3.3

Velocities and velocity deviation in gap material . . . . . . . . 225 Fluid flow across profile b − b0 . . . . . . . . . . . . . . . . . . . 225 Total velocity deviation . . . . . . . . . . . . . . . . . . . . . . 231

5.4

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

5.5

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

6 Synthesis and conclusions 6.1

241

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

A Geological maps

243

B Image analysis

245

B.1 Phase separation and grain identification . . . . . . . . . . . . . . . . . 245 B.1.1 Common user input . . . . . . . . . . . . . . . . . . . . . . . . 245 B.2 Gray scale thresholding . . . . . . . . . . . . . . . . . . . . . . . . . . 247 B.2.1 Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

Light processing . . . . . . . . . . . . . . . . . . . . . . . . . . 250 Heavy processing . . . . . . . . . . . . . . . . . . . . . . . . . . 255 B.3 Watershed method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 B.3.1 Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 C Program code

267

C.1 Image_analysis.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 C.1.1 Thresholding_heavy.m . . . . . . . . . . . . . . . . . . . . . . . 276 C.1.2 Watershedding.m . . . . . . . . . . . . . . . . . . . . . . . . . . 282 List of figures

a

List of tables

g

Bibliography

i

Index

i

Preface This study investigates deformation processes that occur in the vicinity of the northern margin of the Hornelen Basin. The approach encompasses quantitative methods like image analysis and numerical modeling that build on the classical geological tools of field work and textural analysis. The motivation for this study is the outstanding questions of the tectonic origin of this basin margin; in particular the origin of the “sandstone dikes”. The general view of this margin has been that of an unconformity. However, the results presented in this thesis suggest that it is much more tectonized than previously recognized.

1

Chapter 1

Geological Setting 1.1

Regional Geology

The continent-continent collision between Baltica and Laurentia in Silurian to early Devonian period (Caledonian-Appalachian orogen) (Bryhni and Andréasson, 1985) resulted in thrusting of nappes southeastwards onto the Baltic craton (Hossacker and Cooper, 1986; Fossen, 1992). The tectonostratigraphy consists of: (i) An autochthonous Baltic basement and cover overlaid by a Lower Allochthon consisting of Baltic basement and sediment cover. (ii) a Middle Allochthon, derived from closer to the western Baltic continental margin, consisting of high grade Precambrian crystalline rocks and their cover; (iii) an Upper Allochthon derived from the outer rifted margins of Baltica and oceanic/arc terranes; and (iv) an exotic Uppermost Allochthon lacking Baltic affinities. Roberts and Gee, 1985 and Roberts 1988 suggested that the Uppermost Allocthon may represent part of Laurentia. Bremangerlandet consists of several nappes containing Precambrian gneiss with meta-sedimentary cover rocks and exotic terrains belonging to the Middle and Upper Allochthon. The nappes are strongly attenuated eastwards towards the Vetvika Shear Zone (VSZ). The VSZ marks the boundary between the upper and lower plates of the basement, and is the upward continuation of the NSDZ (Hartz et al., 1994). It puts the mylonintic gneisses of the Precambrian basement complex belonging to the WGR, in tectonic contact with heterogeneous Precambrian quartzo-felspathic genisses (Hartz et al., 1994). The Upper Allochthon at Bremangerlandet consists of two parts with different metamorphic grade that are separated by a major E-W trending thrust fault. The lower 3

1.1. Regional Geology

Chapter 1. Geological Setting

part is amphibolite facies and consists of amphibolites and garnet mica schists. The upper part is greenschist facies and consists of the Kalvåg Mélange which is intruded by the Bremanger Granitoid Complex (BGC) and the Gåsøy intrusion (Bryhni and Lyse, 1985; Hansen et al., 2002). To the south, the low grade Upper Allochthon is unconformably overlain by Devonian conglomerates belonging to the Hornelen basin. The sedimentary basins were deposited in half-graben developments above the generally W-dipping Norfjord-Sogn Detachment Zone (NSDZ) during the late Devonian Caledonian extensional collapse (Osmundsen and Andersen, 2001). Later the area between Sogn and Nordfjord was folded in a NW -to WSW-trending folds with amplitudes and wavelengths on the order of several kilometers during the Cretaceous opening of the North Atlantic (Osmundsen and Andersen, 2001). In the anticlines the metamorphic rocks of the Western Gneiss Region (WGR) and the Baltoscandian nappes are exposed while the Devonian basins are located in the synclines (Osmundsen et al., 1998; Osmundsen and Andersen, 2001).

1.1.1

The Kalvåg Mélange

The Upper Allochthon of the Scandinavian Caledonides is derived from the outer rifted Baltic margin and oceanic/arc terranes. On the Bremangerland peninsula the Upper Allochthon is exposed as the Kalvåg Mélange which is a pelitic mélange containing olistoliths of banded chert, greywacke, calcareous rocks, basalt lavas, andesites, rhyolites, rhyolitiv ignimbrites and quartz porphyry (Cuthbert, 1991; Ravnås and Furnes, 1995). The groundmass consist of mainly matrix-rich sedimentary breccias, mudstones, black shales and interbedded conglomerates and sandstones (Ravnås and Furnes, 1995). The chert layeres consist of 80% quartz and minor amounts of biotite, garnets and chlorite (Steen and Andresen, 1997). The Kalvåg Mélange is related to the Sunnfjord Mélange which formed during obduction of the Solund Stafjord ophiolite complex (Ravnås and Furnes, 1995). Ravnås and Furnes (1995) suggests it was deposited in a back-arc basin along an oceanic magmatic arc. The Kalvåg Mélange is intruded and hornfelsed by the Bremanger Granitoid Complex (BGC) and the gabbronoritic/dioritic Gåsøy intrusion in the southwest prior to thrust emplacement.(Ravnås and Furnes, 1995; Hartz and Andresen, 1997; Hansen et al., 2002). 4


1.1.2


The Bremanger Granitoid Complex

The Bremanger Granitoid Complex (BGC) is also exposed on the Bremanger peninsula. It is unconformably overlain by Devonian sediments along its southern border and it intrudes the olistromal Kalvåg Mélange. It is dated to 440±5 Ma by Hansen et al. (2002) using

206 Pb/206 Pb

and U-Pb zircon dating. The age is coeval with the

nearby Solund Stavfjord Ophiolite Complex (443±3 Ma; Dunning and Pedersen, 1988) and the subduction related Gåsøy Intrusion (443±4 Ma; Furnes et al., 1989) and it is belied they all formed at the same margin of the Iapetus Ocean (Hansen et al., 2002). The BGC is believed to have formed by partial melting of the mafic lower parts of an oceanic magmatic arc, inside which also the Solund Stavfjord Ophiolite Complex was forming (Hansen et al., 2002). The BGC is a sheet-like intrusion which strikes approximately 070° and dips ∼60° to the SSE. The thickness range from 1.3 - 2.6 km and the length extends ∼17 km along strike (Hansen et al., 2002). The BGC consists of mostly homogenous white granodiorites/tonalites with locally tens of meters wide bodies of greyish-white granitoids and small bodies or veins of white granites (Hansen et al., 2002). The internal contacts vary from sharp to transitional. The intrusive relationship with the Kalvåg Mélange is evident from the appearance of dikes and xenoliths of melange in the granodiorite. Hansen et al. (2002) emphasizes that the Bremanger Granitoid Complex is mostly undeformed, apart from near the basin contact: “Apart from rare shear zones and some cataclastic deformation which occur mostly in the vicinity of contacts with the Devonian rock, the Bremanger Granitoid Complex is not significantly affected by deformation (Hansen et al., 2002).”

1.1.3

Hornelen basin

The Hornelen basin is the largest of three Devonian basins situated in Western Norway. They are related to the Devonian collapse of the Caledonides and have formed as stepwise coarsening- to fining upwards (CUFU) sequences of sand, silt, and conglomerate. The basins were formed during late- to post-orogenic extension of the over-thickened Caledonian crust mainly due to a relative change in plate motion between Laurentia and Baltica (Osmundsen and Andersen, 2001). This change resulted in the development of westerly-dipping major detachments between Nordfjord and Sognefjorden (NSDZ) which is a 3 km thick extensional mylonite zone. The hanging wall of the NSDF consists of a suite of Calendonian nappe 5



rocks unconformably overlain by Devonian sedimentary rocks, while the footwall consists of metamorphic rocks belonging to the WGR (Osmundsen and Andersen, 2001). The CUFU sequences gives the Hornelen basin its step-like topography. The steps are a product of erosion, where the fluvial sandstones are easily to erode compared to the coarser conglomerates. Each CUFU sequence is coupled with an episode of faulting. When a movement occurred it created a space in which first lakes and rivers entered, which were flanked by alluvial fans, tallus cones and debris flows located at the fault scarps. In time the conglomerates and breccias from the sides spread further towards the center and buried the sandstones under coarse grained conglomerates/breccias. Renewed faulting created another CUFU unit on top of the former (Osmundsen and Andersen, 2001). The NSDZ has a listric geometry, hence continued stretching and westward movement of the hangingwall caused the half-grabens to gradually increase their tilt towards east (Hartz et al., 1994). The facies distribution in the Hornelen basin is asymmetric. Along the northern boundary, step fans are dominated by large flow debris deposits. While in south the the induvidual fans have a larger radius and are dominated by stream-transported conglomerates (Bryhni, 1964). In the Hornelen basin, NE- and NW-dipping normal faults dominate the low stratigraphic levels (Hartz et al., 1994) and the basins are bordered by a large, low-to moderate angle normal fault (NSDZ) and a steeper transfer fault sub-parallel to the extension direction (Osmundsen and Andersen, 2001). To the west the Hornelen basin rests unconformably on the exotic Allochthon nappes (Bremangerlandet) and to the north and south steep syndepositional to postdepositional, oblique-extensional faults bounds the basin (Hartz and Andresen, 1997). The eastern margin is bounded by brittle low-angle normal fault (Norton, 1986).

6

Chapter 2

Fieldwork 2.1

Introduction

The field study of the Bremanger Granodiorite Complex (BGC) has been carried out by Torbjørn Ersjord Bjørk and Solveig Feen Rønjom under the supervision of professor Håkon Austrheim. The field work included geological mapping and sample collection and emphasized on the deformed rocks of the BGC found in the vicinity of and at the contact to the Hornelen basin. The purpose of the field work was to determine the degree of tectonic deformation related to the basin development. The geological map is found in Appendix A Geological maps on page 243.

2.2

Observed deformation features

The different observed deformation features can be been divided into two main categories: • Ductile deformation – Shear zones. – Shear zones with brecciated core. – Shear zones with alternating parallel and kinked bands. • Brittle deformation – Comminution breccia. – Jig-saw breccia. – Faults and fault zones. 7

2.2. Observed deformation features

Chapter 2. Fieldwork

– Ultra cataclasites – Fractures/faults with granulated material. – Sandstone dikes. – Marginal breccia. The field work emphasized on the deformation features found in the granodiorite in the vicinity and adjacent to the basin margin. The deformation features will be described with respect to the their location to the basin margin. However, also the features of the the basin conglomerate and the mélange rock will be described. By deformation features observed in the granodiorite it is implied a geographical location up to 600 m from the basin margin. But deformation features are also found as far as 1.8 km away from the basin margin. By deformation features observed in the granodiorite adjacent to the basin margin its implied that the geographical location is approximately 0 - 20 m away from the basin margin.

2.2.1

Deformation features observed in the Granodiorite (BGC)

Shear zones There are several shear zones in the granodiorite near the basin margin. They generally have a E-W strike and dip mainly towards the South (figure 2.1). The inclination is generally between 30 and 90◦ . The shear zones have the same orientation and and inclination as the foliation found locally in the the granodiorite. We also notice that the shear zones also follow the basin margin. However this might be a topographical feature. They range in thickness from 2 cm to 15 m and show both dextral and sinistral relative motion. Some can be traced for less than 30 m, while others can be traced for several hundred meters. Locally they are displaced by large bodies of brecciated granodiorite (figure 2.2 and 2.5). We also observe shear zones with a brecciated core, 5 - 10 cm wide, with rotated foliated clasts (figures 2.3 and 2.4). Others have alternating bands with planar foliation that follows the general strike and oblique kinked bands (figure 2.5).

8



Equal Area

Granodiorite Foliation/Shear Zones (n = 97)

Figure 2.1: The shear zones and foliation in the granodiorite have a general E-W oriented strike. They dip mainly towards the South.

Figure 2.2: Foliated granodiorite cut by N-S oriented body of brecciated material. Locality BE8803.

9



Figure 2.3: Shear zone with cataclastic core. Top to the right sense of shear.

10



(a) 1 m thick shear zone with a approximately 5 (b) Shear zone with 1,5 - 5 cm thick cataclascm thick cataclastic core. The shear zone can be tic core. The shear zone is oriented 070/64 and followed approximately 15 - 20 m and display no we observe top to the left sense of shear. It can visible sense of shear. From locality BRE4805.

be followed approximately 25 m. From locality BRE1905.

Figure 2.4: Shear zones with cataclastic cores.

(a)

(b) S-C fabric indicating top to the left sense of shear. Close-up of (a).

Figure 2.5: Shear zone with alternating parallel and in inclined foliation (S-C fabrics). The shear zone is oriented 050-56/58 and has top to the left sense of shear. It is approximately 40 cm thick and can be followed approximately 15 - 20 m. To the right the shear zone is cut by a body of brecciated material with rotated foliated clasts. Locality BRE0205.

11



Breccias Several types of breccias are found at the contact with the basin and in the granodiorite. The size of the bodies vary from 3 - 10 m in width. The breccias are often found as N-S oriented bodies that cuts and displaces E-W oriented shear zones. There are found two main types of breccias: Matrix supported and clast supported. They are categorized on the basis of appearance and degree of deformation. Matrix dominated breccias are characterized by white rotated foliated clasts of granodiorite in a green epidote rich matrix. They can cover large areas, up to 100 m2 and are found at several places along the contact. Matrix dominated breccias are mainly characterized by a bimodal grain size distribution. Centimeter to tens of centimeter large clasts of granodiorite in a fine grained matrix. Locally you find meter larger clasts of granodiorite. Sometimes the deformation has been so extensive that the matrix is so fine that the grains cannot be observed and the breccia is categorized as a comminution breccia (figure 2.6). The second type of breccia is clast supported breccia. Here the clasts inteconnect througout the breccia. The clasts range in size from centimeters to tens of centimeter. There are breccias with relatively small deformation where fragments are at most only separated by a small zone of matrix (figures 2.7 and 2.8). The brecciation is not so intensive and it is possible to piece together the individual clasts and the breccia is referred to as a jig-saw breccia. However, we do also find localized deformation along fractures where we observe rotated foliated fragments of granodiorite (figure 2.7(b) and (d)). We also observe displacement of passive markers that indicate that the breccia has been created by faulting (figure 2.9). In other clast supported breccias the deformation has been more substantial and we find clasts of a variety of sizes (figure 2.10). Some times we observe granulation of material and rotation of fragments locally in the breccia with which creates openings at other places where we find a fine grained epidote rich matrix with flow structures (figure 2.11). We also notice an earlier ductile deformation event with a top right sense of shear is superseded by this brittle deformation.

12


(a)


(b)

Figure 2.6: Comminution Breccia. We observe rotated foliated clast of granodiorite in a green epidote rich matrix. From locality BRE1005.

(a)

(b)

(c)

(d)

Figure 2.7: Jig-saw breccia breccia. The foliated granodiorite is brecciated and fragmented (a and c). We also observe that the deformation localized a long fractures zones where we find rotated foliated fragments (b and d).

13



(a)

Figure 2.8: Jig-saw breccia. The granodiorite has been brecciated and fragmented within a meter wide zone. The deformation is not so intensive and it is possible to piece together the individual clasts. From locality BE7503.

(b) Close-up of a.

Figure 2.9: Brecciated granodiorite. Notice granulated material in the faults that displace the quartz vein. The displacement is about 2 cm for the topmost faullt. From locality BRE5505.

14



(a) Brecciated granodiorite near the contact with (b) Brecciated granodiorite from locality BE65. the Kalvåg mélange. From locality BR9803.

Figure 2.10: Clast supported breccia with a clasts with a variety of different sizes. From localities BE7503.

(a)

(b)

(c)

(d)

Figure 2.11: Breccia with granulation and rotation of fragments. An earlier ductile deformation event have created a top to the right sense of shear (a). Later the rock was fragmented. The localized crushing of the rock (b and c) opens the rock at other places were we find a brown epidote rich matrix with flow structures (d)

15



Faults, fractures with granulated material and ultra cataclastic veins There are several faults and joints in the granodiorite. The majority of the faults and joints are oriented approximately NNW-SSE and have a relatively steep inclination (figure 2.13). The fault show both left -and right lateral displacement (figure 2.12). There are also several fractures with granulated material. These fractures are mainly oriented NNW-SSE in the Vågane area and NNE-SSW in the Smørhamn area (figure 2.14, see Appendix A Geological maps on page 243 for locations). The fracures with granular material have a steep inclination, 60 - 90◦ , but for the majority of the fractures we do not see the third dimension and the inclination is not possible to measure. Hence the structural data is displayed using rose diagrams. The fractures with granulated material range from narrow fractures only mm thick to 2 cm thick (figures 2.15 - 2.20). We also observe displacement along these fractures (figures 2.18(a) and 2.19) Some places we observe ultra cataclastic dikes where it not possible to distinguish any fragment (figure 2.20). Locally we observe conjugated fracture sets (figure 2.17(a)). Fractures with granulated material is found as far as 1.8 km away from the basin margin, but the deformation in mostly found within 200 - 600 m away from the basin margin (see Appendix A Geological maps on page 243). However, we do not always observe an increase of fractures with granulation towards the basin. A 120 m long profile perpendicular to the basin margin was investigated (see Appendix A Geological maps on page 243). Starting at the basin margin every 20 m the fractures found inside 1 m2 was measured (figure 2.21). We observe that the number of fractures was very small near the basin margin. In fact no fractures were found adjacent to the basin margin. The number of fractures increase to a maximum of 45, 60 meters away from the basin margin (P3), before the number of fractures become approximately constant at 25 -30 fractures per m2 for the rest of the profile (P4 - P7). However, this could be a local characteristic as at other places we observe severe deformation 0 - 20 m from the basin margin (see below). Also, the granodiorite is foliated in this area. Quartz veins and quartz net-veining Locally we find bodies of quartz in the granodiorite. In the breccias the quartz veins often appear deformed (figure 2.23). The quartz veins in the more undeformed granodiorite appear either as veins with thickness ranging from cm to tens of cm (pegmatites) (figure 2.24(a)) with a preferred ordination or as mm to cm thick net-veins pattern (figure 2.24(b)). The quartz veins have a general N-S orientation with inclination 70 80◦ (figure 2.22). 16


(a) Fault zone in the granodiorite.


(b) On the fault planes we find a thin layer of quartz with slickensides that indicate relative displacement.

The fault plane in the picture has

moved to the right. Close-up of the fault plane in (a).

Figure 2.12: Fault in the granodiorite oriented approximately 320/70. The slickenside striations (b) indicate that the faults are left lateral. However, the fault in the granodiorite are have both left -and right lateral movement. From locality BR2103.

Equal Area

Equal Area

Granodiorite Joints (n = 50)

Granodiorite Faults (n = 25)

Figure 2.13: Stereoplots of faults and joints in the granodiorite. The fault and joint are generally oriented NNW-SSE with a relatively steep inclination.

17



Equal Area Equal Area

Vågane (n = 246), rose diagram (symmetric):

Smørhavn (n = 124), rose diagram(symmetric): Outer Circle = 13%

Outer Circle = 11%

Mean dir = 36.5, alpha95 = 6.6

Mean dir = 89.7, alpha95 = -1.0

Figure 2.14: Rose diagrams of fractures with granulated material in the granodiorite. The fractures are generally oriented NNE-SSW in the Sørhamn area and NNW-SSE the Vågane area.

Figure 2.15: Network of fractures with granulated material in the granodiorite. Parallel fractures oriented perpendicular to the basin margin with interconnecting fractures. In some of the fractures a 0.5 to 10/15 cm tick granulation zone has been developed (coloured red in the field sketch on the right). From locality BRE5505.

18



(a) The granodiorite is in this locality intensely (b) Fracured granodiorite. Network of fractures fractured. The fractures are generally between 1 1 - 3 mm thick, but some are up to 4 mm thick. and 3 mm wide, but some are as thick as 14 mm. From locality BR6903. From locality BR6203.

(c) The fractured Granodiorite. Network of frac- (d) Fractured granodiorite with granulated matetures with granulated material. The fractures are rial. The fracures are up to 18 mm in thickness. generally 20 - 30 cm long and 1 - 4 mm wide. From From locality BR8003. locality BR7403.

Figure 2.16: Fractures with granulated material in the granodiorite. The granodiorite is locally intensely fractured. The fracture thickness is usually small, 1 - 4 mm. However, some fractures are almost 2 cm thick.

19



(b) Granulated material in the fractures.

Figure 2.17: Intensely fractured granodiorite. The fractures makes conjugate sets (a) and we observe granulated material in the fractures (b). From locality BE4903. (a) Intensely fractures granodiorite. The fractures form conjugate sets.

(a) Fractured granodiorite.

(b) Fine grained matrix and relatively large fragments of granodiorite in the fracture. Close-up of (a).

Figure 2.18: Fractured granodiorite. Notice displacement along fracture (a) and opening and accumulation of fine grained material in other fractures (b). From locality BRE5505.

20



Figure 2.19: Faulted granodiorite with frac-

Figure 2.20: Cataclastic vein found in a

ture network. Two generations of fractures

fault zone. The grain are so small that their

observed. Notice that the horizontal fracture

not distinguishable. The green colour indi-

is displaced at two places and a breccia have

cates fluid infiltration and accompanied saus-

been formed along the faults.

suritiation of feldspars into epidote. From locality BR2503.

P3

P4

18 0

16 0

18 18

0

0 16 16

0

0 14

0 12

14

0

0 12

0

80

10

60

40

0

18 0

16 0

14 0

12 0

10 0

0

6 4 2 0

20

Frequency

5

80

0

P7

P6

60

10

Foliated,n = 24

10

40

0

Foliated, n = 27

20

80

60

0

40

0

18

0 16

0 14

0 12

0 10

80

60

40

20

0

14 0

5

0

0

10

20

5

0

12 0

P5

10

Frequency

Frequency

10 0

Foliated, n = 45

Foliated, n = 14

Frequency

80

60

40

18 0

16 0

14 0

12 0

10 0

80

60

40

20

0

0

15 10 5 0

20

Frequency

Frequency

P2 5 4 3 2 1 0

Weak foliation, n = 28

Foliated, n = 29

Figure 2.21: A 120 m long profile perpendicular to the basin margin. Starting at the basin margin every 20 m the fractures found inside 1 m2 was measured. No fractures were found adjacent to the basin margin (P1). The number of fractures increase to a maximum of 45, 60 meters away from the basin margin (P3), before the number of fractures become approximately constant at 25 - 30 fractures per m2 for the rest of the profile (P4 - P7). From locality P1-P7.

21



Equal Area

Granodiorite quartz-veins (n = 36)

Figure 2.22: Orientation of quartz veins in Figure 2.23: Quartz veins in granodiorite the granodiorite is generally N-S with inclina- breccia. The breccia contains fragments of rotion 70 - 80◦ .

tated foliated clasts of granodiorite (2 - 3cm large) and up to 1.5 m long ductilely deformed quartz vein. From locality BR0403.

(a) The area with quartz vein is approximately 50 (b) Quartz net-veining. long and 10 - 15 m wide.

Figure 2.24: N-S oriented quartz veins in the granodiorite. The quartz veins range from cm tick veins to tens cm tick bodies of pegmatitic quartz. Net-vein pattern of thin quartz veins is also observed. From locality BR7203.

22



. Deformation features observed in the granodiorite near the basin margin Near the basin margin (< 20 m) sandstone dikes are found oriented perpendicular to the basin margin. The sandstone dikes are 2 - 3 m long and up to 25 cm thick. The majority of the sandstone dikes are oriented NW-SE to N-S and we also observe that the thickness is greatest for this orientation (figure 2.28). We also observe that several of these sandstone dikes are pull-apart fractures (figure 2.25). In the sandstone dikes we observe flow structures (figure 2.26). However, we do at the same time observe granulation in fractures that are connected to the sandstone dikes. Faulting and fracturing which involves development of kink folds in the granodiorite which get ruptured is also observed (figure 2.27). Closer to the basin margin we observe a mixture of both sand from the basin and rotated foliated clasts of granodiorite. Adjacent to the margin the fracturing is locally so intense that a marginal breccia with fractures oriented both parallel and perpendicular have formed (figure 2.29). The parallel fractures terminate at the basin margin. No displacement along the fractures is aobserved. We also observe matrix supported breccias with angular clasts at the basin margin (figure 2.30). The clasts are of relatively uniform size when the matrix fraction is large (figure 2.30(a)), as apposed to when the matrix fraction is relatively small (figure 2.30(b)). The fragments are less angular when the matrix fraction is small. We also note that there are flow structures in matrix with a relatively large fraction of matrix (figure 2.30(a)).

23


(a) Pull-apart fracture.


(b) Pull-apart fracture. The displacement is approximately 10 cm.

Notice that fractures with

granulated material is connected to the pull-apart fracture.

Figure 2.25: Sandstone dikes found near the basin margin. The fracture interface is relatively straight and fracture material display flow structures. Notice the fractures are pull-apart fault and that the relative displacement is oriented perpendicular to the basin margin. From locality BRE5905. Figure 2.26:

Close

up Area B in figure 2.25(b). Fracture

Granulated material

oriented sub-normal to the basin margin filled with granulated material is connected with sandstone dike with flow structures. Coin for scale.

Flow structures

24

25

the basin margin we observe a mixture of both sand from the basin and rotated foliated clasts of granodiorite.

Figure 2.27: Fracturing near the basin margin. Notice that the foliated granodiorite has developed a kink fold which has ruptured. Closer to

Chapter 2. Fieldwork 2.2. Observed deformation features



Sandstone cracks,n = 82 n = 82, bin-size = 10 deg.

14

14

12

12

10

10

Frequency

Frequency

Sandstone cracks,n = 82 n = 82, bin-size = 10 deg.

8 6

8 6

4

4

2

2

0

0 10

20

30

40

50

60

70

80

90

100 110 120 130 140 150 160 170 180

10

20

30

Strike

40

50

60

70

80

90

100 110 120 130 140 150 160 170 180

Strike

(a) Frequency of sandstone dikes against orienta- (b) Maximum width of sandstone dikes against tion.

orientation.

Figure 2.28: Histogram of frequency and maximum width of sandstone dikes against orientation. Both the frequency and orientation i largest for NW-SE to N-S (135 - 180◦ ) oriented sandstone dikes.

Figure 2.29:

Marginal

breccia near basin margin. The fracturing is so intense that a marginal breccia with fractures both oriented both parallel an perpendicular have formed. The parallel fractures terminate at the basin margin. No displacement is observed along the fractures.

26



(a) Matrix supported breccia with a relatively (b) Matrix supported breccia with relatively small large fraction of angular clasts. The clast are of fraction of matrix and less angular clasts. The relatively uniform size. Note flow structures in the clasts are of a variety of sizes. matrix.

Figure 2.30: Matrix supported breccia with angular clasts found at the basin margin. The clasts are of relatively uniform size when the matrix fractions is large (a). As apposed to when the matrix fraction is relatively small (b). The fragments are less angular when the matrix fraction is small. We also note that there are flow structures in the breccia with a relatively large fraction of matrix (a). Both breccias are from the Vågane area, exact locality unknown.

27


2.2.2


Features and deformation observed in conglomerate near the basin margin

The conglomorate is deposited as an unconformity above the granodiorite. The layer are generally oriented NNE-SSW and the inclination is generally 30 - 50◦ .(figure 2.31). We observe layering with alternating bands of fine grained material and larger fragments. The fine grained material display often lamination. Some of the largest fragments are the size of small boulders. At the margin there is often observed a approximately 30 cm thick basal conglomerate without ant layering. The conglomerate is fractured. The fracures are generally oriented NNW-SSE (figure 2.32). The inclination of the fractures is steep, generally 60 - 90◦ .

Equal Area

Equal Area

Conglomerate Bedding (n = 32)

Conglomerate joints (n = 65)

Figure 2.31: Orientation of conglomerate Figure 2.32: Orientation of fractures in the bedding. The orientation of the strata is gen- conglomerate.

The orientation of the frac-

erally NNE-SSW. The inclination is approxi- tures is generally NNW-SSE. The inclination mately 30 - 50◦ .

is generally steep, generally 60 - 90◦ .

28



(a) Basal conglomerate on top of marginal breccia.

(b) Transition from basal conglomerate to layered (c) The conglomerate is layered with alternating conglomerate.

sequences of small and large fragments.

Figure 2.33: Transition from marginal breccia to layered conglomerate with a layer of basal conglomerate in between. From locality BRE2905.

2.2.3

Deformation features observed in the mélange

The mélange has a strongly developed foliation. The orientation of the foliation is generally E-W with inclination 30 - 70◦ (figure 2.34). Locally we find large lenses of mélange in the granodiorite. From meters to tens of meters large (figure 2.35). However, there are some which are between 50 and 50 m long. We observe lenses of granodiorite in the mélange. These lenses are deformed by faulting (figure 2.36). Also folding of the mélange is observed (figure 2.37). 29


2.2.4


Summary

The presented field evidence shows that the granodiorite is much more extensively deformed than earlier recognized in the vicinity of the Hornelen Basin. Both ductile and brittle deformation is observed. The majority of the shear zones have a E-W strike and dips generally toward the basin margin. From the presented field observation it evident that the shear zones predate the brittle deformation. We observe both foliated clasts in breccias and also bodies of breccias that cut and possible displaces shear zones. We also observe cataclastic cores in the center of some shear zones that can be traced for 5 to 20 m. The brittle deformation consists of faults, fractures with granulated material, breccias, and sandstone dikes (pull-apart fractures). We note that orientations of these brittle deformation structures are comparable and may be related to the basin development. I now quantify these structures more closely and conduct numerical simulations to gain insight into possible deformation/implacment mechanism and their controlling factors.

30



Equal Area

Melange Foliation (n = 42)

Figure 2.34: Orientation of mélange folia- Figure 2.35: Lens of foliated mélange (dark) tion is generally E-W with inclination 30 - in the granodiorite. The lens is approximately 70◦ .

4 - 5 m long and 3 m wide. Vein of granodiorite on the right. From locality BR9003.

Figure 2.36: Lens of granodiorite in the Figure 2.37: Folded mélange From locality mélange. We observe that the lens is displaced BR6505. by a fault. From locality BR9603.

31

Chapter 3

Methods 3.1

Field work

The field study of the Bremanger Granodiorite Complex (BGC) has been carried out by Torbjørn Bjørk and Solveig Rønjom under the supervision of prof. H. Austrheim. The field work included geological mapping and sample collection and was emphasized on the deformed rocks of the BGC found at the contact to and in the vicinity of the Hornelen Devonian basin. The purpose of the field work was to determine the degree of tectonic deformation in the footwall related to the basin development. A detailed report describing the field work is found in the previous chapter.

33

3.2. Image analysis

3.2

Chapter 3. Methods

Image analysis

To extract quantitative particle size and shape information from images of thin sections a new image analysis technique was developed. The technique uses either gray scalethresholding or a “watershed” method to separate mineral phases and individual grains. The gray scale thresholding method worked best on the back scatter electron (BSE)images and a brief description of the technique is given below. The program “Gray scale image analysis” (Bjørk, 2006) is written in MatLab and uses MatLab’s Image Processing Toolbox (The MathWorks, 2001). The code is open-source and allows for individual user alterations if needed. The program requires a minimum amount of user input and is able to semi-automatically produce reliable results for a large amount of images. A more detailed description of the technique and accompanying program scripts is given in Appendix B Image analysis and Appendix C Program code on page 245 and 267, respectively.

3.2.1

Particle size and shape determination

Gray scale BSE-images are read directly into MatLab. The images are obtained from scanning electron microscope (SEM) studies of the thin sections. The intensity of BSEs is proportional to the atomic mass unit of the element. That is, minerals with relatively dense and light elements show up as bright and dark objects, respectively. This phenomena is used to distinguish the different mineral phases and identify the individual grains. The BSE-image has a gray-scale range of 256, where black equals 0 and white equals 255 (fig. 3.1). By plotting a histogram of the gray scale intensity you can identify the different phases (fig. 3.2). The histogram displays three distinct areas. The the lower values including the first peak corresponds to plagioclase & quartz (0 80). Plagioclase & quartz have the same intensity and are not possible to separate by this method1 . They are therefore treated as one phase, even though the rheological properties of the two minerals are different. The range from the second to the third valley corresponds to the K-feldspar phase (80 - 140). The final peak correspond to the epidote phase (140 - 190). It is possible to use the data cursor as a guide to finding the appropriate gray scale ranges for individual phases. When the data cursor mode

1

It is possible to separate plagioclase and quartz by using the SEM’s EDS-analyzer to map the

sodium content and using that as guide to separate the two phases. But this is a very time demanding procedure and high resulting mapping is needed to adequately separate the two phases. Therefore it is not a possible tool to use on many images.

34

Chapter 3. Methods

3.2. Image analysis

Figure 3.1: Original BSE-image.

Figure 3.2: Histogram of gray-scale intensity.

35

3.2. Image analysis

Chapter 3. Methods

is enabled you can click on the grains in the image and data values of the x− and y− coordinates, and gray scale intensity value is displayed. An important point to note is that other accessory minerals, e.g. mica, zircon, apatite and titanite, can on the basis of light intensity be included in the grains of one the principal mineral phases. At low magnification these minerals will generally belong to the matrix, due to their small size and be automatically removed (see below). At high magnifications these minerals need to be dealt with or removed manually (see Appendix B Image analysis on page 252). However, it is a source of error which one needs to be aware of. In samples with coarse grains and fine grained matrix the matrix material is usually darker and easily separated from the particles together with any small grains of accessory minerals. For these samples the “watershed” method is usually better (See Section B.3 Watershed method on page 256). When the appropriate gray scale range is found for a phase of interest the image is thresholded and turned into a binary image (fig. 3.3(b)). That is, pixels within the range are set to 1 (white) and the background to 0 (black) to isolate the phase of interest. Several morphological operations are used on the image to further separate the individual grains. All morphological operations inspect the pixels in a 3x3 pixel-environment. First image noise is reduced by removing any single isolated white pixels. To separate touching grains, H-bridges and spurs are identified and removed automatically. An H-bridge pixels is the a pixels that has two opposite 0 and 1 nearest neighbours pairs (see Appendix B fig. B.2.1 on page 251). A spur is a pixel with only one next nearest neighbor and seven 0 neighbours (see Appendix B fig. B.2.1 on page 251). Finally inclusions in the grains are filled. This will potentially add particles a different phase to that grains. To remove any remaining matrix and to set a limit for the minimum grain size detectable on each magnification image, all grains with area less than 200 pixels are removed. Note, this is a reasonable limit, smaller matrix threshold would add error to calculations of particle shape (see below) .The connecting white pixels (i.e. the grains) are then labeled so that each grain is specified with an positive integer (fig. 3.3(c)). The connectivity of the pixels is chosen nearest neighbour only. That is, in a 3x3 pixel-environment only nearest neighbours pixels are considered neighbours and next nearest neighbour pixels are not (see Appendix B fig. B.2.1 on page 252). This is done to avoid that grains which touch each other only at the vertices of two pixels are regarded as one single grains. High resolution images of 2560x1920 pixels are used so 4-connectivity instead of 8-connectivity does not affect the size and shape of the final particles. Grains that are located at the edges of the image are removed since we do not know their entire size and shape (fig. 3.3(d)). The area of 36

Chapter 3. Methods

3.2. Image analysis

the edge-grain is also calculated and subtracted from the total area of the processed image. The edges of the separated particles are usually much rougher than the edges of the real particle in the original image because of pixel level image noise. To remove this artifact the grains are eroded and dilated one or several times in a process called morphological closing (fig. 3.4). From the labeled image (such as fig. 3.3(d)), size and shape properties of the particle can easily be calculated. For a detailed description of the size and shape parameters and the use of them see Section 4.3.1 Particle size on page 83 and 4.3.2 Shape descriptors on page 90.

37

3.2. Image analysis

Chapter 3. Methods

(a) Original BSE-image.

(b) Thresholded B&W-image of phase of interest.

(c) Labeled image after morphological processing. (d) Labeled-image with edge grains removed Individual grains are randomly coloured to highlight their separation from grain neighbours.

Figure 3.3: Development of the phase differentiation and identification grain.

38

Chapter 3. Methods

(a) Before.

3.2. Image analysis

(b) After.

Figure 3.4: Morphological closing to make grain boundaries smoother. The grain boundaries are superimposed onto the original image for comparison.

39

3.2. Image analysis

3.2.2

Chapter 3. Methods

Critical evaluation of image analysis technique

The main processing problems with this technique arise from insufficient contrast between the particles and matrix, and/or particles of different phases. To alleviate this issue the “Gray scale image analysis” (Bjørk, 2006) has several an options for improving image contrast. Both spreading the intensity values over the full range of the image and saturating of values at low and high intensities. If that is not adequate it is possible to adjust the image contrast using a standard image-editing software. In the end, the success of the image processing technique relies on the contrast in the image. If the contrast between particles and matrix, or particles of different phases is too small, they cannot be properly differentiated. The best way to deal with this problem is to fine tune the contrast so that sufficient contrast exist between particles and matrix, and/or between different phases. Also both particles and matrix consists of a range of gray scale intensities which may be overlapping. This will add noise to the image, but this is solved by the removal of any isolated white pixels. As mentioned above, the particle edges detected with the image analysis technique are rougher than the edges of the real particle. Often the edge of a grain has a slightly different gray scale intensity than the rest of the grain, especially at large magnifications. The dilation and erosion functions used to remove pixel noise from grain edges will introduce an effective grain smoothing error. This is minor provided that these tools are not used repeatedly (see Appendix B fig. ?? on page ??). Finally there is the challenge of separating the individual particles. This is relatively easy when the sample material consists of isolated grains or grains with nearly touching boundaries. When grains of the same phase have a long connecting grain boundary, the grain separation becomes more advanced and requires more processing. Although the tool separates the grains and shows a good fit when the grain boundaries are superimposed onto the original image, it is important to remember that every morphological operation affects the size and shape of the particle slightly and is therefore a systematic source of error. To minimize errors, it is important to use high resolution images. This helps with the phase differentiation and particle separation. As well as minimizing the effect the morphological operations have on particle size and shape. Although the program is semi-automatically and requires minimal user input, it is not fail proof. Therefore it is important to evaluate the image processing to check for any misfit of grains boundaries or overlap between phases. To evaluate the result of the thresholding and particle separation, the borders of the individual particles are superimposed on the original BSE-image (fig. 3.6(a) and (b)). To evaluate particle overlap the particles of 40

Chapter 3. Methods

3.2. Image analysis

the different phases are given value equal to 10eN , where N = 0, 1 & 2, and added to a single image matrix. That means that any overlap will have a value different than for the principal phases and will be clearly visible when the image of all the phases combined is displayed and colour-mapped (fig. 3.6(c)). Overlapping particles can be removed manually with a graphical user-interface where the grains are clicked on with the mouse cursor and removed. The open-source code makes it is very easy to alter the settings, if so is desired, for different images. A series of images from SEMs taken at different magnifications may have different contrasts. Samples of different material is best processed by either the thresholding or the “watershed” method. In this study the image analysis technique was used to analyze fracture material from extensional fractures and fault gouges. But the technique is also applicable to other types of images and other studies where quantification of particle size, shape, and orientation can be useful for understanding the geological process (e.g. growth of crystal needles in magmatic phenocrystals, stress oriented crystals, analogue experiments of granular material with glass beads, fracture patterns, and etc.(see Appendix B fig. B.20 on page 264). For separating grains the sequence of morphological operations can easily by altered. The number of repetitions of a single morphological operation can easily be altered or canceled. The technique also allows for plotting an artificial image where any shape parameter can be color-scaled and the orientation of the major axis plotted as an arrow radiating from center of the particle (fig. 3.5). In this way one can compare particle size and any geometrical relations with particle shape and orientation. To the authors knowledge this has not been reported in the literature. In the samples analyzed in this study comparison of the particle geometry to shape and orientation did not show any underlying structure. Although the image analysis technique is not fully automatic and requires some manual adjustments for optimization of individual images, it can successfully differentiate phases and individual grains with a minimum amount of user-input. It gives a good approximation of particle size, shape, and orientation. Any misfit of grain boundaries or overlap is easily detected and can be corrected manually. Once all the individual particles are identified particle size, shape and orientation is easily calculated for individual particles and size -and shape distributions for a given phase and the entire sample can be determined. Finally, it is important to note that for a truly thorough testing of the end result one should compare the end result to a different phase and particle recognition scheme to see how sensitive the particle size and shape distributions are to the applied method. 41

3.2. Image analysis

Chapter 3. Methods

Figure 3.5: Artificial image where the convexity (surface roughness) is plotted as a colour gradient. The different phases are plotted with differen colours: Epidote (green), K-feldspar (blue), and quartz & plagioclase (red). A Dark colour indicate a smooth particle, whereas a light colour indicates a irregular object. The arrows show the orientation of the particles major axis.

42

Chapter 3. Methods

3.2. Image analysis

(a) Epidote grain boundaries are superimposed (b) Plagioclase & quartz grain boundaries are suonto the original image for comparison.

perimposed onto the original image for comparison.

(c) Phasemap. Epidote (green), plagioclase & quartz (red), matrix (blue), and grain boundaries (white). Note that K-feldspar grain have not been identified as is regarded as matrix in this image.

Figure 3.6: Graphical evaluation of the result. Automatically determined grain boundaries are superimposed onto the original image for comparison (a and b). The grain boundaries will have the same color if the grains are touching , but they will still be regarded as separate grain. This is an artifact from the actual plotting of the grains boundaries and will not affect the size, shape or number of grains. The labeled images of the different phases are combined to a single image where any overlap will be displayed as a colour different from the phases.

43

3.3. Numerical modeling

3.3 3.3.1

Chapter 3. Methods

Numerical modeling Introduction

The molecular nature of structure matter is well established and not debatable. However, for many investigations of material behavior the individual molecule or or even smaller scale units are of no concern and and only the material as a whole is important. That is, the macroscopic behavior of the material is explained by disregarding considerations such as the heterogeneous microstructure of atoms. Instead the material is assumed to be continually distributed throughout its volume and to completely fill out the space it occupies. This continuum concept of matter is a fundamental postulate of continuum mechanics and provides a framework for studying the behavior of materials like solids, fluids and gases. The simplifying approximation that physical quantities (such as mass and momentum) can be handled in the infinitesimal limit, makes it possible to employ differential equations in solving problems in continuum mechanics. Some of the differential equations capture fundamental physical laws, such as conservation of mass or conservation of momentum and are called conservation laws. Other are specific to the materials being investigated and are called constitutive equations. The (universal) physical laws do not depend on the coordinate system in which they are observed. Continuum mechanics thus uses tensors, which are mathematical objects that are independent of coordinate system. However, these tensors can be expressed in coordinate systems, for computational convenience. A efficient tool for studying material behavior is the finite element method (FEM). The FEM is a numerical technique for solving problems which are described by partial differential equations in the form Ax = b. By combinding conservation laws and constitutive relationships (rheology) it is possible to make a set of equations with as many unknowns as equations that can be solved. The domain of interest is divided into an assembly of finite elements (discretization of the continuum). In other words, the continuous physical problem is transformed into a discretized finite element problem with unknown nodal values. Interpolation (polynomials) functions are used interpolate the field variables over the element in terms of the nodal values. The degree of polynomial depends of the number of nodes per element. The matrix equation for the finite element relates the nodal values of the unknown function to other parameters. This is done by Galerkin’s method. To solve the global equation system one combines the local element equations for all elements used for discretization. Several book have been written upon the subject of continuum mechanics and the FEM (Mase (1970); Hughes (1987); Scheid (1988); Buchanan (1995); Kwon and Bang (2000); Zienkiewicz 44

Chapter 3. Methods


and Taylor (2000); Turcotte and Schubert (2002); Zienkiewicz et al. (2005) and others). I will in this section derive the quantities and relations needed for the FEM formulation.

3.3.2

Two dimensional stress and strain (rates)

The concept of stress and strain can be confusing. Partly because they require several numbers to represent them, and partly because the nomination to present them are unfamiliar. In this section I will briefly review the concept of stress and strain (rates) that are used in the numerical model. Stress tensor Following Sonin (2003a) I will derive the components of the stress tensor. To define the stress tensor we need to start with force. Force is a basic concept. It is a vector which has a magnitude and a direction. There are two types of forces: (i) Body forces which act on the material, independent of surrounding material (force of gravity). (ii) Surface forces which arise from interaction between bodies or from action of one part of the body on another part of the body across some internal surface. In a coordinate system forces can be represented as components parallel to the coordinate axis. If a force is acting on a surface the intensity of the force depends of the area it is distributed over. This called a traction (fig. 3.7(a)) and is defined as ∆F T~ = lim ∆A→0 ∆A

(3.1)

The traction is proportional with the magnitude of the force and inversely proportional with the area the force is distributed over. However, in the limit ∆A → 0 the traction is independent of the magnitude of the area, but will depend on the orientation, which is specified by the surface normal ~n. I.e. T~ = T~ (~r, t, ~n). The component perpendicular to the surface is called normal traction and the traction component parallel to the surface is called shear traction. In order to satisfy equilibrium, Newton’s third law, two opposite traction acting on opposite sides of the surface must exist. That is, at a given point the surface stress on a plane with orientation ~n through the point, must be equal in magnitude, and opposite in direction to that of the surface stress with an opposite orientation of −~n. This pair define the surface stress (fig. 3.7(b)): ~σ (~r, t, ~n) = −~σ (~r, t, −~n)

(3.2)

Besides the magnitude of the force, it is clear that the components of the surface 45


(a) Traction

Chapter 3. Methods

(b) Surface stress

Figure 3.7: Traction and stress pairs acting on a surface. The components perpendicular to the surface are called normal components and the components parallel to the surface is called shear components (Modified from Twiss and Moore (1992)).

stress, depends on both the orientation of the force and the surface it acts on. That is, the stress state of a point varies with the orientation of the plane through the point. It seems that one apparently has to deal with six independent variables (x, y, z, t and the two that define ~n), rather than four. Thus, for knowing the stress state at a point, we must be able to determine the surface stress of any plane through the point. Luckily it is sufficient to calculate the surface stress for the planes whose normals make up the Cartesian coordinate system. All other values can med calculated by linear superposition of the principal values. If we look at the stresses acting onto a given point ~r at time t, the values of the stresses exserted onto to the surfaces in the positive x, y -and z-direction(fig.) can be expressed in the terms of their components ~σ (~i) = σxx~i + σxy~j + σxz~k

(3.3)

~σ (~j) = σyx~i + σyy~j + σyz~k

(3.4)

~σ (~k) = σzx~i + σzy~j + σzz~k

(3.5)

The component σij acts in the direction of the j-th coordinate axis and on a surface perpendicular to the i-th coordinate axis. By using Newton’s third law, we can show that the stress exserted onto a surface of any orientation ~n can be expressed by the components of the principal stresses. Consider a infinitesimal tetrahedron fluid element which at time t is at distance ~r(t) from the origin (fig. 3.8). Three of tetrahedrons faces are normal to the x, y -and z-axis with areas, δAx , δAy , and δAz , respectively. The forth face has an arbitrary surface normal ~n and area δA. The areas of the three 46

Chapter 3. Methods


orthogonal faces are related to the fourth one by the following geometrical relationship:

n

δAy δAx z δA δAz

x y

Figure 3.8: Tetrahedron shaped fluid particle. (Modified from Sonin (2003a)).

δAx = cos θnx δA = nx δA

(3.6)

δAy = cos θny δA = ny δA

(3.7)

δAz = cos θnz δA = nz δA

(3.8)

where θni is the angle bewteen ~n and the i-th axis and ni is the i-th component of ~n. When thetrahedon is shrunk towards the point ~r which it is centered around, both the linear momentum (m~v ) and the body force (m~g ) becomes arbitrary small compared to the surface forces and are thus negligible. I.e in the limit as the tetrahedron is shrunk towards the point ~r(t), it follows from Newton’s third law that the surface forces must balance: ~σ (n)δA + ~σ (i)δAx + ~σ (j)δAj + ~σ (k)δAz = 0

(3.9)

We know that stress in a surface pointing in the +i, +j and +k directions is equal in magnitude and opposite of the stress pointing in the −i, −j and −k directions, respectively (eq. 3.2). Using this relationship and the geometrical relationship of the areas (eqs. 3.6-3.8) the force balance equation (eq. 3.9) becomes ~σ (n)δA + ~σ (i)nx δA + ~σ (j)ny δA + ~σ (k)nz δA = 0

(3.10)

~σ (n) = −~σ (i)nx − ~σ (j)ny − ~σ (k)nz

(3.11)

~σ (n) = ~σ (i)nx~σ (j)ny ~σ (k)nz

(3.12)

This result can be expressed in the terms of the component of the principal stresses (eqs. 3.3-3.5) 47


Chapter 3. Methods

σx (~n) = σxx nx + σyx ny + σzx nz

(3.13)

σy (~n) = σxy nx + σyy ny + σzy nz

(3.14)

σz (~n) = σxz nx + σyz ny + σzz nz

(3.15)

The result can be written in matrix form 

 σxx σxy σxz

 σ(~n) =   σyx σyy σzx

σzy

 nx

    σyz    ny  σzz nz

(3.16)

where he nine principal stress components make up the stress tensor. In two dimensions the stress tensor reduces to

"

σxx σxy

# (3.17)

σyx σyy

The components perpendicular to the surface (σxx , σyy ) are called normal stresses and the components parallel to the surface (σxy , σyx ) are called shear stresses. The stress is considered negative for compressive stress and positive for tensile stress2 .

The symmetry of the stress tensor The stress tensor needs to be symmetric. That is, σij = σji . This can be proved by looking at a infinitesimal cube with sides of lengths δx, δy, and δz (fig. 3.9). If the cube rotates with an angular velocity θ˙ and rotates as a solid body, the net torque around the z-axis (assuming there is no external body force exserting any torque) is given by Tz = Iz

dθ˙ dt

where Iz is the moment of inertia of the cube around the z-axis and

(3.18) dθ˙ dt

is the angular

acceleration around the z-axis. The moment of inertia is expressed as Z Z Z Iz =

ρ(x2 + y 2 )dV

This can be expressed as 2

Hence the negative sign convection for pressure.

48

(3.19)

Chapter 3. Methods


σyy σyx y + δy δx σxy

θx

σxx

δy

σxx z σxy y

y x

x + δx σyx

x

σyy

Figure 3.9: Symmetry of the stress tensor. Stresses acting on the faces of the infinitesimal fluid element with sides δx, δy, and δz. The element is rotated an infinitesimal amount around its z-axis with an angular velocity θ˙ (modified from Sonin (2003b)).

Z Iz =

− δx 2 δy 2

Iz Iz Iz Iz Iz Iz

Z

δy 2

Z

− δy 2 δz 2

δz 2

− δz 2

ρ(x2 + y 2 )dxdydz

# µ ¶ 1 δx 3 δx 2 = ρ 2 + 2 y dydz δz 3 2 2 − δy − 2 2 ¸ Z δy Z δz · 3 2 2 δx 2 = ρ + δxy dydz 12 − δy − δz 2 2 µ ¶ µ ¶ # Z δz " 2 δx3 δy 1 δy 3 = ρ 2 + 2δx dz 12 2 3 2 − δz 2 ¸ Z δz · 3 2 δx δy δxδy 3 + dz = ρ 12 12 − δz 2 · ¸ δx3 δy δz δxδy 3 δz =ρ 2 +2 12 2 12 2 · 3 ¸ δx δyδz δxδy 3 δz =ρ + 12 12 · 2 ¸ 2 δx + δy =ρ δxδyδz 12 Z

Iz

δx 2

Z

(3.20)

"

(3.21) (3.22) (3.23) (3.24) (3.25) (3.26) (3.27)

Substituting with last equation into eq. 3.18 we now obtain · Tz = ρ

¸ δx2 + δy 2 dθ˙ δxδyδz 12 dt 49

(3.28)


Chapter 3. Methods

The net torque in the z-direction around the particles center can also be expressed in terms of force and distance from the center of the particle δx δy σyx δyδz − 2 σxy δxδz 2 2 Tz = (σyx − σxy ) δxδyδz Tz = 2

(3.29) (3.30)

Combining eqs. 3.28 & 3.30 we obtain ·

σyx − σxy

¸ δx2 + δy 2 θ˙ =ρ 12 dt

(3.31)

As the cube is shrunk towards ~r(t) which it is centered around dx and dy → 0 and this yields σxy = σyx

(3.32)

This means that the stress tensor only have three or six independent components in two -and three-dimensions, respectively. To sum up, the stress tensor is simply a quantity that allows us to describe stress state of the material in a continuum in terms of quantities that depend on time and position, and not orientation of the surface which the stress is exserted. Physically the components of the stress tensor represents the components of the principal stresses at point ~r(t) at time t. The principal stresses are chosen to be the three stresses on the infinitesimal element whose normals point in the positive direction of the axes of the coordinate system. The stresses that are exserted on a plane of any possible orientation can be calculated by linear superposition of the principal values (eqs.3.13-3.15). So what does the stress state of a material actually mean? Simply put, it means that if you deform a body and cut out an infinitesimal cube, the stress state is represented by the forces needed to prevent the cube from changing its size or shape. Strain (rate) tensor Strain is used to quantify the deformation. By deformation we mean change of shape (distortion) or change is volume (dilatation)3 . Strain can be grouped into two classes: (i) Normal strain which is associated with dilation, as appose to shear strains which are not associated with dilation, but alter angles between lines that where mutually perpendicular before deformation. The normal strains can in their simplest for be 3

This do not include rigid body translation and/or rotation.

50

Chapter 3. Methods


defined as the change in length of the side parallel to one of the axis divided by their original length δx0 − δx δx δy 0 − δy = δy 0 δz − δz = δz

εxx =

(3.33)

εyy

(3.34)

εzz

(3.35)

where δx, δy, and δz is the length of the side in the unstrained form and δz 0 , δz 0 , and δz 0 is the length of the side after deformation. The shear strain is defined to be one-half of the decrease in angle ABC (fig. 3.11). εij =

1 (φ1 + φ2 ) 2

(3.36)

As with the stress tensor the strain tensor is symmetric and εij = εji . However, it can be shown that strain is in fact the spatial derivative of the displacement vector ~u. It is a measurement of how the displacement vector varies throughout the body. For small strains there is no difference between strain and strain rate. The latter is simply the time derivative of the former. Following Turcotte and Schubert (2002) I will in this section derive the strain tensor components. Normal strain Consider a infinitesimal rectangular element ABCD with sides of lengths δx, and δy (fig, 3.10). After an infinitesimal amount of deformation the element is in position A0 B 0 C 0 D0 . The components of displacement of corner A to A0 is ux (x, y) = x0 − x

(3.37)

uv (x, y) = y 0 − y

(3.38)

The strain is considered positive for extension and negative for shortening to agree with the sign convention used for stress. Corner B is displaced from x + δx, y to B 0 at x0 + δx0 , y 0 . The displacement in x-direction is ux (x + δx, y) = x0 + δx0 − (x + δx)

(3.39)

Similarly corner D is displaced from x, y + δy to D0 at x0 , y 0 + δy 0 . The displacement in x-direction is 51


Chapter 3. Methods

y

D’

y uy + (δuy/δy)δy y + δy

C’ C

D

δx’

δx δy’

δy y’ A’

B’

uy y A

B ux

ux + (δux /δx)δx x x’

x

x + δx

x’ + δx’

Figure 3.10: Deformation a rectangular fluid particle ABCD into the rectangular A0 B 0 C 0 D0 (modified from Turcotte and Schubert (2002)).

uy (x, y + δy) = y 0 + δy 0 − (y + δy)

(3.40)

Since the deformation is infinitesimally small we can write δx0 − δx ∂ux = δx ∂x ∂uy δy 0 − δy = δy ∂y

(3.41) (3.42)

Eqs. 3.39 & 3.40 then become ¸ δx0 − δx ux (x + δx, y) = x − x + δx δx ∂ux ux (x + δx, y) = ux (x, y) + δx ∂x £

¤

0

·

¸ δy 0 − δy δy uy (x, y + δy) = y − y + δy ∂uy uy (x, y + δy) = uy (x, y) + δy ∂y £

¤

0

(3.43) (3.44)

·

(3.45) (3.46)

If we substitute eq. 3.44 into eq. 3.39 and subtract eq. 3.37 we obtain £

¸ · ¤ £ ¤ ∂ux δx − [ux (x, y)] x0 + δx0 − (x + δx) − x0 − x = ux (x, y) + ∂x ∂ux δx0 = δx + δx ∂x 52

(3.47) (3.48)

Chapter 3. Methods


Similarly if we substitute eq. 3.46 into eq. 3.40 and subtract eq. 3.38 we obtain £

· ¸ ¤ £ 0 ¤ ∂uy y + δy − (y + δy) − y − y = uy (x, y) + δy − [uy (x, y)] ∂y ∂uy δy 0 = δy + δy ∂y 0

0

(3.49) (3.50)

From the definitions of the strain components (eqs. 3.33-3.35) we now obtain x δx + ∂u δx0 − δx ∂ux ∂x δx − δx = = δx δx ∂x ∂uy 0 δy + δy − δy ∂uy δy − δy ∂y = = = δy δy ∂y

εxx =

(3.51)

εyy

(3.52)

Shear strain Shear strains change the shape of the body and alter angles of planes that were mutually perpendicular before deformation. Consider the infinitesimal fluid particle ABCD in fig. 3.11. After deformation it has been distorted into a parallelogram. The angles φ1 and φ2 are have the following geometrical relationships y

D’

y uy + (δuy/δy)δy y + δy

C’ C

D

δx’

δx δy’

δy y’ A’

B’

uy y A

B ux

ux + (δux /δx)δx x

x

x’

x + δx

x’ + δx’

Figure 3.11: Deformation of rectangular fluid particle ABCD into a parallelogram AB 0 C 0 D0 (modified from Turcotte and Schubert (2002)).

uy (x + δx, y) δx ux (x, y + δy) tan(φ2 ) = δy

tan(φ1 ) =

53

(3.53) (3.54)


Chapter 3. Methods

Since the deformation is infinitesimal we can assume that tan(φi ) = φi . We can then write uy (x + δx, y) δx ux (x, y + δy) φ2 = δy

φ1 =

(3.55) (3.56)

We can express the displacements as ∂ux δy ∂y ∂uy uy (x + δx, y) =uy (x, y) + δx ∂x ux (x, y + δy) =ux (x, y) +

(3.57) (3.58)

For simplicity we assume that ux (x, y) = 0 and uy (x, y) = 0 we obtain ∂ux δy ∂y ∂uy δx uy (x + δx, y) = ∂x ux (x, y + δy) =

(3.59) (3.60)

If we now substitute eqs. 3.59 & 3.60 into eqs. 3.55 & 3.56 we get uy (x + δx, y) φ1 = = δx ux (x, y + δx) φ2 = = δy

∂uy ∂x δx

∂uy δx ∂x ∂ux δy ∂ux ∂y = δy ∂y =

The equation for shear strain (eq. 3.36) can now be expressed as µ ¶ 1 ∂uy ∂ux εxy = + 2 ∂x ∂y

(3.61) (3.62)

(3.63)

In numerical modeling the shear strain or angular strain is in often expressed as µ ¶ ∂uy ∂ux γxy = 2εxy = + (3.64) ∂x ∂y Hence we now have derived all the components for the strain tensor. This can be expressed in general terms as 1 εij = 2

µ

∂uj ∂ui + ∂xj ∂xi

¶ (3.65)

As mentioned above the strain rate is the time derivative of the strain for small incremental strains and the strain rate tensor can be expressed as · µ ¶¸ µ ¶ εij ∂uj ∂vj d 1 ∂ui 1 ∂vi ε˙ij = = + = + dt dt 2 ∂xj ∂xi 2 ∂xj ∂xi where v =

∂u ∂t

is the displacement velocity. 54

(3.66)

Chapter 3. Methods

3.3.3


Conservation laws

The modeling of physical phenomena is based on that material systems behave according to universal physical laws. For out purpose this mean that important physical properties like mass, linear motion and angular motion must be invariant (i.e. remain unchanged during transformation). This is done by applying conservation laws to the properties. In this section I will derive the conservation equations needed to formulate the quantitative problem needed for implementing the numerical code. The laws are stated in terms of a material volume (fig. 3.12(a)). A material volume is defined as by closed bounding surface that envelops the its material particles at a certain time. Material particles are infinitesimal particles so small that insintric properties are essentially uniform within it (fig. 3.12(b)). That is, the properties only change with time.

55


Chapter 3. Methods

z

V(t+ ∆t)

V(t) x

y

(a) The material volume V moves with the material particles it envelopes. z

v(t + ∆t)

v(t)

F(t) r(t) x

y

(b) The motion of a material particle between time t and t + ∆t. A force F (t) acts on the material particle at time t, forcing it to change its velocity v(t) to v(t + ∆t). The material particle is so small that its insitric properties only change with time.

Figure 3.12: Motion of a material volume and a material particle between time t and t + ∆t (modified from Sonin (2003b)).

56

Chapter 3. Methods


Conservation of mass - The continuity equation Every material particle in the material has a mass. The mass of a material particle is given by

Z m=

ρ(x, t)dV

(3.67)

V

where ρ(x, t) is the density field and dV = dxdydz, represents a volume element inside the material volume. The conservation of mass requires that the mass of a specific material particle remains constant: dm d = (ρδV ) = 0 dt dt

(3.68)

To derive a conservation law for the entire material volume the law is applied to each of the material particles that comprise the material volume, and summing. This sum can be viewed as an integral over the volume of mass which is expressed as a function of position ~x and time t (field function): d dt

Z

Z ρ(~x, t)dV =

V (t)

V (t)

∂ρ dV + ∂t

Z ρ~v dS = 0

(3.69)

S(t)

V(t) and S(t) signifies integration over the material volume and surface, respectively, at time t. The second term can be converted into a volume integral using Gauss’s/ divergence theorem: Z

Z ρ~v dS =

S(t)

(∇ · ρ~v ) dV,

V = dS

(3.70)

V (t)

Hence equation eq. 3.69 becomes d dt

Z

µ

Z ρ(~x, t)dV =

V (t)

V (t)

¶ ∂ρ + ∇ · (ρ~v ) dV = 0 ∂t

(3.71)

Sine the material volume V is arbitrary and eq. 3.71 must hold for any material volume, the integrand must be zero ∂ρ + ∇ · (ρ~v ) = 0 ∂t

(3.72)

This can be expressed as ∂ρ + ~v · ∇ρ + ρ∇ · ~v = 0 ∂t The total derivative can expressed as ∂ρ dρ = + ~v · ∇ρ dt ∂t 57

(3.73)

(3.74)


Chapter 3. Methods

Substituting eqs. 3.73 & 3.74 into eq. 3.71 the conservation of mass yields d dt

Z

Z ρ(~x, t)dV =

V (t)

V (t)

dρ + ρ∇ · ~v dV = 0 dt

(3.75)

This equation is often referred to as the continuity equation. For incompressible fluids the density does not change with time and and the continuity equation reads Z ∇ · ~v dV = 0

(3.76)

V (t)

But how does this actually relate to a incompressible fluid where the material flow? If we consider a two-dimensional4 infinitesimal rectangular element with sides δx and δy and look a the flow in the xy-plane (fig. 3.13), the conservation of mass simply states that there is no net flow in or out of our element. Following Turcotte and Schubert v + (δv/δy) δy

y + δy δx

u

y

δy

u + (δu/δx) δx

y x

x + δx

u

x

Figure 3.13: Fluid flow across a two-dimensional infinitesimal rectangular element (Modified from Turcotte and Schubert (2002)).

(2002) the flow rate per unit area in the x-direction at x is u. At x + δx the flow rate per unit are is u(x + δx) = u +

∂u δx ∂x

(3.77)

The net flow in x-direction between x and x + δx is 4

Since the numerical model is 2D, from here on the analysis will be conducted using 2-dimensional

material particles/elements

58

Chapter 3. Methods


u+

∂u ∂u δx − u = δx ∂x ∂x

(3.78)

Similarly the flow rate per unit area in the y-direction at y is v. The flow rate per unit area at y + δy is v(y + δy) = v +

∂v δy ∂y

(3.79)

Hence the net flow in y-direction between y and y + δy is v+

∂v ∂v δy − v = δy ∂y ∂y

(3.80)

The net outward flow in the x -and y-direction is then found by multiplying the net flow with the area of the face across which the flow occurs ∂u δxδy ∂x ∂v δyδx ∂y

(3.81) (3.82)

The total net outward flow out of the element with area δA = δxδy is then ³

∂u ∂x

+

∂v ∂y

´ δyδx

δyδx

=

∂u ∂v + ∂x ∂y

(3.83)

If there is no density variations, there can be no net flow in or out of the infinitesimal rectangular element and the conservation of fluid reads ∂u ∂v + =0 ∂x ∂y

(3.84)

Conservation of momentum - The equations of equilibrium Change in the material particles momentum can accomplished by exerting forces onto the surface of the particles and/or by body forces advecting momentum. First, there is the stress that acts onto the surfaces of the two-dimensional infinitesimal material particle with local form f~ = ~σ dS. The second force acting upon the material particle ~ = ρ~g , where ~g is the acceleration of gravity. For the two-dimensional is gravity, G infinitesimal infinitesimal rectangular element of fig. 3.14 the forces in x and y-direction are X

µ ¶ µ ¶ ∂σxy ∂σxx Fx = σxx + ∂x ∂y − σxx ∂y + σxy + ∂y ∂x − σxy ∂x = 0 (3.85) ∂x ∂σy 59


Chapter 3. Methods

(σyy + δσyy/δy) δx (σyx + δσyx/δy) δx y + δy δx (σxy + δσxy/δx) δy (σxx + δσxx/δx) δy

δy

σxx δy σxy δy ρg δxδy

y

y x

x + δx σyx δx

x

σyy δy

Figure 3.14: Forces acting on a two-dimensional infinitesimal rectangular element (Modified from Turcotte and Schubert (2002)).

X

µ ¶ µ ¶ ∂σxy ∂σyy Fy = σxy + ∂x ∂y − σxy ∂y + σyy + ∂y ∂x − σyy ∂x + ρ~g ∂x∂y = 0 ∂x ∂σy (3.86)

σxy = σyx to conserve angular momentum. If they were not equal in magnitude, the material particle would rotate around it own axis and accelerate towards infinity (see 3.3.2 The symmetry of the stress tensor on page 48). Simplification yields the equations of equilibrium (force balance equations): ∂σxx ∂σxy + =0 ∂x ∂y ∂σxy ∂σyy + + ρ~g = 0 ∂x ∂y

3.3.4

(3.87) (3.88)

Viscous rheology - The constitutive equation

We now have equations for mass conversation, force balance, stress, strain and strain rates. However, we have more unknowns than we have equations. In order to solve anything the system needs to be closed. This is done by relating the stress generated by deformation to the strain rates. This is called a constitutive relationship or more familiar term, rheology. The equation are then called the constitutive equations and thy relate the total stress σ to the strain rate ε. ˙ In this section I will, following Sonin (2003a), derive the Newtonian stress tensor components. 60

Chapter 3. Methods


Newtonian fluid In contrast to a elastic continuum, where stresses and strains are linearly related and strain is reversible up to a certain yield strength, fluids cannot store energy applied by an external force and have no “memory” of their shape or form prior to any applied deformation. Deformation is irreversible and once the external force is removed the stresses drop. However, fluids tend to resist the rate of deformation. The higher the applied shear stress, the higher the shear strain rate5 . The simplest fluid model is the Newtonian model. A Newtonian fluid have the following characteristics (Sonin, 2003a): - The shear stress is proportional to the rate of shear strain. - The shear stress is zero when the rate of shear is zero. - The stress and strain rate have a linear realtionship which is isotropic. Shear stress We recall that the incremental angular strain (eq. 3.64) is δγxy = 2δεxy =

∂ux ∂uy + ∂y ∂x

(3.89)

The strain rate which is the time derivative of the strain can be written as

γ˙ xy =

∂δγxy ∂ = ∂t ∂t

µ

∂ux ∂uy + ∂y ∂x

¶ =

∂vx ∂vy + ∂y ∂x

(3.90)

The two first assumptions require that µ σxy = µγ˙ xy = µ


¶ (3.91)

where coefficient µ is the viscosity of the fluid. In general terms this can be written as µ σij = µ

5

∂vj ∂vi + ∂xj ∂xi

¶ ,

i 6= j

(3.92)

In general fluids show this relationship, although for some fluids the stress is inversely proportional

to the strain rate.

61


Chapter 3. Methods

Normal stress To derive constrictive equations for normal stresses and normal strains is not a simple as with shear stresses, but can be done by noting that normal and shear deformations generally occur hand in hand and see how the normal stresses and strains relate to the shear stresses and strains. If we look at a fluid element with sides h at time t and let the fluid element shrink to its center (fig. 3.15). The fluid element if deformed between time t and t + ∆t as the figure shows. The sides AB and AD have rotated by unequal amounts which cause shear deformation in the fluid particle. This shear deformation will cause the extend AC diagonal and shorten the BD diagonal. In a coordinate system (x0 , y 0 ) which rotated 45◦ to the principal coordinate system (x, y), the deformation of the diagonals will be linear. From the previous section we now know the relationship between the shear stress and the shear strain rate in the principal coordinate system (x, y). Since the principal coordinate system is arbitrary, the stresses and strains must relate to the rotated coordinate system in general form. This is done by looking at the forces that are exserted on to triangular halves of the fluid particle, ∆ABD and ∆ACD (fig. 3.16(a)). Force balance in the x-direction on ∆ABD (fig. 3.16(b)) requires 0 0 −σxx BD cos θ + σyx BD sin θ + σxx AD + σyx AB = 0

AB AD + σyx =0 BD BD 0 0 −σxx cos θ + σyx sin θ + σxx cos θ + σyx sin θ = 0 0 0 −σxx cos θ + σyx sin θ + σxx

0 0 σxx cos θ − σyx sin θ = σxx cos θ + σyx sin θ

t

C

D

(3.96)

t + ∆t

h B’

θ

(3.95)

h

x’

y’

(3.94)

D’

C’ y

(3.93)

45o x

A’ A

B

Figure 3.15: Relation between normal and shear strain. The infinitesimal rectangular element is deformed between time t and t + ∆t. (Modified from Sonin (2003a)).

Similarly force balance in the y-direction requires 62

Chapter 3. Methods


0 0 −σxx BD sin θ − σxy BD cos θ + σyy AB + σxy AD = 0

AB AD + σxy =0 BD BD 0 0 −σxx sin θ − σxy cos θ + σyy sin(theta) + σxy cos θ = 0 0 0 −σxx sin θ − σxy cos θ + σyy

0 0 σxx sin θ + σxy cos θ = σyy sin(theta) + σxy cos θ = 0

(3.97) (3.98) (3.99) (3.100)

Multiplying eq. 3.96 with cos θ and eq. 3.100 with sin θ, and adding the result we get 0 0 σxx cos2 (θ) − σyx sin θ cos θ = σxx cos2 (θ) + σyx sin θ cos θ (3.101) 0 0 σxx sin2 θ + σxy sin θ cos θ = σyy sin2 (theta) + σxy sin θ cos θ = 0 (3.102) ¡ 2 ¢ 0 σxx sin θ + cos2 θ = σxx cos2 θ + σyy sin2 θ + σxy sin θ cos θ + σyx sin θ cos θ (3.103)

Since sin2 θ + cos2 θ = 1, σxy = σyx , and 2 sin θ cos θ = sin 2θ eq. 3.103 reduces to 0 σxx = σxx cos2 θ + σyy sin2 θ + σxy sin 2θ

(3.104)

If we look at ∆ACD, the force balance in the x-direction on ∆ACD (fig. 3.16(b)) requires 0 0 −σyy AC sin θ + σyx AC cos θ + σxx AD − σyx CD = 0

(3.105)

AD CD − σyx =0 AC AC 0 0 −σyy sin θ + σyx cos θ + σxx sin θ − σyx cos θ = 0

(3.107)

0 0 σyy sin θ − σyx cos θ = σxx sin θ − σyx cos θ

(3.108)

0 0 −σyy sin θ + σyx cos θ + σxx

(3.106)

Similarly the force balance in y-direction on ∆ACD requires 0 0 σyy AC cos θ + σxy AC sin θ − σyy CD + σxy AD = 0

AD CD + σxy =0 AC AC 0 0 σyy cos θ + σxy sin θ − σyy cos θ + σxy sin θ = 0 0 0 σyy cos θ + σxy sin θ − σyy

0 0 σyy cos θ + σxy sin θ = σyy cos θ − σxy sin θ

(3.109) (3.110) (3.111) (3.112)

Multiplying eq. 3.108 with sin θ and eq. 3.112 with cos θ, and adding the result we get

0 0 σyy sin2 θ − σyx sin θ cos θ = σxx sin2 θ − σyx sin θ cos θ

(3.113)

0 0 σyy cos2 θ + σxy sin θ cos θ = σyy cos2 θ − σxy sin θ cos θ

(3.114)

0 σyy

2

2

= σxx sin θ + σyy cos θ − σxy sin 2θ 63

(3.115)


Chapter 3. Methods

∆ACD

∆ABD a

t + ∆t b

a

b

a+d D

t + ∆t

D t

C

b +c

t

d A

A

B c

y x’

y’ θ

45o x

(a) Deformation of the two triangles ∆ABD and ∆ACD.

∆ACD

∆ABD

σyyCD σyxCD D

D σxyAD

θ

σxx’ sinθ BD σxx’ cosθ BD

σxxAD

C

σxyAD

σxx’BD

σxxAD

θ σyy’AC

A σyyAB σyx’BD

σyy’ sinθ AC

A

B

σyy’ cosθ AC

σyxAB σyx’ cosθ AC

θ

σyx’AC

σyx’ sinθ AC σyx’BD

σyx’BD

θ y x’

y’

o

θ

45

x

(b) Stresses exserted on the two triangles ∆ABD and ∆ACD at time t.

Figure 3.16: The infinitesimal rectangular element is deformed between time t and t + ∆t. (Modified from Sonin (2003a)).

64

Chapter 3. Methods


With θ = 45◦ (for simplicity) eqs. 3.104 & 3.115 reduces to σxx + σyy + σxy 2 σxx + σyy = − σxy 2

0 σxx =

(3.116)

0 σyy

(3.117)

Combining eqs. 3.116 & 3.117 we obtain 0 0 σxx − σyy = 2σxy

(3.118)

Substituting eq. 3.91 into this result we then get µ 0 0 σxx − σyy = 2µγ˙ xy = 2µ


¶ (3.119)

which relates the diagonal stress tensor terms in the (x0 , y 0 ) coordinate system with the angular strain in the (x, y) coordinate system. What remains is to relate the angular strain rate in the (x, y) coordinate system to the strain rates in the (x0 , y 0 ) coordinate system. The displacements a, b, and c (fig. 3.16(a)) are related to the incremental normal strains δε and angular strains δγ in the (x, y) coordinate system by c h a = h b+d = h

δεxx =

(3.120)

δεyy

(3.121)

δγxy

(3.122)

The normal strains in the (x0 , y 0 ) coordinate system can be calculated by the stretching and shortening of the diagonals. Since ∆ACD and ∆ABD are isoscoles6 at time t, and the deformation between time t and t + ∆t is infinitesimally small we obtain √ √ 2h2 = h 2 p δ(AC) = (a − d)2 + (b − c)2 AC =

δε0xx

δAC = = AC

(3.123) (3.124)

µ ¶ √ + b+c 1 c a b+d 1 2 √ = + + = (δεxx + δεyy + δγxu ) (3.125) 2 h h h 2 h 2

a+d √ 2

√ √ 2h2 = h 2 p δ(BD) = (a − d)2 + (b − c)2 BD =

δε0yy = 6

δBD = BD

a−d √ 2

√ + c−b 1 √ 2 = 2 h 2

(3.126) (3.127) µ

c a b+d + − h h h

Triangles with two sides of equal length.

65

¶ =

1 (δεxx + δεyy − δγxu ) (3.128) 2


Chapter 3. Methods

Subtracting eqs. 3.128 from 3.125 we get 1 1 δε0xx − δε0yy = (δεxx + δεyy + δγxu ) − (δεxx + δεyy − δγxu ) 2 2 0 0 δεxx − δεyy =δγxy

(3.129) (3.130)

This expression shows that the angular strain in the (x, y) coordinate system is equal to the difference of the normal strains in the (x0 , y 0 ) coordinate system. The relation between the strain rates then becomes Dγxy Dε0xx Dε0yy − = ⇐⇒ ε˙0xx − ε˙0 = γ˙ xy Dt Dt Dt

(3.131)

When this equation is inserted into our expression relating the diagonal stress tensor terms in the (x0 , y 0 ) coordinate system with the angular strain in the (x, y) coordinate system (eq. 3.119), we obtain 0 σxx

−

0 σyy

= 2µγ˙ xy

¡ ¢ = 2µ ε˙0xx − ε˙0yy = 2µ

µ

∂vy0 ∂vx0 − ∂x0 ∂y 0

¶ (3.132)

Similarly for the (x0 , z 0 ) plane we can get µ 0 σxx

−

0 σzz

= 2µ

∂vx0 ∂vz0 − ∂x0 ∂z 0

¶ (3.133)

Adding the two last equations we then obtain µ

¶ ∂vy0 ∂vx0 ∂vx0 ∂vz0 − + − = 2µ − + − ∂x0 ∂y 0 ∂x0 ∂z 0 µ ¶ 0 ∂vy ∂vx0 ∂vx0 ∂vz0 0 0 0 0 3σxx = σxx + σyy + σzz + 2µ 3 0 − − − ∂x ∂x0 ∂y 0 ∂z 0 µ ¶ 0 0 0 0 σxx + σyy + σzz ∂vy ∂v 0 ∂vx0 2 ∂vz0 0 + 2µ x0 − µ σxx = + + 3 ∂x 3 ∂x0 ∂y 0 ∂z 0 0 σxx

0 σyy

0 σxx

0 σzz

(3.134) (3.135) (3.136)

Since the (x0 , y 0 ) coordinate system was chosen arbitrarily, this last expression must hold for any orientation. Expressed in more general terms we obtain our final result σxx = −p + 2µ where p = −

σxx +σyy +σzz 3

∂vx 2 − µ∇ · ~v ∂x 3

(3.137)

is the pressure. We can now combine eqs. 3.92 & 3.137 for a

general expression for all the terms of the Newtonian stress tensor 2 σij = −pδij − µ∇ · ~v δij + µ 3 66

µ


¶ (3.138)

Chapter 3. Methods


δij is the Kroenecker delta. The Kroenecker delta and is basically an identity matrix with i roms and j columns.

½ δij =

1,

i=j

0,

i 6= j

(3.139)

The second and third terms in eq. 3.138 are the volumetric and deviatoric stress components. That, is the stress components responsible for dilatation and distortion, respectively. For a incompressible fluid the volumetric strain component is zero. From the equation of continuity (eq. 3.76) we recall that ∇ · ~v = 0

(3.140)

Hence the constitutive equation for an incompressible fluid reads µ σij = −pδij + µ


¶ (3.141)

σij = −pδij + 2µεdev ij

(3.142)

σij = −pδij + τij

(3.143)

where σ is the total stress and τ is the deviatoric stress. However, for the finite element formulation the assumption that the material is totally incompressible is not truly valid. The trick is to use a mixed formulation and assume that the volumetric bulk strain can be be described by a elastic rheology p = −K∆V

(3.144)

where K is the bulk compressibility and ∆V is the volume change. The pressure becomes larger when volume is reduces and less when the volume is increased. The pressure gradient then becomes ∂p = −K∆V˙ = −K ∂t

µ

∂vx ∂vy + ∂x ∂y

¶ (3.145)

We now have a alternative expression for the continuity equation (eq. 3.84). The compressibility is set to a very high value making the divergence of the velocity approach zero. This can be viewed as making the elastic bulk rheology of the material very stiff and practically incompressible. This is often referred to as a penalty approach for incompressible flow (Hughes, 1987). We recall from eq. 3.66 that the total the strain rate tensor7 is 7

For the finite element formulation we use γ˙ ij = 2ε˙ij =

67

∂vi ∂xj

+

∂vj ∂xi

for angular strain.


Chapter 3. Methods " ε˙tot ij

=

∂vx ∂x ∂vx ∂y

+

∂vx ∂y

∂vy ∂x

+ ∂vy ∂y

∂vy ∂x

# (3.146)

The total strain rate ε˙tot can be split into a volumetric and deviatoric part 1 ε˙tot = ε˙vol + ε˙dev = T r ε˙tot + ε˙dev 3

(3.147)

Hence the deviatoric strain rate can then be expressed in terms of the total strain rate 1 ε˙dev = ε˙tot − T r ε˙tot 3 " dev

ε˙

=

∂vx ∂x

∂vx ∂y

+

∂vy ∂x

(3.148)

∂vx ∂y

+ ∂vy ∂y

∂vy ∂x

#

1 − 3

"

∂vx ∂x

+

∂vy ∂y

0 ∂vx ∂x

0

#

+

(3.149)

∂vy ∂y

For the finite element method we now use Void notation8 and obtain

{ε˙dev } =

   

   

∂vx ∂x ∂vy ∂y

   1

∂vx ∂x ∂vy ∂y

− ³ ´   3    ∂v y   1 ∂vx +  0 2 ∂y ∂x    ∂vx 2 −1 0   ∂x    1 ∂vy dev  {ε˙ } =  −1 2 0  ∂y  3  x ∂vy 0 0 32  ∂v ∂y + ∂x

    (3.150)

      

(3.151)

  

Inserting the last equation into eq. 3.142 the constitutive equation for a incompressible Newtonian fluid can now be expressed as     σxx σyy    σ xy

       1 4 −2 0      1     = −p + µ −2 4 0  1  3          0   0 0 3  {z } |    

∂vx ∂x ∂vy ∂y ∂vx ∂y

+

    ∂vy ∂x

  

(3.152)

Ddev

where Ddev is the deviatoric material property matrix. In more general terms this reduces to σij = −pδi j + Ddev εdev ij = −pδi j + τij

(3.153)

We now have 6 unknowns (stresses (τxx , τyy , and τxy ), velocities (vx and vy ), and pressure (p)), but only 5 equations (the equations of equilibrium (eqs. 3.87 & 3.88) and the constitutive equation (eq. 3.153)). However, by using the alternative formulation 8

Vector ({σ}) representation of matrices ([σ]).

68

Chapter 3. Methods


for the continuity equation (eq. 3.145) we have the additional equation we need. The equations of equilibrium (eqs. 3.87 & 3.88) can now be expressed as ∂τyx ∂σxx ∂σyx ∂ ∂p ∂τxx ∂τyx + = (−p + τxx ) + =− + + ∂x ∂y ∂x ∂y ∂x ∂x ∂y ∂σyy ∂σxy ∂τxy ∂τxy ∂ ∂p ∂τyy + = (−p + τyy ) + =− + + ∂y ∂x ∂x ∂x ∂y ∂y ∂x

(3.154) (3.155)

The alternative continuity equation (eq. 3.145) and the equilibrium equations (eqs. 3.154 & 3.155) are used in the finite element formulation were we seek an equation set in the form of Ax = b, where A is an equation matrix, x and b are column vector containing the unknowns (velocities and pressures) and constants, respectively. The solution of Ax = b is simply x = A/b. However, before we are able to that we need need to define iso-parametric elements and shape functions (see below).

3.3.5

Two-dimensional iso-parametric elements

Iso-parametric elements are based on parametric definition of both coordinate and displacement functions (see Hughes (1987); Kwon and Bang (2000); Zienkiewicz and Taylor (2000) or Zienkiewicz et al. (2005) for a detailed introduction). Shape functions are defined in terms of the ideal (orthogonal) coordinate system (−1 ≤ ξ ≤ 1, −1 ≤ η ≤ 1) and are used for both the specification of the element shape and interpolation (approximation) of the displacement -and pressure field (hence the name iso-parametric elements): x ˜≈ v˜x (x, y) ≈ p˜(x, y) ≈

X X X

Niv (ξ, η)xi Niv (ξ, η)vx,i

y˜ ≈ v˜y (x, y) ≈

Nip (ξ, η)pi

X X

Niv (ξ, η)yi

(3.156)

Niv (ξ, η)vy,i

(3.157) (3.158)

where vi and pi are the velocity -and pressure components of a point with local coordinates (ξ, η). v˜i and p˜ are the displacements and pressure values, respectively, of the nodes of the finite element. x ˜, y˜ are the point coordinates in the physical coordinate system and xi , yi are the coordinates of the element nodes. The model used in this thesis uses 7-node triangular elements (fig. 3.17). Note that the inclined edge is 1 − ξ − η. A function ζ is then defined as ζ = ζ(ξ, η) = 1 − ξ − η

(3.159)

The triple coordinates ξ, η, and ζ define the triangular coordinates. The triangular 69


Chapter 3. Methods

η 1x3

ζ=1−ξ−η 5

6 7

0x 0

1

2 x 1

4

ξ

Figure 3.17: Local reference element used for numerical solution. 7 nodes for velocity degrees of freedom (filled circles) and 3 for the pressure degree of freedom (crosses).

coordinates are 0 along the edge and 1 at the opposite vertex. The high-order shape functions for the triangle are defined by the triangular coordinates. This is done by Lagrange type interpolation (Hughes, 1987). The velocity shape functions are expressed as

N1v = ζ(2ζ − 1) + 3ζηξ

(3.160)

N2v = η(2η − 1) + 3ζηξ

(3.161)

N3v = ξ(2ξ − 1) + 3ζηξ

(3.162)

70

Chapter 3. Methods


N4v = 4ηξ − 12ζηξ

(3.163)

N5v = 4ζξ − 12ζηξ

(3.164)

N6v = 4ζη − 12ζηξ

(3.165)

N7v = 27ζηξ

(3.166)

and their derivatives with respect to their local coordinates ∂N1v ∂η ∂N2v ∂η ∂N3v ∂η ∂N4v ∂η ∂N5v ∂η ∂N6v ∂η ∂N7v ∂η

= 1 − 4ζ + 3ζξ − 3ηξ = −1 + 4η + 3ζξ − 3ηξ = 3ζξ − 3ηξ = 4ξ + 12ηξ − 12ζξ = −4ξ + 12ηξ − 12ζξ = 4ζ − 4η + 12ηξ − 12ζξ = −27ηξ + 27ζξ

∂N1v ∂ξ ∂N2v ∂ξ ∂N3v ∂ξ ∂N4v ∂ξ ∂N5v ∂ξ ∂N6v ∂ξ ∂N7v ∂ξ

71

= 1 − 4ζ + 3ζη − 3ηξ

(3.167)

= +3ζη − 3ηξ

(3.168)

= −1 + 4ξ + 3ζη − 3ηξ

(3.169)

= 4η − 12ζη + 12ηξ

(3.170)

= 4ζ − 4ξ − 12ζη + 12ηξ

(3.171)

= −4η − 12ζη + 12ηξ

(3.172)

= 27ζη − 27ηξ

(3.173)


Chapter 3. Methods

The derivatives of the velocity shape functions are converted to the physical coordinate system are calculated by the means of the inverse Jacobian(see below). There is only three pressure shape functions (fig. 3.17) and they are expressed as

3.3.6

N1p = 1 − ξ − η

(3.174)

N2p = ξ

(3.175)

N3p = η

(3.176)

Finite element formulation

To develop a finite element formulation we use Galerkin’s method of weighted residuals (Hughes, 1987; Kwon and Bang, 2000; Zienkiewicz and Taylor, 2000; Zienkiewicz et al., 2005). The Galerkin’s method is based upon assuming an approximate solution for the differential equation which one wants to solve. Since the assumption is an approximation, the differential will not be satisfied and there will be a (error) residual. The residual is optimized with respect to some parameter by the weighed residual method. Given a differential equation F which can be approximated as F ≈ F˜ = a0 + a1 x + a2 x2 + a3 x3 + . . .

(3.177)

where ai are unknown constants. The approximated solution must satisfy the boundary conditions and contain ate least one more constant than there are boundary conditions. The residual is the difference between the exact solution F and the approximation F˜ R = F − F˜

(3.178)

Since F 6= F˜ for all values of x within in the domain the residual will never vanish. A weighting function w is then selected and the weighted average of the residual over the domain is set to 0.

Z Ω

wi (x)R(x, ai )dV = 0

(3.179)

The Galerkin’s method uses the shape functions as constants for the approximation and as weighting functions and the last equation becomes Z Ni (x)R(x, Ni )dV = 0

(3.180)

Ω

To find the finite element formulation we apply this method to our force balance equations and continuity equation (eqs. 3.154 & 3.155). The force balance equations is weighted by 7 velocity shape functions N v and becomes 72

Chapter 3. Methods

3.3. Numerical modeling Z ( N1v

+

Ω

∂τxx ∂x ∂τxy ∂x

+

Ω

∂τxx ∂x ∂τxy ∂x

Z ( N2v

Z ( N7v

Ω

+

+

∂τxy ∂y ∂τyy ∂y ∂τxy ∂y ∂τyy ∂y

− − − −

∂p ∂x ∂p ∂y ∂p ∂x ∂p ∂y

.. . ∂τxx ∂x ∂τxy ∂x

+ +

∂τxy ∂y ∂τyy ∂y

− −

∂p ∂x ∂p ∂y

) dΩ = 0

(3.181)

dΩ = 0

(3.182)

dΩ = 0

(3.183)

)

)

Applying integration by parts gives us µ

Z

¶

Z

Z

∂Niv σxx dΩ (3.184) ∂x Ω Z ZΩ ∂Niv + Niv σxy ny dx − σxy dΩ (3.185) Ω ∂y Z Z ∂Niv − Niv pnx dy + pdΩ (3.186) Ω ∂x R R R Since the surface terms Niv τxx nx dy = − Niv σxy ny dx, and Niv pnx dy which is a Niv

∂τxx ∂τxy ∂p + − ∂x ∂y ∂x

Niv τxx nx dy

dΩ =

−

projection of the velocity function upon the surface must be 0 to conserve mass, our equation set becomes Z ( Ω

∂N1v ∂x τxx ∂N1v ∂x τxy

Ω

∂N2v ∂x τxx ∂N2v ∂x τxy

Z (

Z ( Ω

+ + + +

∂N1v ∂y τxy ∂N1v ∂y τyy ∂N2v ∂y τxy ∂N2v ∂y τyy

− − − −

∂N1v ∂x p ∂N1v ∂x p ∂N2v ∂x p ∂N2v ∂x p

.. . ∂N7v ∂x τxx ∂N7v ∂x τxy

+ +

∂N7v ∂y τxy ∂N7v ∂y τyy

− −

∂N7v ∂x p ∂N7v ∂x p

) dΩ = 0

(3.187)

dΩ = 0

(3.188)

dΩ = 0

(3.189)

)

)

In matrix form we can write:      Z     Ω     

∂N1v ∂x

0 ∂N2v ∂x

0 .. . ∂N7v ∂x

0

0 ∂N1v ∂y

0 ∂N2v ∂y

.. .

0 ∂N7v ∂y

∂N1v ∂y ∂N1v ∂x ∂N2v ∂y ∂N2v ∂x

.. .

∂N7v ∂y ∂N7v ∂x



       τxx    τyy   τ  xy    

73

           −         

∂N1v ∂x ∂N1v ∂y ∂N2v ∂x ∂N2v ∂y

.. .

∂N7v ∂x ∂N7v ∂y

         p dΩ = 0      

(3.190)


Chapter 3. Methods

Substitution with the constitutive equations (eq. 3.153) gives us:      Z     Ω     

∂N1v ∂x

∂N1v ∂y ∂N1v ∂x ∂N2v ∂y ∂N2v ∂x

0 ∂N1v ∂y

0 ∂N2v ∂x

0 ∂N2v

0 .. .

∂y

.. .

∂N7v ∂x

.. .

∂N7v ∂y ∂N7v ∂x

0 ∂N7v ∂y

0





       h  i  dev D          

∂vx ∂x ∂vy ∂y ∂vx ∂y

+

∂vy ∂x

           −         

∂N1v ∂x ∂N1v ∂y ∂N2v ∂x ∂N2v ∂y

.. .

∂N7v ∂x ∂N7v ∂y

         p dΩ = 0      

(3.191)

The velocities and pressure are interpolated using the shape functions introduced in the previous section (eqs. 3.157 & 3.158):

v˜x (x, y) ≈

7 X

Niv (ξ, η) vx,i

(3.192)

Niv (ξ, η) vy,i

(3.193)

Nip (ξ, η) pi

(3.194)

i=1

v˜y (x, y) ≈ p˜ (x, y) ≈

7 X i=1 3 X i=1

Hence the velocities can be expressed as

(

v˜x

)

" ≈

v˜y

N1v

0

0

N1v

N2v

0

0

N2v

N7v

··· ···

0

  vx,1      vy,1      # v   x,2 0 vy,2 N7v   ..   .       vx,7     v y,7

                            

= [N v ] {v}

(3.195)

The strain rate vector can now be expressed as:       

   

∂vx ∂x ∂vy ∂y ∂u ∂y

+

∂v ∂x

  

  = 

∂N1v ∂x

0 ∂N1v ∂y

0 ∂N1v ∂y ∂N1v ∂x

∂N2 1v ∂x

0 ∂N2v ∂x ∂N2v ∂x

0 ∂N2v ∂y

Similarly the pressure can be expressed as 74

···

∂N7v ∂x

···

0

···

∂N7v ∂y

 0 ∂N7v ∂x ∂N7v ∂x

  {v} = [B] {v}  (3.196)

Chapter 3. Methods

3.3. Numerical modeling     p    1  p p p p˜ ≈ [N1 N2 N3 ] = [N p ] {p} p2     p  

(3.197)

3

Hence our equation set becomes Z Ω

h where BG =

[B]T [D] [B] {v} + [BG ]T [N p ] {p} dΩ = 0

∂N1v ∂N1v ∂N2v ∂N2v ∂x ∂y ∂x ∂y

...

∂N7v ∂N7v ∂x ∂y

(3.198)

i .

The derivatives of the shape functions are expressed through the local coordinates ξ and η, while the displacement matrix contains the derivatives with respect to the global coordinates x, y. The derivatives are converted from one coordinate system to the other by means of the chain rule of derivation ∂Niv ∂x ∂Niv ∂y ∂Niv = + ∂ξ ∂x ∂ξ ∂y ∂ξ ∂Niv ∂x ∂Niv ∂y ∂Niv = + ∂η ∂x ∂η ∂y ∂η

(3.199) (3.200)

In matrix form we can write (

∂Niv ξ ∂Niv η

)

" =

∂x ∂ξ ∂x ∂η

∂y ∂ξ ∂y ∂η

#(

∂Niv x ∂Niv y

)

( = [J]

∂Niv x ∂Niv y

) (3.201)

where J is the Jacobian transformation matrix . The Jacobian transformation matrix and its components are calculated using derivatives of the shape functions with respect to the local coordinates ξ, η and global coordinates of element nodes xi , yi 7

7

∂x X Niv = xi ∂ξ ξ

∂y X Niv = yi ∂ξ ξ

7

∂y X Niv = yi ∂η η

j=1

(3.202)

j=1 7

∂x X Niv = xi ∂η η j=1

(3.203)

j=1

The derivatives with respect to the global coordinates are calculated with the use of the inverse Jacobian matrix: ( ∂N v ) " i

x ∂Niv y

=

∂ξ ∂x ∂η ∂x

∂ξ ∂y ∂η ∂y

#(

∂Niv ξ ∂Niv η

)

( = J −1

∂Niv ξ ∂Niv η

) (3.204)

The determinant of the Jacobian matrix |J| is used for the transformation integrals from the global system to the local coordinate system 75


Chapter 3. Methods

dΩ = dxdy = dξdη|J|

(3.205)

Hence our equation set (eq. 3.198) becomes Z Ω

[B]T [D] [B] {v} + [BG ]T [N p ] {p} dξdη |J| = 0

(3.206)

The pressure formulation is the continuity equation (eq. 3.145) is discretized so that

∂p ∂t

=

pN −pO ∆t .

The continuity equation the reads µ

∂vx ∂vy + ∂x ∂y

¶ +

pN pO = ∆T K ∆T K

(3.207)

Similarly as with the force balance equations, the continuity equation is weighted. However, since we only have three pressures we only need 3 pressure shape functions N p. ¶ Z PN ∂vx ∂vy p + + dΩ = Ni ∂x ∂y ∆T K Ω Ω ¶ Z Z µ N ∂v P ∂v y x + + dΩ = Nip N3p ∂x ∂y ∆T K Ω Ω ¶ Z Z µ N P ∂vx ∂vy N3p + + dΩ = Nip ∂x ∂y ∆T K Ω Ω Z µ

N1p

PO dΩ ∆T K PO dΩ ∆T K PO dΩ ∆T K

(3.208) (3.209) (3.210)

Substituting with the approximations (eqs. 3.195 & 3.197) using the shape functions we get Z · Nip

Ω

¸ Z ∂N1v ∂N1v ∂N2v ∂N2v ∂N7v ∂N7v NP PN NP PO ... {v} + dΩ = dΩ (3.211) ∂x ∂y ∂x ∂y ∂x ∂y ∆T K Ω ∆T K

Hence eqs. 3.208-3.210 can be written in matrix form as 

N1p



· ¸ v ∂N v  p  ∂N1v ∂N1v ∂N2v ∂N2v ∂N 7 7  N  −  2  ∂x ∂y ∂x ∂y . . . ∂x ∂y dΩ Ω N3p   p N Z  1p  [N p ]{pN }  N  −  2  ∆T K dΩ Ω N3p  Np Z  1p =−   N2 Z

Ω

76

N3p

  [N p ]{pO }   ∆T K dΩ (3.212)

Chapter 3. Methods


Which reduces to Z − Ω

[N p ]T BG −

[N p ]T [N p ]{pN } dΩ = − ∆T K

Z Ω

[N p ]T [N p ]{pO } dΩ ∆T K

(3.213)

We now see that the first term in the last equation contains the transpose matrix product of the last term in eq. 3.206. Hence our eqs. 3.206 & 3.213 can be written as.       

K

Q M − ∆tK

QT where K =

3.3.7

R

ΩB

T D dev BdΩ,

Q=

                  v    0  =                 pN    − M pO ∆tK

R

T p Ω BG N Ω,

and M =

R

Ω [N

            

(3.214)

p ]T N p dΩ.

Numerical integration

The stiffness matrix KQ is evaluated numerically using Gauss-Legrendre quadrature (Hughes, 1987; Kwon and Bang, 2000; Zienkiewicz and Taylor, 2000; Zienkiewicz et al., 2005) over the quadrilateral elements. The Gauss-Legrendre quadrature can integrate a polynomial function of order 2n − 1 using n-point quadrature exactly. In numerical integration a finite number of calculations are used and the integral is approximated as Z

b

f (x)dx ≈ a

n X

f (xi )wi

(3.215)

i=1

where n is the number of integration points an w is the weighting coefficient. The weighting coefficient wi can be thought as the width of the rectangular area with height f (xi ) (fig. reffig:gauss). The integration domain is normalized such that −1 ≤ x ≥ 1 in order to derive standard values for integration points. 7 node triangular elements are used with 7 velocity degrees of freedom and 4 pressure degrees of freedom (fig. 3.19). The velocity and pressure shape function are quadratic and constant, respectively, and 6 integration points are more than sufficient. The Gauss-Legendre quadrature formula for the integration of the the full matrix takes the form: Z

1

Z

1

[K Q] (ξ, η)∂ξ∂η |J| ≈ −1

−1

n X n X i=1 j=1

77

[K Q] (ξi , ηj ) |J| wi wj

(3.216)

3.3. Numerical modeling Z

1

Z

1

· T

Q −1

−1

Chapter 3. Methods

¸ ¸ n X n · X M M T − (ξ, η)∂ξ∂η |J| ≈ Q − (ξi , ηj ) |J| wi wj ∆tK ∆tK i=1 j=1

(3.217) Z

1

Z

1

·

¸

− −1

−1

M O p (ξ, η)∂ξ∂η |J| ≈ ∆tK

n X n · X

¸

−

i=1 j=1

M O p (ξi , ηj ) |J| wi wj ∆tK (3.218)

Where n is the number of integration points, ξi and ηj are the integration point (or sampling point) and wi and wj are the weighting coefficients. Since the central pressure node is not shared, the calculation of pressure requires a lot of time. For optimization of the code static condensation is performed. Static condensation removes the pressure variables at the boundaries that the elements share and therefore must have the same value before the global matrix is computed. In this f(x)

f(xi)

a

wi

b

x

Figure 3.18: Numerical integration. In numerical integration a finite number of calculations is used a the integral is approximated as sum of the rectangular elements below the graph: Rb Pn f (x)dx ≈ i=1 f (xi )wi where n is the number of integration points and and wi is the a weighting coefficient (modified from Kwon and Bang (2000)).

78

Chapter 3. Methods


η 1x3

ζ=1−ξ−η 5

6 7

0x 0

1

2 x 1

4

ξ

Figure 3.19: Local reference element used for numerical solution. 7 nodes of velocity degrees of freedom (filled circles), 3 pressure degrees of freedom (crosses) and 6 integration points (circles).

way of sub-structuring the number of unknowns to be computed are reduced. This optimizes the code yielding faster calculation, but at then same time it gives the same results than the one would obtain from the analysis of whole structure.

3.3.8

Boundary conditions

The domain has free top surface and the bottom boundary is kept fixed. Pure shear deformation is applied at the right left sides of the domain with boundary conditions εxx = 0.5. Different material properties are assigned to the matrix, the boudins, and the gaps between the boudins. The boundary conditions are applied such that the full matrix A

9

(eq. 3.219) is kept symmetric to optimize the calculations.          0 x A1,1 A1,2 A1,3 · · · A1,17     1               A2,1 A2,2 A2,3 · · · A2,17   x2   0    =   ..   ..  ..    .  .  .                  b  x A A A ··· A 17,1

17,2

17,3

17

17

17,17

(3.219)

To prescribe velocities for any node, the equation for the appropriate velocity degrees of freedom is replaced with the constraint. E.g. if the velocity x2 = 0.5 , each entry in the second row apart from the diagonal is set to zero and we obtain " 9

For simplicity the full matrix is written A =

M pO ∆tK

K

Q

QT

M − ∆tK

}T .

79

# , x = {vpN }T , and b = {0 0 · · · 0 −

3.3. Numerical modeling       

Chapter 3. Methods

A1,1

A1,2

A1,3

···

A1,17

0

1

0 .. .

···

0

A17,1 A17,2 A17,3 · · ·

A17,17

    x1    x2   ..  .      x 17

      

    0         0.5   = ..      .            0  

(3.220)

However, this has the disadvantage that the modified A-matrix is no longer symmetric. It is preferable to modify the matrix, to retain symmetry. This is done by eliminating the constrained degrees of freedom from all rows of the stiffness matrix. Each entry in the second column apart from the diagonal to zero to by subtracting appropriate multiples of the second row.       

A1,1

0

A1,3

···

A1,17

0

1

0 .. .

···

0

A17,1 0 A17,3 · · ·

3.3.9

A17,17

   x1     x2   ..  .      x 17

  0 − A1,2 x2      0.5 = ..     .       0−A x   17,2 17       

            

(3.221)

Mesh generation

The domain is meshed using a triangular mesh generator (Shewchuk, 1996) and is re-meshed when the element quality is below a critical value. The element quality is 1 for a equal sided triangle and approaches 0 when angles between the triangle sides are allowed to approach 180◦ (Shewchuk, 2002). The triangular mesh allows for complex shapes and therefore the boudins have slightly rounded corners to avoid stress singularities. Also the re-meshing allows for a unlimited amount of strain to be accomplished.

3.3.10

FEM code

The Matlab scripts used in this thesis for input generation, numerical calculations and result post-processing have been developed by the author. However, the actual FEM simulations were done with MILLAMIN developed by M. Dabrowski and D. Schmid. The reason for this is that MILLAMIN is capable of dealing with power-law rheologies, a feature that was not implemented in the code developed by the author of this thesis. Also, MILLAMIN is capable of setting up, solving, and post-processing problem of approximately one million degrees of freedom per minute, which allows for faster analysis of the investigated systems.

80

Chapter 4

Petrography 4.1

Introduction

Field observations suggest that in the vicinity of the basin margin the deformation processes in the Bremanger Granodiorite Complex (BGC) change with respect to the distance from the margin. Five distinct brittle deformation structures are observed. (i) marginal breccias, (ii) sandstone dikes, and (iii) faults, (iv) fractures with granulated material, and (v) breccias. Description of (ii) and (iii) will form the basis of this chapter and from now on termed sandstone dike and fault gouge, respectively. The marginal breccia will be studied in Chapter 5 Intrusion between rigid plates on page 205. There are several unanswered questions; (i) how far does the extensional fracturing and fracture filling with basin material extend into the basement rock, (ii) when do faulting and accompanied cataclasis dominate the deformation process, and (iii) is there a sharp transition or a transition zone where both processes operate. To determine how the deformation process in the BGC change as a function of distance to the basin margin, we need to identify characteristics of the sandstone dikes (i.e. pull-apart fractures) and cataclastic end-member processes in particular the fault gouge or when fine grained material is filling the fractures; and compare those to localities in which both processes appear to have been active. To do so, localities were selected in which only the end-member processes seem to have been active, and localities with signatures of both processes. Both the qualitative characteristics obtained from microscopy and quantitative textural characteristics obtained from grain size and shape analysis were included in this study. To obtain quantitative measurements of grain size and shape from backscatter electron (BSE)-images a new technique was developed using the Matlab Image Processing Toolbox (The MathWorks, 2001). The technique uses gray 81

4.2. Sample preparation

Chapter 4. Petrography

scale-thresholding to identify phases and individual grains, and to convert images into binary images from which size and shape can be autimatically calculated. For a detailed description see Section 3.2 Image analysis on page 34.

4.2

Sample preparation

Samples of the fault gouge and sandstone dike from locality BRE59-05 and BRE60051 were drilled, sectioned and later impregnated with low viscosity resin to produce polished microscope slides. The slides were carbon coated an analyzed with optical Olympus BX41 microscope and Jeol JSM 6460LV scanning electron microscope. Back scatter electron BSE)-images with magnifications ranging in factors of 2 from 100x 1000x, and in addition at 1500x were taken to study grain size and shape of the dike infill material.

4.2.1

Back scatter electron imaging

BSE-imaging is a technique used to detect areas with different chemical compositions. Back scattering of electrons from the top surface of the thin section increases with the average atomic number of the bombarded material I.e. minerals with a relatively high average atomic number appear as bright objects compared to minerals with a low average atomic number.

4.3

Petrography

Cataclasis is defined as the granulation of individual grains by fracturing and rigidbody rotations of grains and fragments (Engelder, 1974). The term gouge generally is restricted to material which cataclastic deformation is so severe that any surviving grain are completely surrounded by a fine-grained matrix. The term cataclastic is used for material that has a cataclastic texture, but are less severely deformed than gouge. For the rest of the thesis, when describing internal strucures, I adopt the definition by An and Sammis (1994): “The width of the crushed zone, the distribution of its constituent fragments and the presence of any internal structures.” 1

See Appendix A Geological maps on page 243 for geographical location.

82


4.3.1

4.3. Petrography

Particle size

Particle size determinations are undertaken to obtain information about the size characteristics of an ensemble of particles. Determining the size of a regular shaped particle, e.g. a sphere, is quite simple. It is uniquely defined by its diameter. But for crosssections of irregular shaped particles the size depends on the way it is defined. The most commonly used measurement of particle size is equal area diameter; which is the diameter of a circle having the same area as the projected particle (Brittain, 2001). Of course, different shaped object will have an influence on the circle equivalent diameter, but it is a useful approximation since it is a single number that gets larger or smaller as the particle does and its measurement is objective and repeatable (Malvern Instruments, 2006).

Particle size distribution Particle size distribution (PSD) of fragmented geological material like the filling of fractures is useful to qualify (and quantify) aspects of the process responsible for the formation. Many methods have been proposed for describing PSDs in both two and three dimensions. They include both incremental and cumulative relations between mass/volume/number and increasing or decreasing mass/size (Blenkinsop, 1991). It has been demonstrated by Sammis et al. (1987); Blenkinsop (1991) that best way to characterize PSD is by using the fractal relationship of particle frequency by size. A summary of the literature given by Blenkinsop (1991) shows that most of usual relationships between mass/volume/number and mass/size that are used to describe PSDs, can be expressed as power-law coefficients. In order that the results from this study can be effectively compared to other data sets where other relationships have been used, and vice versa. In this study distribution of particle frequency by size is used. Also for fractal/power-law relationships there exist a simple solution to the problem of converting two-dimensional measurements of PSDs to three-dimensional equivalents for isotropic figures (see Section 4.3.1 2D versus 3D measurements on page 85). The definition of a fractal distribution is that the number of objects N with a characteristic size greater than S scales with the relation given by equation 4.1 (Turcotte, 1989; Blenkinsop, 1991). N (S) ∼ S −D

(4.1)

The coefficient D is the power-law coefficient2 and is determined from a graph of 83

4.3. Petrography


log N (S) against log S (figure 4.1). The fractal dimension D provides a measure of the relative importance of large objects versus small objects. A large D value implies that small grains dominate, and vice versa. A fractal set is one in which the sizefrequency relation is independent of scale over an infinite range. For any physical application, i.e. a distribution with a finite range, there will be upper and lower limits to the applicability of the distribution (Turcotte, 1989; Blenkinsop, 1991) (figure 4.1). The upper and lower limit are defined as the limits over which the distribution obeys a scale-independent size frequency relation. The upper limit is generally controlled by the size of the object or region being fragmented, in this case the size of the thin section. The lower limit is likely to be controlled by the grain size, the resolution that can be measured, and other sampling technique effects (see Section 4.3.1 2D versus 3D measurements on the next page). Hence power-law exponent should be accompanied by the range for which the relation is a good fit, the power-law range, bounded by the upper and lower power-law limits. The power-law range should be well within measurement range so the PSD is not biased by lack of detection (Blenkinsop, 1991). Earlier studies showed that gouge particles obey a log-normal or self-similar distribution (Engelder, 1974; Sammis et al., 1986). However later studies of natural and simulated gouge found a power-law PSD relation (Sammis et al., 1987; Sammis and Biegel, 1989; Marone and Scholz, 1989; Blenkinsop, 1991; An and Sammis, 1994; Blenkinsop and Fernandes, 2000; Storti et al., 2003). Although the entire PSD is not described by a single relationship, in general an excess of small particles are consistent with a power-law distribution (Sammis et al., 1986; Blenkinsop, 1991; An and Sammis, 1994).

2

Calling the distribution fractal is not always strictly correct because this implies that the distribu-

tion is self similar and constant over many orders of magnitude, hence the term power-law distribution is used here.

84


4.3. Petrography

1000 Measurement Range

1.5 mm

Number of particles > S

Fractal Range 100 Fractal Dimension, D

10

0 0.001

1

0.003

0.01

0.03 Lower

0.1

0.3

1.0

Upper Fractal Limit

Size of particles, S, mm

Figure 4.1: For any physical application, there will be upper an lower limits to the applicability of the fractal distribution (Turcotte, 1989; Blenkinsop, 1991). The upper and lower limit are defined as the limits over which the distribution obeys a scale-independent size frequency relation. The power-law range should be well within measurement range so the PSD is not biased by lack of detection (Blenkinsop, 1991). Modified from Blenkinsop (1991).

2D versus 3D measurements Two-dimensional measurements do not give the same absolute particle size or distribution, as three-dimensional measurements. In a 2D-section there is a greater probability of intersecting larger particles and likewise a lower probability of intersecting smaller particles (due to a sampling sampling effect) (Blenkinsop, 1991). Also a section trough a particle will, in general, show less than the maximum particle dimensions (due to the truncation effect) (Blenkinsop, 1991). A disintegration procedure of the fractured material with sieving and weighing as used by sedimentologists allows for three-dimensional measurements. But this procedure is not gentle and Sammis et al. (1987) questions if this technique may be responsible for more comminution than existed in the original sample. Also with this technique all geometrical relations between particles are lost together with any information they could provide. Under the assumption that spherical particles adequately describes other geometries, Sammis et al. (1987) show that when two-dimensional self-similarity is observed over many binary or85

4.3. Petrography


ders of magnitude, the three dimensional structure must have the same property. This is a general result of fractal theory that states that the fractal dimension of an isotropic three-dimensional figure is larger by unity than a two-dimensional cross-section of it (Turcotte, 1989). Also PSDs of gouge material from the Lopez Fault derived from careful disintegration and 3D counting of particles by An and Sammis (1994) compares to earlier studies of this gouge by Sammis et al. (1987) and Sammis and Biegel (1989) using 2D thin sections. This implies that the gouge material is isotropic and homogenous. Since this is currently recognized as a reasonable way to approximate 3D distributions from 2D measurements I adopt this method. Hence the power-law coefficients in this study are given as: D3D = 1 + D2D . PSD in labs and nature, PSD evolution, and gouge comminution models Three models have been proposed to explain the evolution of PSD in fault gouge. They are expressed directly in fractal terms. In the “Pillar Strength Model” by Allegre et al. (1982) failure occurs whenever fragile domains are arranged such that no pillars of strong material exist. Turcotte (1986) showed that this model predicts a D-value3 of 1.97. Similarly in the “Plane of fragility model” by Turcotte (1986) failure occurs when a plane of fragile domains are formed. This model predicts a D-value equal to 2.84. An important feature of these models is that they do not differ between constrained and unconstrained comminution. In contrast, Sammis et al. (1987) proposes a beam model for “constrained comminution”. They argue that unconstrained models where each particle can be regarded as acting independently of its neighbours is incorrect for gouge material inside a fault zone. In their model, instead of size dependent fracturing, smaller particles lower the axial loading of larger particles and in this way cushion them from concentrated and destructive stresses. This geometry can exist over all particle sizes and the particle fracture probability is independent of it size, but is strongly dependent of its neighbour. As particles are continuously moving in relation to each other during deformation, neighbours of same size are most likely to fracture and the comminution mechanism continuously rearranges the geometry to maintain self-similarity. This models predicts a D-value equal to 2.60. Abundant observations on natural and experimentally deformed gouge indicate that the final PSD depends on shear strain, confining pressure, rock type, and initial PSD. Shear tests on quartz sandstone, granite and granodiorite (Sammis et al., 1986; Biegel et al., 1989; Marone and Scholz, 1989) show that they have tendency to 3

From here on D3D is denoted D for simplicity when nothing else is specified.

86


4.3. Petrography

develop D-value of 2.6, supporting the constrained comminution model. Experiments on both natural and experimental gouges show finer particle size with increasing confining pressure and strain. Also, observations by Marone and Scholz (1989) support the idea that one of the principal comminution mechanisms is crushing of similar-sized neighbour particles and comminution is the result of shear displacement between particles. Biegel et al. (1989); Marone and Scholz (1989) showed that D-values equal to 2.6 may represent a steady-state reached after large amount of shear displacement where there is a transferal of strain from the bulk material to shear bands. Within the shear bands further comminution is accompanied by a preferential fragmentation of large particles resulting in increased D-values in excess of 3. The “constrained comminution model” predicts that particles of similar size should not be nearest neighbours. However unless large particles are added to the gouge zone during shear, the mean size of the particles should decrease with displacement; i.e. with sufficient displacement large particles will eventually come into contact with each other and fracture. Larger particles should fracture and produce smaller particles, which should in turn fracture, keeping the relative portions within each size range constant, in which the y-intercepts would decrease, but the D-value would not change. Marone and Scholz (1989) explains the particle size reduction and loss self-similarity by a lowering of the upper fractal limit to the preferential loss of large particles which is consistent with small grains having a greater fracturing strength than large particles. Thus the smaller particles observed in the shear band represent a lower bound for the size-independent strength model by Sammis et al. (1987). An and Sammis (1994) suggest the observation is explained by a “grinding limit” as observed in commercial processes. It becomes increasingly difficult to fragment particles smaller than some limiting size because of a transition from brittle to completely plastic behavior at small particles sizes and an equilibrium is formed in which small particles are aggregating to form larger particles as fast as larger particles are disintegrated. Also, the number and size of natural flaws decrease with particle size until there are no nucleation sites. This causes a pile up of smaller particles, resulting in an increase in D-values. This idea was also supported by computer simulations by An and Sammis (1994). Preferential fragmentation of larger particles during shear localization have also been suggested by (Blenkinsop, 1991; Blenkinsop and Fernandes, 2000) as a possible explanation for the occurrence of D-values of significantly higher than 2.6 in cataclastic rocks. Blenkinsop (1991) also pointed out the rock type, alteration, initial PSD, and fracture mode have an effect on the final PSD. However, there is some controversy regarding this topic. Storti et al. (2003) suggests that shear localization and resulting D-values higher than 2.6 - 2.7 are 87

4.3. Petrography


characterized by an preferential increase of smaller particles due to particle abrasion, rather than by the selective decrease of larger particles. With progressive strain there is change in the dominant deformation mechanism from fracturing to abrasion. The increment of smaller particles produced by surface abrasion is much greater than the decrement caused by fracturing of larger particles. Experiments done by Engelder (1974) compares to natural formed gouge in quartz sandstone with a log-normal distribution, while experiments by Sammis et al. (1986) showed that simulated quartz sandstone gouge is parameterized by a mixed log-normal and is texturally similar to the disaggregated protolith, with little if any comminution beyond disaggregation. Fractal dimension measured from natural formed gouges are more controversial. Results from faults in crystalline basement rocks gave D values averaging 2.6 Sammis et al. (1986, 1987); Sammis and Biegel (1989); An and Sammis (1994), supporting the constrained comminution model. While observations by Marone and Scholz (1989); Blenkinsop (1991); Blenkinsop and Fernandes (2000); Storti et al. (2003) suggests that fractal dimensions of cataclastic rocks do not cluster around a specific value but are distributed around a wide interval from 1.8 - 5.5. Blenkinsop (1991) showed that the fractal dimension show a correlation with type of fracture. Dilatant and extensional fractures generally have low D-values of 2.32 and 2.37, respectively. Shear fractures have significantly higher D-values of 2.6 - 2.7. This was also recognized by Marone and Scholz (1989) who observed that particles within hydrostatically loaded quartz sandstone layers have D values of 1.8 and show significantly less overall comminution than sheared layers. This suggests that comminution is driven by relative movement between particles. Storti et al. (2003) observed relation between the position of the sample within the fault core and their D-values. D-values increase from the boundary region toward the shear bands. This have also been recognized by Blenkinsop (1991). Also the study by Storti et al. (2003) show complete overlap between D-values obtained from both strike-slip and extensional fault cores, implying that the fault kinematics does not influence the fractal dimension significantly. Blenkinsop and Fernandes (2000) recognized that PSDs and microstructures of cataclastites show a progressive evolution with strain from a intact and samples with extensional microfractures with a curved PSD to a linear power-law relationship in microfaults (figure 4.2). The grain size also decrease during this evolution.

88


4.3. Petrography

10 Intact Extension Fracture

N/n

1 0.001

Shear Fracture

0.01

S

0.1

1

Figure 4.2: PSDs of cataclastites show a progressive evolution with strain from a intact and samples with extensional microfractures with a curved PSD to a linear power-law relationship in microfaults. Modified from Blenkinsop and Fernandes (2000).

Even though there is some dispute on the the observed D-values and the deformation mechanisms, both theoretical, experimental and natural observations agree on the following: The fractal dimension show an inverse correlation with mean grain size and is proportional with the amount of shear, displacement, number of fracturing events and confining pressure (Engelder, 1974; Sammis et al., 1986; Marone and Scholz, 1989; Blenkinsop, 1991; An and Sammis, 1994; Blenkinsop and Fernandes, 2000; Storti et al., 2003). The fractal dimension show a correlation with type of fracture and placement in fault zone. Dilatant and extensional fractures generally have low D-values while shear fractures have significantly higher D-values Marone and Scholz (1989); Blenkinsop (1991). PSDs show a progressive evolution with strain from a intact and samples with extensional fracturing with a curved PSD to a linear power-law relationship in faults (Blenkinsop and Fernandes, 2000). The fractal dimension increases from the boundary regions towards the shear bands Storti et al. (2003). Gouge thickness increases with shear strain (Engelder, 1974). At large shear strains, some large survivor grains become complectly incorporated by the fine matrix (Engelder, 1974; Ozkan and Ortoleva, 2000). Shear localization is responsible from development of D-values significantly higher than 2.6 (Marone and Scholz, 1989; Blenkinsop, 1991; An and Sammis, 1994; Blenkinsop and Fernandes, 2000; Storti et al., 2003). The shear bands consistently contain fewer large particles and have a smaller mean size compared with the bulk material. 89

4.3. Petrography

4.3.2


Shape descriptors

PSDs gives variable information about the size characteristics of an ensemble of particles, but it does not say anything about the shape of the particles. I.e. particles with the same PSD do not necessary show the same shape characteristics. Measuring size alone can be insufficient to identify important, but subtle differences. Hence shape measurements in addition to PSDs may provide a more effective descriptor than just PSD comparison alone. There are several different ways of quantitatively describing the shape of a particle depending on the aspect of interest. In general they should obey three criteria (Crompton, 2005): (i) It should be intuitive. (ii) It should be normalized to values between zero and 1, thus making interpretation easier. And (iii), it should be sensitive to deviations. It is important to remember that it is unlikely for one single shape descriptor to perfectly discriminate and characterize all applications and different combinations of shapes (figure 4.3). Hence in this study three different shape parameters have been chosen to discriminate and characterize the grain shapes. Elongation Elongation is a measurement of aspect ratio (AR = length/width) relationship and is defined as ε=1−

1 AR

(4.2)

Elongation has has values between 0 and 1. Shapes symmetrical in all axis like circles and squares will have an elongation value equal to 0, whereas shapes with large aspect ratios will have elongation values closer to 1. The aspect ratio is calculated from the major and minor axis lengths of an ellipse that have the same point of gravity as the particle. From figure 4.3(a) we can see that the elongation is an indicator of form and is unaffected by surface roughness. Convexity Convexity4 is a measurement of surface roughness and is calculated by dividing the convex hull perimeter by the actual particle perimeter (equation 4.3). ζ=

2 PCH

PP2 article

(4.3)

The convex hull is the smallest convex polygon that can contain the particle. It is 4

Convexity is equivalent to the PARIS factor used by Herwegh et al. (2005).

90


4.3. Petrography

best visualized by the area enclosed by an elastic band stretched around the particle. Convexity also has values in the range of 0 to 1. A smooth particle has convexity equal to 1 as the convex hull perimeter is equal to the actual perimeter, whereas a irregular particle has convexity closer to 0 (figure 4.3(b)). Unlike elongation, convexity is un effected by overall form and symmetry, it is only affected by surface roughness. Circularity Circularity is a measurement of the ratio of the perimeter of the particle and the perimeter of an equal area circle (equation 4.4). ψ=

4πA 2 PEAC

(4.4)

As the the name suggests, circularity is a measure of how close the particle shape is to a circle. Circularity also have values in the range of 0 to 1. A circle has circularity 1, whereas irregular objects have circularity closer to 0. Circularity is both sensitive to overall shape and symmetry, and surface roughness (figure 4.3(c)). In order to optimize the convexity and circularity descriptors for subtle variations in shape, the terms in the numerator and denominator in equations 4.3 and 4.4 are squared. Finally, also the orientation of the major axis of the particle is measured (hereby called orientation). The orientation is the angle (in degrees) between the x-axis and the major axis of the ellipse that has the same point of gravity as the particle. The thin sections are oriented such that 0◦ is perpendicular to the fracture orientation and −90◦ and 90◦ is parallel. Statistical tools To uncover underlying structure, extract important variables and detect outliers and anomalies, statistical tools are employed to the shape descriptors and orientation. These include both graphical representations like scatter plots and histograms and qualitative techniques. To allow comparison between magnifications and different samples the shape descriptors are plotted as number of events are normalized by the total P number of events (total number of particles) in the histograms, N (i)/ 1i=0 N . To estimate a location parameter for the distribution; i.e., to find a typical or central value that best describes the data, the mean value is calculated. The mean value is the sum of the data points divided by the number of data points N (equation 4.5). The peak value is another measurement of location. It is the value of the sample that occurs with the greatest frequency. It is not necessarily unique and is typically used in a qualitative fashion. To characterize the spread of the distribution the standard 91

4.3. Petrography


deviation is calculated (equation 4.6). The standard deviation is the square root of the variance which is roughly the arithmetic average of the squared distance from the mean. Y¯ =

N X Yi i=1

N

v uN uX (Yi − Y¯ )2 σ=t N −1

(4.5)

(4.6)

i=1

Skewness is a measure of symmetry, or more precisely, the lack of symmetry. Negative values for the skewness indicate data that are skewed left and positive values for the skewness indicate data that are skewed right. By skewed left the data are spread out more to the left of the mean than to the right. That is, the left tail is long relative to the right tail. Similarly, skewed right means that the right tail is long. The skewness of a normal or any perfectly symmetric distribution is 0. It looks the same to the left and right of the center point. The skewness of a distribution is defined as S=

N X (Yi − Y¯ )3 (N − 1)σ 3

(4.7)

i=1

Kurtosis is a measure of whether the data are peaked or flat relative to a normal distribution, how outlier-prone the distribution is. The kurtosis for a standard normal distribution is 3. Distributions with kurtosis higher than 3 tend to have a distinct peak near the mean, decline rather rapidly, and have heavy tails. They are more outlierprone. As oppose to distributions with kurtosis less than 3 which tend to have a flat top near the mean rather than a sharp peak. The kurtosis is defined as K=

N X (Yi − Y¯ )4 (N − 1)σ 4

(4.8)

i=1

5

The values reported here are the ones the author of this thesis has calculated in Matlab using the

equations stated above (equations 4.2 - 4.4). They generally compare to the original values reported in this figure (Crompton, 2005). However, the original convexity values of approximately 0.7 for both the “spiky” objects, which clearly are wrong, have been corrected.

92


4.3. Petrography

Elongation = 0

Elongation = 0.82

Elongation = 0

Elongation = 0.79

Elongation = 0.28

Elongation = 0.83

(a) Elongation is a measurement of aspect ratio, length/width relationship. Shapes symmetrical in all axis like circles and squares will have an elongation value equal to 0, whereas shapes with large aspect ratios will have elongation values closer to 1. It is an indicator of form and symmetry and is unaffected by surface roughness.

Convexity = 1

Convexity = 1

Convexity = 1

Convexity = 1

Convexity = 0.39

Convexity = 0.50

(b) Convexity is a measurement of surface roughness and unlike elongation it is unaffected by overall form and symmetry. A smooth particle will has convexity equal to one, whereas a irregular particles has convexity closer to zero

Figure 4.3: Shape descriptors. (a) elongation, (b) convexity, and (c) circularity. It is unlikely for one single shape descriptor to perfectly discriminate and characterize all applications and different combinations of shapes. Therefore three different shape parameters have been chosen to discriminate and characterize the particle shapes. Modified from Crompton (2005)5 .

93

4.4. Fault gouge


Circularity = 1

Circularity = 0.61

Circularity = 0.89

Circularity = 0.68

Circularity = 0.45

Circularity = 0.36

(c) Circularity is both sensitive to overall shape and symmetry, and surface roughness. A circle has circularity 1, whereas irregular objects have circularity closer to zero.

Figure 4.3: (Cont.) Shape descriptors. (a) elongation, (b) convexity, and (c) circularity. It is unlikely for one single shape descriptor to perfectly discriminate and characterize all applications and different combinations of shapes. Therefore three different shape parameters have been chosen to discriminate and characterize the particle shapes. Modified from Crompton (2005).

4.4 4.4.1

Fault gouge Locality BRE60-05

Numerous joints and faults with a NW-SE orientation occur in the granodiorite adjacent the Hornelen basin. However the general lack of passive markers makes it impossible to quantify the apparent displacement on these. But in this area the granodiorite has a strongly developed foliation, 070/62, with quartz-veins parallel to the foliation and the displacement along faults becomes apparent (figure 4.4). Several faults are found in this area, both sinistral and dextral. The apparent displacement is between 1.5 and 18 cm. And over a 10 m long profile oriented approximately perpendicular to the faults, 10 faults are encountered and their accumulated displacement is about 60 cm. The locality is approximately 200 meters away from the contact with the basin (see figure A.1 on page 244).

94


4.4. Fault gouge

Figure 4.4: Locality BRE60-05. The granodiorite has a strongly developed foliation and the apparent offset becomes visible from the displaced quartz veins. Sample BRE1505 was drilled from the fault marked with the red dotted circle. Coin for scale.

Sample BRE1505-2 Wall-rock The wall-rock of the fault is highly fragmented with fractures mainly sub-parallel and sub-normal to the fault (figure 4.5). The wall-rock consist of relatively large aggregates of quartz grains and small grains of plagioclase that have partly or completely decomposed to epidote and chlorite (figure 4.6). The decomposition is strongest near fractures and the plagioclase, epidote and sericite (white mica) forms a mesh structure surrounding the larger quartz aggregates. The epidote is found as small green radiating needles in the albitic plagioclase or as spherical grains in the fractures (figure 4.7). The epidote grains in the faults are very fragmented and display a complex texture with irregular shape, black inclusion of quartz, plagioclase and voids, and zonation with Fe-rich rims (figure 4.8). This indicates on growth during cataclasis. The sub-parallel fractures are between 1 and 5 mm in length and the thickness range from 0.01 to 0.4 mm. They are filled with small spherical grains of epidote 95

4.4. Fault gouge


and quartz and have a relatively low interface roughness compared to the sub-normal fractures (figure 4.9). We observe that the sub-parallel fractures displace or crush parts of plagioclase grains (figures 4.7 & 4.11), and displace sub-perpendicular fractures (figure 4.10). As mentioned above, the sub-perpendicular fractures are cut by the sub-parallel fractures. They are between 1 and 2.5 mm in length and between 0.01 and 0.05 mm in thickness. They are filled with spherical grains of green and brown coloured epidote and colorless quartz. Some of the fractures are undulating and filled with small quartz grains (figure 4.9). This might possible be refractory quartz left from pressure solution or grains formed by granulation. Thin seams perpendicular to the sandstone dike are cut by the parallel oriented faults (figure 4.12). These seams are enriched in zircon, titanite and apatite. The remnant foliation is seen as alternating bands, perpendicular to the fault zone, alternating with fine grained plagioclase, epidote and sericite, and homogenous areas with quartz and plagioclase. (figure 4.13). The sericite grains have a prismatic shape and fibrous texture and are found as either inclusions in the plagioclase, or it dominates the mineral phase assemblage in small regions in the wall-rock (figure 4.14).

96


4.4. Fault gouge

Figure 4.5: Thin section BRE1505-2 (TLa ). The thin section is approximately 2 cm wide. The orientation of the fault is 388/90. a

Transmitted light.

97

4.4. Fault gouge


(a) Transmitted light (TL)

(b) Crossed Nichols (CN).

Figure 4.6: The wall-rock consists of aggregates of quartz surrounded by a mesh texture of plagioclase, epidote and chlorite. Image from Site E in figure 4.4.

98


4.4. Fault gouge

Figure 4.7: Plagioclase grain displaced by sub-parallel fracture. Epidote is found as needles is the plagioclase and as spherical grains is the fault. TL-image from Site A in figure 4.4.

(a)

Figure 4.8: The epidote grains in the wall-rock faults display a complex texture with black inclusions of quarts, plagioclase and voids, zonation and fragmented appearance. The zonation is du to different amounts of iron, which gives a relatively high intensity in BSE-images. This indicates on growth during cataclasis. BSEimages from Site A in figure 4.4.

99

4.4. Fault gouge


Figure 4.8:

(Cont.)

The epidote grains in the wall-rock faults display a complex texture with black inclusions of quarts, plagioclase and voids, zonation and fragmented appearance. The zonation is du to different amounts of iron, which gives a relatively high intensity in BSEimages.

This indi-

cates on growth dur-

(b)

ing cataclasis.

BSE-

images from Site A in figure 4.4.

Figure

4.9:

Sub-parallel

The fractures

(vertical) have a relatively straight interface compared

to

the

sub-perpendicular fractures. The sub-normal fractures have a more undulating and small

interface

contain

often

quartz

grains

(refractory

quartz

from pressure solution or grains formed by granulation?).

CN-

image from Site F in figure 4.4.

100


4.4. Fault gouge

Figure 4.10: Sup-parallel fracture have displaced sub-normal fracture filled with quartz and sericite. CN-image from Site B in figure 4.4.

Figure 4.11: The sigmoidal clast of albitic plagioclase is missing its lower tail. Large aggregate of quartz is displaced in the fault and has crushed part of the plagioclase grain. CN-image from Site D in figure 4.4 (Note that the image is rotated 45◦ clockwise).

101

4.4. Fault gouge


Figure 4.12: Seams normal to the fault gauge are enriched in epidote, apatite, titanite and zircon. BSE-image from Site A in figure 4.4.

Figure 4.13: The remnant foliation is seen as alternating bands, normal to the fault zone, alternating with fine grained plagioclase, epidote and sericite, and homogenous areas with quartz and plagioclase. Note that the K-feldspar is only found in the fault zone. This suggests that the K-feldspar is formed from the fluid. BSE-image from Site B in figure 4.4.

102


4.4. Fault gouge

Figure 4.14: Sericite (white mica) is usually found as a accessory phase formed from plagioclase. But locally in the wall-rock sericite is more abundant. CN-image from Site G in figure 4.4.

Fault gouge On the right hand side there is a thin transition zone with large fragments of the wall-rock breaking loose from the wall (figure 4.15). The fragments have a prismatic elongated shape with their longest axis parallel with the fault. In the fault gouge the grains size distribution appears bimodal with mainly a fine grained matrix and few large survivor grains (matrix supported, large matrix/clast ratio). The matrix consists of all the mineral phases found in the wall-rock and in addition K-feldspar (figure 4.13). However, the survivor grains are predominantly quartz. The fault zone is also cut by thin light coloured bands of mainly quartz and accessory epidote (figure 4.15). Several generations (2) of these bands which displace each other is observed. The bands continue into the “mylonitic” transition zone on the left hand side (figure 4.15). This suggests that the bands are “internal faults” in the fault zone formed after the fault gouge material had lithified. The “mylonitic” zone is banded with dark and light coloured bands (figure 4.16(a), TL). The dark bands consists of mainly epidote. The light coloured bands appear as relatively large grains, but with crossed Nichols, we observe that they consist of small fragments of mainly quartz, feldspar and minor amounts of epidote (figure 4.16(b)). 103

4.4. Fault gouge


The epidote in the fault gouge display a complex structure similar to that of the epidote in the faults in the wall-rock (figure 4.8), indicating on growth during cataclasis (figure 4.17). The epidote grain in the fault gouge are fragmented with irregular shapes, black inclusion of quartz, plagioclase and voids, and zonation with Fe-rich rims. BSE-images was taken from Site E in the fault gouge (figure 4.4) to study grain size and shape (see below). The “mylonitic” zone was not dealt with using the image analysis techniques.

104


4.4. Fault gouge

(a) Transmitted light.

(b) Back scatter electron.

Figure 4.15: Fault zone. Transition zone on the right with relatively large prismatic fragments breaking loose from the wall-rock. The fault gouge have an apparent bimodal grain-size distribution with a fine grained matrix and few large survivor grains. The fault zone is cut by several generations (2) of light coloured quartz bands. This suggests that the bands are “internal faults” in the fault zone formed after the fault gouge material had lithified. Images from Site C in figure 4.4.

105

4.4. Fault gouge


(a) Transmitted light.

(b) Crossed Nichols.

Figure 4.16: Close-up of the “mylonitic” transition zone on the left hand side of the fault zone in figure 4.15(a). The “mylonitic” zone is banded with dark and light coloured bands. The dark bands consists of mainly epidote. The light coloured bands appear as relatively large grains, but with crossed Nichols, we observe that they consist of small fragments of mainly quartz, feldspar and minor amounts of epidote.

106


4.4. Fault gouge

(a)

(b) Close-up of (a).

Figure 4.17: The epidote grains in the fault gouge display a complex structure indicative of growth during cataclasis, similar to that of the epidote in the faults in the wall-rock (figure 4.8). The epidote grain in the fault gouge are fragmented with irregular shapes, black inclusion of quartz, plagioclase and voids, and zonation with Fe-rich rims. Images from Site H in figure 4.4.

107

4.4. Fault gouge

4.4.2


Image analysis results sample BRE1505-2

Particle size distribution Table 4.1 on the next page summarizes particle size data for sample BRE1505-2. The abundant phase in modal percentage in fault gouge sample BRE1505-2 is plagioclase & quartz (table 4.1), both when it comes to and number of particles and relative porhyroclast proportion6 ∼7 45% of the number of particles are plagioclase & quartz. Similarly ∼ 38% of the particles are K-feldspar, whereas only ∼ 17% of the particle are epidote. The porphyroclast proportion for all of the phases combined increases from ∼ 35% at lowest magnification to a peak value of ∼ 70% at 600x magnification before a decrease to ∼ 60% at highest magnification. This is also observed for the relative porphyroclast proportion of K-feldspar which increases from ∼ 23% to a maximum of ∼ 41% and then decreases towards ∼ 13%. As opposed to the relative porphyroclast proportion of plagioclase & quartz which decreases from ∼ 72% to a minimum of ∼ 40% and then increases to ∼ 67%. The relative porphyroclast proportion of epidote increases with increased magnification from ∼ 5% to ∼ 20%. K-feldspar, and plagioclase & quartz have particles in the range of ∼ 2 µm to ∼ 190 µm and ∼ 410 µm, respectively (table 4.1). While epidote is limited to relatively small particles between ∼ 2 and∼ 90 µm. Both the individual phases and all of the phases combined have a smaller peak values than mean values. K-feldspar has a mean and peak particle size of 29, 89 µm and 3.28 µm, respectively. Similarly plagioclase & quartz have mean and peak particle size of ∼ 42.24 µm and 3.82 & 4.43 µm, respectively. However epidote has a relatively small mean particle size of 19.70 µm and large peak particle size of 15.58 µm. The particle size distribution is shown by number in figure 4.18. Fault gouge sample BRE1505-2 has D values of 2.9 - 3.2 for the majority of the particle size range (denoted D1 ). For all of the the phases combined the D1 -value is 2.9977 (table 4.1 and figure 4.18). The range for the slope fit is 3.07−115.6 µm. Similar D1 -values and ranges are observed for K-feldspar (2.9998) and plagioclase & quartz (2.8644)(figures 4.18(c) and 4.18(d)). However, the largest fraction of the particles for all of the phases combined, K-feldspar, and plagioclase & quartz can be fitted to a slope with a D2 -value larger than that of their respective D1 -values (table 4.1 and figure 4.18). For all of the phases combined the largest fraction of particles between 29.06 − 245.0 µm have a D2 -value of 3.7111 (figure 4.18(a)). K-feldspar has a even larger 6

Clast to matrix ratio (area percentage of particles).

7

Approximately is abbreviated ∼ for simplicity.

108


4.4. Fault gouge

D2 -value of 4.1209 (38.68 − 315.0 µm) (figure 4.18(c)), while plagioclase & quartz has a smaller D2 -value of 3.3477 (24.47 − 260.90 µm) (figure 4.18(a)). Whereas K-feldspar and plagioclase & quartz show similar PSD characteristics which in turn influencers the PSD characteristics of all of the phases combined. This is of course as one would expect. However, it masks the PSD characteristics of epidote, the minor of the constitutive phases. Epidote has a higher D1 -value of 3.2323 and a upper boundary of 59.36 µm (figure 4.18(b)). And as oppose tho the other phases it is the smallest fraction of epidote grains between 2.26 − 11.57 µm that can be fitted to a smaller D2 -value of 1.8020. Sample BRE1505-2 Power-law Phases Epidote

Particle size (µm)

coefficient, D

Mean

Peak

Min.

Max.

19.70

15.58

2.27

92.72

Small fraction K-feldspar

29.89

3.28

2.42

190.48

Large fraction Plag. & quartz

42.24

3.82,

3.07

Range (µm) Lower

410.22

Upper

3.2323

3.52

59.36

1.8020

2.26

11.57

2.9998

4.46

120.3

4.1209

38.68

315.0

2.8644

4.43

110.1

3.3477

24.47

260.9

2.9977

3.07

115.6

3.7111

29.06

254.0

4.43 Large fraction Combined

33.65

8.01

2.26

410.22

Large fraction

Number of Porphyroclast proportion8 (%) 100x

200x

400x

600x

800x

particles 1000x

1500x

Epidote

1.65

2.97

5.01

6.17

4.97

6.25

12.26

265

17.30%

K-feldspar

8.09

19.71

20.92

29.47

13.20

11.78

7.98

581

37.92%

Plag. & quartz

25.60

27.52

30.59

35.58

35.65

35.65

40.49

686

44.78%

Combined

35.34

50.20

56.52

71.22

53.81

53.67

60.74

1532

100.00%

Matrix

64.66

49.80

43.48

28.78

46.19

46.33

39.26

100.00

100.00

100.00

100.00

100.00

100.00

100.00

Relative porphyroclast proportion (%) 100x

200x

400x

600x

800x

1000x

4.67

5.92

8.86

8.66

9.23

11.64

K-feldspar

22.89

39.26

37.01

41.38

24.53

21.94

13.14

Plag. & quartz

72.44

54.82

54.12

49.96

66.24

66.41

66.67

100.00

100.00

100.00

100.00

100.00

100.00

100.00

Epidote

1500x 20.19

Table 4.1: Particle size data from fault gouge sample BRE1505-2.

109

4.4. Fault gouge

10

10

(N(S) > S)/area

10

10

10

10

10


−1

−2

D1 −3

100x 200x 400x 600x 800x 1000x 1500x BLF D1 = 2.9977 D2 = 3.7111

−4

−5

−6

−7

10

0

D2

10

1

10

2

10

3

S [µm]

(a) Phases combined. 10

−2

D2

(N(S) > S)/area

10

10

10

10

10

−3

D1

−4

100x 200x 400x 600x 800x 1000x 1500x BLF D1 = 3.2323 D2 = 1.8020

−5

−6

−7

10

0

1

10 S [µm]

10

2

(b) Epidote.

Figure 4.18: PSD fault gouge sample BRE1505-2. Low D-values ranging from 1.6673 to 2.3759 for plagioclase & quartz, and epidote, respectively. D-value of 2.2282 for all the phases combined.

110


10

4.4. Fault gouge

−2

D1

(N(S) > S)/area

10

10

10

10

10

−3

−4

100x 200x 400x 600x 800x 1000x 1500x BLF D1 = 2.9998 D2 = 4.1209

−5

−6

−7

10

0

D2

10

1

10

2

10

3

S [µm]

(c) K-feldspar. 10

−2

D1

(N(S) > S)/area

10

10

10

10

10

−3

−4

100x 200x 400x 600x 800x 1000x 1500x BLF D1 = 2.8644 D2 = 3.3477

−5

−6

−7

10

0

D2

10

1

10

2

10

S [µm]

(d) Plagioclase & quartz.

Figure 4.18: (Cont.) PSD fault gouge sample BRE1505-2. Low D-values ranging from 1.6673 to 2.3759 for plagioclase & quartz, and epidote, respectively. D-value of 2.2282 for all the phases combined.

111

3

4.4. Fault gouge


Particle shape The size and shape characteristic of the different phases and the phases combined i presented in figures 4.19 - 4.27. Epidote Elongation vs. size As mentioned above epidote has particle sizes between approximately ∼ 2 and ∼ 90 µm, however approximately 90% of the particles are smaller than ∼ 30 µm (figure 4.19). The elongation of epidote is between ∼ 0 and ∼ 0.8 and is evenly distributed throughout the range. Particles smaller than ∼ 50 µm have elongation values spread over the entire range. But particles larger than ∼ 50 µm are limited to elongation values between ∼ 0.2 and ∼ 0.7. The maximum elongation decreases with increasing particle size from ∼ 0.8 to ∼ 0.7. The minimum elongation is constant S)/area

10

10

10

10

10

4.5. Sandstone dike

−1

−2

−3

−4

100x 200x 400x 600x 800x 1000x 1500x BLF D = 3.2139

−5

−6

−7

10

0

10

1

10

2

10

3

S [µm]

(a) Phases combined 10

(N(S) > S)/area

10

10

10

10

10

−2

−3

−4

100x 200x 400x 600x 800x 1000x 1500x BLF D = 3.2316

−5

−6

−7

10

0

1

10 S [µm]

10

2

(b) Epidote

Figure 4.43: PSD sandstone dike sample BRE1305A1. Low D-values ranging from 2.6673 to 3.2316 for plagioclase & quartz, and epidote, respectively. D-value of 2.2282 for all the phases combined.

151

4.5. Sandstone dike

10

(N(S) > S)/area

10

10

10

10


−2

−3

−4

100x 200x 400x 600x 800x 1000x 1500x BLF D = 3.0203

−5

−6

10

0

1

10 S [µm]

10

2

(c) K-feldspar 10

(N(S) > S)/area

10

10

10

10

10

−2

−3

−4

100x 200x 400x 600x 800x 1000x 1500x BLF D = 2.6673

−5

−6

−7

10

0

10

1

10

2

10

3

S [µm]

(d) Plagioclase & quartz

Figure 4.43: (Cont.) PSD sandstone dike sample BRE 1305A1. Low D-values ranging from 2.6673 to 3.2316 for plagioclase & quartz, and epidote, respectively. D-value of 3.2139 for all the phases combined.

152


4.5. Sandstone dike

Particle shape The size and shape characteristic of the different phases and the phases combined i presented in figures 4.44 - 4.52. Epidote Elongation vs. size

As mentioned above epidote has particles up to ∼ 100 µm, but

more than approximately 95% of the particles are smaller than ∼ 50 µm (figure 4.44). There is also a concentration of grains smaller than ∼ 15 µm. The elongation of epidote is between ∼ 0 and ∼ 0.8 and the elongation values are evenly distributed throughout the range (figure 4.44). There is no systematic relationship between size and elongation. Convexity vs. size

The convexity of epidote shown in figure 4.44 is between

∼ 0.4 to ∼ 1.0, but approximately 70% of the particles have convexity larger than ∼ 0.8 (figure 4.44). The convexity is concentrated around high values of ∼ 0.8 − 1.0 for particles smaller than ∼ 50 µm. For particles larger than ∼ 50 µm the maximum convexity drops to ∼ 0.8 and decreases with increasing size to ∼ 0.6. The minimum convexity is approximately constant for all particles sizes at ∼ 0.4. Circularity vs. size The circularity of epidote is between ∼ 0.2 and ∼ 1.0 (figure 4.44). Overall the distribution of the circularity of the particles is even throughout the range. But for particles smaller than ∼ 10 − 15 µm the circularity is spread over ∼ 0.4 − 1.0. For particles between ∼ 15 µm and ∼ 50 µm the maximum and minimum circularity decreases from ∼ 1.0 to ∼ 0.8, and ∼ 0.4 to ∼ 0.2, respectively. For the particles larger than ∼ 50 µm the maximum circularity the circularity decreases from ∼ 0.4 to ∼ 0.2, except for an outlier at ∼ 0.6. The minimum circularity is approximately constant at ∼ 0.2 for all particle sizes. To summarize, for epidote both the convexity and the circularity of the particle decrease with increasing particle size. The small epidote particles are more smooth and spherical than large particles. Orientation vs. size The orientation of epidote grains are between ∼ −90◦ and ∼ 90◦ and show no relationship with either size, elongation, convexity or circularity (figure 4.45). Elongation, convexity, and circularity

When the elongation of epidote is

plotted against its convexity, the data set spreads out evenly (figure 4.46(a)). There is no obvious relationship between elongation and convexity. The maximum convexity 153

4.5. Sandstone dike


is constant at ∼ 1.0 for all elongation values. The minimum convexity decreases from ∼ 0.5 to ∼ 0.2 at elongation equal to ∼ 0.3 before it increases to ∼ 0.6. The maximum an minimum elongation varies between ∼ 0 − 0.1 and ∼ 0.7 − 0.8, respectively with increasing convexity. The circularity of epidote is evenly distributed when it is plotted against elongation (figure 4.46(b)). The maximum circularity is approximately constant at ∼ 1.0 for elongation ∼ 0 − 0.3. For elongation larger than ∼ 0.3 the maximum circularity decreases linearly towards ∼ 0.4. The minimum circularity decreases from ∼ 0.5 to ∼ 0.15 at elongation ∼ 0.3. For elongation larger than ∼ 0.3 the minimum circulation is approximately constant at ∼ 0.15. For small circularity values between ∼ 0.1 and ∼ 0.5, the maximum elongation varies between ∼ 0.6 and ∼ 0.8. Overall the circularity decreases with increasing elongation as expected by equation 4.2 and 4.4. The more elongated the epidote grains are, the more angular they get. The convexity of epidote is proportional with the circularity (figure 4.46(c))as expected by the definitions of the terms (equation 4.3 and 4.4).

154


4.5. Sandstone dike

1

Elongation ε

0.8 0.6 0.4 0.2 0 1

Convexity ζ

0.8 0.6 0.4 0.2 0 1

Circularity ψ

0.8 0.6 0.4 0.2

Orientation θ [°]

0 90

45

0

−45

−90 0

50

100

150

200 250 Size S [µm]

300

350

400

Figure 4.44: Scatter plots of size vs. elongation, convexity, circularity and orientation of epidote. Sandstone dike sample BRE1305A1.

155

4.5. Sandstone dike


400 350

Size S [ µm]

300 250 200 150 100 50 0 1

Elongation ε

0.8 0.6 0.4 0.2 0 1

Convexity ζ

0.8 0.6 0.4 0.2 0 1 0.9 Circularity ψ

0.8 0.7 0.6 0.5 0.4 0.3 0.2 −90

−45

0 Orientation θ [°]

45

90

Figure 4.45: Scatter plots of orientation vs. size, elongation, convexity, and circularity of epidote. Sandstone dike sample BRE1305A1.

156


4.5. Sandstone dike

1 0.9 0.8

Convexity ζ

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.2

0.4

0.6 Elongation ε

0.8

1

0

0.2

0.4

0.6 Elongation ε

0.8

1

0

0.2

0.4

0.6 Convexity ζ

0.8

1

1 0.9 0.8

Circularity ψ

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

1 0.9 0.8

Circularity ψ

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

Figure 4.46:

Scatter plots of elongation, convex-

ity, and circularity of epidote. BRE1305A1.

157

Sandstone dike sample

4.5. Sandstone dike


K-feldspar Elongation vs. size

In determining K-feldspar shape and size data we see that

K-feldspar has two groupings of particles sizes (figure 4.47). Approximately 70% of the particles are smaller than ∼ 15 µm, whereas the rest of the particles are spread over ∼ 15 − 65 µm. The elongation of K-feldspar is between ∼ 0.1 and ∼ 0.9 (figure 4.47). Not considering size, the elongation of K-feldspar is evenly distributed throughout the. The maximum elongation decreases with increasing size from almost ∼ 0.9 to ∼ 0.6. The minimum elongation decreases from ∼ 0.2 to ∼ 0.1 at approximately ∼ 25 µm before it increases back to ∼ 0.2. Convexity vs. size

The convexity is between ∼ 0.4 and ∼ 1.0 (figure 4.47).

But only 2 particles have convexity values smaller than ∼ 0.6. The convexity is concentrated around ∼ 0.8 − 1.0 for particle smaller than approximately ∼ 50 µm. For particle larger than ∼ 50 µm the maximum convexity decreases linearly to ∼ 0.6. The minimum convexity is approximately constant at ∼ 0.6 for all particle sizes. Also noticeable is that throughout the particle size range, the convexity values are skewered towards the maximum value. Circularity vs. size The circularity is between ∼ 0.2 and ∼ 1.0 and is is evenly distributed throughout the range (figure 4.47). But for particles larger than ∼ 15 µm the maximum circularity decreases towards ∼ 0.7. The minimum circularity is approximately constant at ∼ 0.25 for all particle sizes, except an outlier at ∼ 0.15. Thus, the convexity and circularity is approximately constant for the majority of particles. But for particles larger than ∼ 50 µm both the convexity and circularity decreases although the increase in particle size is only ∼ 15 µm. Thus K-feldspar particle smaller than ∼ 50 µm are smooth and spherical, whereas particles ∼ 50 µm become more rough surfaced and angular with increasing size. Orientation vs. size

The orientation of K-feldspar grains are between ∼ −90◦

and ∼ 90◦ and show no relationship with either size, elongation, convexity or circularity (figure 4.48). Elongation, convexity, and circularity When plotted against elongation the convexity spreads out evenly (figure 4.49). The maximum convexity is approximately constant at 1.0 for all elongations. The convexity elongation decreases with increasing elongation from ∼ 1.0 to∼ 0.5 at elongation equal to ∼ 0.4 before it increases to ∼ 1.0. The circularity is skewered towards higher values at higher elongations when plotted against elongation (figure 4.49). The maximum circularity is approximately constant at ∼ 1.0 for elongation ∼ 0.1 − 0.25. For elongation larger than ∼ 0.25 158


4.5. Sandstone dike

the circularity decreases to ∼ 0.3. The minimum circularity decreases from ∼ 0.8 to ∼ 0.3 at elongation equal to ∼ 0.25. For elongation larger than ∼ 0.25 the minimum circularity varies slightly around ∼ 0.2. In general the circularity decreases with increasing elongation This indicates that elongated K-feldspar grains are more angular than symmetric grains. The convexity of K-feldspar is proportional with the circularity (figure 4.49) as expected by de definitions of the terms (equation 4.3 and 4.4).

159

4.5. Sandstone dike


1

Elongation ε

0.8 0.6 0.4 0.2 0 1

Convexity ζ

0.8 0.6 0.4 0.2 0 1

Circularity ψ

0.8 0.6 0.4 0.2

Orientation θ [°]

0 90

45

0

−45

−90 0

50

100

150

200 250 Size S [µm]

300

350

400

Figure 4.47: Scatter plots of size vs elongation, convexity and circularity of K-feldspar. Sandstone dike sample BRE1305A1.

160


4.5. Sandstone dike

400 350

Size S [ µm]

300 250 200 150 100 50 0 1

Elongation ε

0.8 0.6 0.4 0.2 0 1

Convexity ζ

0.8 0.6 0.4 0.2 0 1

Circularity ψ

0.8 0.6 0.4 0.2 0 −90

−45


45

90

Figure 4.48: Scatter plots of orientation vs. size, elongation, convexity, and circularity of K-feldspar. Sandstone dike sample BRE1305A1.

161

4.5. Sandstone dike


1 0.9 0.8

Convexity ζ

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.2

0.4

0.6 Elongation ε

0.8

1

0

0.2

0.4

0.6 Elongation ε

0.8

1

0

0.2

0.4

0.6 Convexity ζ

0.8

1

1 0.9 0.8

Circularity ψ

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

1 0.9 0.8

Circularity ψ

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

Figure 4.49:

Scatter plots of elongation, convexity,

and circularity of K-feldspar. BRE1305A1.

162

Sandstone dike sample


4.5. Sandstone dike

Plagioclase & quartz Elongation vs. size

Size and shape characteristics for plagioclase & quartz are

summarized in figure 4.50. Approximately 70% of the plagioclase & quartz particles are smaller than ∼ 15 µm. The rest is between ∼ 15 µm and ∼ 110 µm (figure 4.50). The elongation of plagioclase & quartz is between ∼ 0 and ∼ 0.75 (figure 4.50). The maximum elongation decreases from ∼ 0.75 to ∼ 0.6 with increasing size. The minimum elongation is approximately constant at ∼ 0 for all particle sizes. Overall the plagioclase & quartz grains get less elongated with increasing particle size. Convexity vs. size The convexity of plagioclase & quartz is between ∼ 0.3 − 1.0 (figure 4.50). But approximately ∼ 70% of the particles have convexity values larger than ∼ 0.9. Particles smaller than ∼ 15 µm are concentrated around convexity values between ∼ 0.8 and ∼ 1.0. Particles between ∼ 15 µm and ∼ 50 µm have the convexity values concentrated around ∼ 0.9 and ∼ 1.0. And for particles larger than ∼ 50 µm the maximum convexity decreases linearly to ∼ 0.7. The minimum convexity increases with increasing particle size from ∼ 0.4 to ∼ 0.7, except for an outlier at ∼ 0.3. Circularity vs. size

The circularity of plagioclase & quartz is spread over

∼ 0.2 − 1.0 and is evenly distributed throughout the range (figure 4.50). Particles smaller than ∼ 15 µm are spread over the entire range. But for particles between ∼ 15 µm and ∼ 50 µm the circularity is concentrated around ∼ 0.7 − 1.0. The maximum circularity is approximately constant for particle sizes smaller than ∼ 30 µm. But for particle larger than ∼ 30 µm the circularity decreases linearly with increasing particle size to ∼ 0.4. The minimum circularity increases from ∼ 0.2 to ∼ 0.6 with increasing particle size, except an outlier at ∼ 0.15. Thus, for particle smaller than ∼ 50 µm the particles are smooth and spherical, whereas particles larger than ∼ 50 µm becomes more rough surfaced and angular. Orientation vs. size The orientation of plagioclase & quartz grains are between ∼ −90◦ and ∼ 90◦ and show no systematic relationship with either size, elongation, convexity or circularity (figure 4.51). Elongation, convexity, and circularity

When plotted elongation the max-

imum convexity is approximately constant at ∼ 1.0 and the convexity is skewered towards high values, for all elongation values (figure 4.52). The maximum elongation increases generally from ∼ 0.6 to a peak value of ∼ 0.75 at convexity equal to 0.9 before it decreases towards ∼ 0.6. The minimum elongation decreases with increasing elongation from ∼ 0.3 to ∼ 0. When the circularity plotted against elongation it is slightly concentrated towards high values (figure 4.52). The maximum elongation is constant 163

4.5. Sandstone dike


at ∼ 1.0 for elongation ∼ 0 − 0.3. For elongations larger than ∼ 0.3 the maximum circularity decreases linearly to ∼ 0.5. The minimum circularity decreases from ∼ 0.6 to ∼ 0.2 as elongation increases from ∼ 0 to ∼ 0.15. For elongations ∼ 0.15 − 0.55 the minimum circularity is constant at ∼ 0.2. For elongations larger than ∼ 0.55 the minimum circularity increases towards ∼ 0.3. The minimum elongation decreases with increasing circularity from ∼ 0.2 to ∼ 0 at circularity equal to ∼ 0.6. For circularity larger than ∼ 0.6 the minimum elongation is constant at ∼ 0. Overall circularity decreases with increasing elongation indicating that elongated plagioclase & quartz grains are more angular than symmetric grains. The convexity of plagioclase & quartz is proportional with the circularity (figure 4.52) as expected by de definitions of the terms (equations 4.3 and 4.4).

164


4.5. Sandstone dike

1

Elongation ε

0.8 0.6 0.4 0.2 0 1

Convexity ζ

0.8 0.6 0.4 0.2 0 1

Circularity ψ

0.8 0.6 0.4 0.2

Orientation θ [°]

0 90

45

0

−45

−90 0

50

100

150

200 Size S [µm]

250

300

350

400

Figure 4.50: Scatter plots of size vs. elongation, convexity, and circularity of plagioclase & quartz. Sandstone dike sample BRE1305A1.

165

4.5. Sandstone dike


400 350

Size S [ µm]

300 250 200 150 100 50 0 1

Elongation ε

0.8 0.6 0.4 0.2 0 1

Convexity ζ

0.8 0.6 0.4 0.2 0 1

Circularity ψ

0.8 0.6 0.4 0.2 0 −90

−45


45

90

Figure 4.51: Scatter plots of orientation vs. size, elongation, convexity, and circularity of plagioclase & quartz. Sandstone dike sample BRE1305A1.

166


4.5. Sandstone dike

1 0.9 0.8

Convexity ζ

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.2

0.4

0.6 Elongation ε

0.8

1

0

0.2

0.4

0.6 Elongation ε

0.8

1

0

0.2

0.4

0.6 Convexity ζ

0.8

1

1 0.9 0.8

Circularity ψ

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

1 0.9 0.8

Circularity ψ

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

Figure 4.52: Scatter plots of elongation, convexity, and circularity of plagioclase & quartz. Sandstone dike sample BRE1305A1.

167

4.5. Sandstone dike


Particle shape distributions Histograms and summary tables are presented to show the shape distributions for different magnifications. To allow comparison between magnifications and different samples the shape descriptors are plotted as number of events are normalized by the P total number of events (total number of particles) in the histograms, N (i)/ 1i=0 N . Elongation The mean value of elongations for the individual phases an all the phases combined show a systematic trend with increasing magnification (figure 4.53). K-feldspar and plagioclase & quartz and the all the phases combined have a general increase in mean with increasing magnification. The mean value of K-feldspar and plagioclase & quartz increases from ∼ 0.4 and ∼ 0.3, respectively, at low magnification to ∼ 0.5. The mean value of the phases combined increases from ∼ 0.36 to 0.45 with increasing magnification. The mean value of epidote varies somewhat with increasing magnification, but has a slight general decrease in mean value from ∼ 0.42 to ∼ 0.40. At low magnifications epidote has the smallest mean value of the individual phases, whereas plagioclase have the smallest mean value at high magnifications. The peak values for epidote are larger than their corresponding mean values at all individual magnifications and all magnifications combined. For K-feldspar, plagioclase & quartz, and all the phases combined the peak values are both smaller and larger. The peak value of epidote generally decrease with increasing magnification from ∼ 0.5 to ∼ 0.4, whereas the peak value of plagioclase & quartz increases from ∼ 0.3 to ∼ 0.5. The peak value of Kfeldspar varies between ∼ 0.3 and ∼ 0.7 with increasing magnification. No systematic trend is observed skewness or kurtosis for either the individual phases or all the phases combined with increasing magnification. The skewness is low and both positive and negative for all individual phases and all phases combined at all magnifications and all magnifications combined. The kurtosis for epidote and K-feldspar varies between 2.0 − 2.7 and 1.8 − 2.0, respectively. For plagioclase & quartz the variation is less, 1.9 − 3.7. For all the phases combined the kurtosis varies is almost constant with increasing magnification at 2.3 − 2.4. At all magnifications combined epidote and plagioclase & quartz have a mean values of ∼ 0.35 − 0.38 similar to that of all the phases combined. K-feldspar has a larger mean value of ∼ 0.5 Also the spread is equal at ∼ 0.16 − 0.17 for all individual phases and all of the phases combined. Plagioclase & quartz is slightly skewered right with magnitude ∼ 0.1. This close to the skewness for all the phases combined, whereas epidote and K-feldspar are more symmetric around

168


4.5. Sandstone dike

their mean values with skewness close to ∼ 0. Similarly the kurtosis of epidote and plagioclase & quarts is ∼ 2.4 and is close to the kurtosis of all the phases combined The kurtosis of K-feldspar is closer to ∼ 2.1. 100x magnification 0.1

N( ε)/ Σ1ε=0N

Skewness Kurtosis Particles

Epidote K-feldspar Plag. & qtz. Combined 0.399 0.301 0.416 0.358 0.165 0.147 0.168 0.166 0.323 0.268 0.544 0.268, 0.323, 0.434 0.0219 0.115 -0.0711 0.124 2.08 2.33 2.67 2.43 52 98 58 208

Phases Mean Std. Peak

Epidote K-feldspar Plag. & qtz. 0.281 0.400 0.365 0.148 0.172 0.163 0.268 0.508 0.508


-0.0468 2.40 58


Epidote K-feldspar Plag. & qtz. Combined 0.522 0.374 0.404 0.404 0.162 0.146 0.155 0.151 0.581, 0.397 0.231, 0.452 0.655 0.268, 0.323, 0.397 -0.470 -0.0821 −0.0396 -0.0978 1.84 2.55 2.43 2.47 17 66 152 69


0.08 0.06 0.04 0.02 0 0

0.1

0.2

0.3

0.4

0.5 ε

0.6

0.7

0.8

0.9

1

200x magnification 0.1

N( ε)/ Σ1ε=0N

0.08 0.06 0.04 0.02

0.112 1.98 18

0.353 2.59 78

Combined 0.336 0.168 0.268, 0.508 0.240 2.39 154

0 0

0.1

0.2

0.3

0.4

0.5 ε

0.6

0.7

0.8

0.9

1


N( ε)/ Σ1ε=0N

0.08 0.06 0.04


0.02 0 0

0.1

0.2

0.3

0.4

0.5 ε

0.6

0.7

0.8

0.9

1

600x magnification 0.1 Phases Mean Std. Peak

N( ε)/ Σ1ε=0N

0.08 0.06 0.04


0.02 0 0

0.1

0.2 Epidote

0.3

0.4

0.5 ε

K−feldspar

0.6

0.7

0.8

0.9

Epidote K-feldspar Plag. & qtz. Combined 0.463 0.335 0.367 0.368 0.171 0.160 0.169 0.165 0.397 0.415 0.213, 0.397 0.287, 0.323, 0.397, 0.489 -0.0429 0.193 0.428 0.270 1.98 2.60 2.60 2.50 21 58 51 130

1

Plagioclase & quartz

Figure 4.53: Elongation sandstone dike sample BRE1305A1. The shape descriptors are plotted as number of events are normalized by the total number of events (total number of P1 particles) in the histograms, N (i)/ i=0 N , to allow comparison between magnifications and different samples.

169

4.5. Sandstone dike



N( ε)/ Σ1ε=0N

0.08

Phases Mean Std. Peak Skewness Kurtosis Particles

0.06 0.04 0.02

Epidote K-feldspar Plag. & qtz. Combined 0.508 0.400 0.374 0.417 0.201 0.156 0.167 0.178 0.397 0.360 0.379 0.379 0.0140 -0.123 0.0999 0.174 2.06 2.24 2.03 2.43 39 70 57 166

0 0

0.1

0.2

0.3

0.4

0.5 ε

0.6

0.7

0.8

0.9

1

1000x magnification 0.1 Epidote K-feldspar Plag. & qtz. Combined 0.449 0.412 0.380 0.444 0.169 0.154 0.134 0.172 0.489 0.489, 0.397, 0.287 0.618 0.415 0.222 0.315 0.193 Skewness 0.553 1.85 2.36 2.35 Kurtosis 2.26 19 78 40 Particles 19 Phases Mean Std. Peak

N( ε)/ Σ1ε=0N

0.08 0.06 0.04 0.02 0 0

0.1

0.2

0.3

0.4

0.5 ε

0.6

0.7

0.8

0.9

1


N( ε)/ Σ1ε=0N


Epidote K-feldspar Plag. & qtz. Combined 0.491 0.450 0.394 0.470 0.114 0.156 0.151 0.181 0.434, 0.379 0.415 0.379, 0.471, 0.692 0.508, 0.526 -0.558 -0.160 0.129 -0.168 3.71 2.33 2.64 1.82 18 60 21 21


Epidote 0.386 0.160 0.415 0.0319 2.45 361


0.08 0.06 0.04 0.02 0 0

0.1

0.2

0.3

0.4

0.5 ε

0.6

0.7

0.8

0.9

1

All magnifications 0.1

N( ε)/ Σ1ε=0N

0.08 0.06 0.04 0.02

K-feldspar Plag. & qtz. Combined 0.449 0.351 0.384 0.181 0.161 0.169 0.397 0.268 0.397 0.0939 0.104 0.134 2.14 2.35 2.44 187 400 948

0 0

0.1

0.2 Epidote

0.3

0.4

0.5 ε

K−feldspar

0.6

0.7

0.8

0.9

1


Figure 4.53: (Cont.) Elongation sandstone dike sample BRE1305A1. The shape descriptors are plotted as number of events are normalized by the total number of events (total number of P1 particles) in the histograms, N (i)/ i=0 N , to allow comparison between magnifications and different samples.

170


4.5. Sandstone dike

Convexity There is a strong peak at ∼ 1 with a corresponding lower mean value at all magnifications and all magnifications combined (figure 4.54). Except for epidote at high magnifications as a result of low number of grains. There is no systematic trend observed for the mean value with increasing magnification for either individual phases or all the phases combined. The mean value are in the range of 0.772 − 0.959. The mean for all individual phases are close to the value for all phases combined. The largest deviation from the mean value of all the phases combined is ∼ 0.040 − 0.050 for K-feldspar and plagioclase & quartz at high magnifications. The peak values are equal to ∼ 1.0 for all individual phases at all magnifications. The skewness is negative for all individual phases and all phases combined at all individual magnifications and all magnifications combined. For epidote the magnitude of the skewness decreases from ∼ 2.8 towards ∼ 0, 3 overall with increasing magnification. For K-feldspar and plagioclase & quartz, and all the phases combined the skewness undulates with increasing magnification. The magnitude of the skewness for is 0.5 − 3.3, 1.0 − 2.0, and 1.2 − 1.8. The kurtosis for epidote has a general decrease from ∼ 7 to ∼ 2.5 with increasing magnification. Whereas the kurtosis for K-feldspar, plagioclase & quartz, and all the phases combined varies with increasing magnification. The kurtosis for Kfeldspar varies between ∼ 2.3 and ∼ 13.4. For plagioclase & quartz and all the phases combined the variation is smaller. Plagioclase & quartz and all the phases combined varies between ∼ 3.1 − 6.9, and ∼ 3.2 − 6.1. For all magnifications combined epidote and K feldspar, plagioclase & quartz and all of the phases combined have mean values of ∼ 0.9. The spread and peak values are similar for all individual phases and all of the phases combined with standard deviations of ∼ 0.12 − 0.13 and peak values of 1.0. All of the individual phases are skewered left. But epidote and K-feldspar have relatively lower magnitudes of ∼ 1.1, whereas plagioclase & quartz have a larger magnitude of ∼ 1.7. This leaves the skewness of all the phases combined at intermediate magnitude of ∼ 1.4. All of the kurtosis values are above 3. But epidote and K-feldspar have relatively small kurtosis values of ∼ 3.6 − 3.8. While plagioclase & quartz have a large kurtosis value of ∼ 5.7 and more sharp peak. For all the phases combined the kurtosis is ∼ 4.4.

171

4.5. Sandstone dike


100x magnification

N( ζ)/ Σ1ζ=0N

0.25 0.2 0.15 0.1


Epidote 0.887 0.127 1.00 -1.89 6.78 58

K-feldspar Plag. & qtz. Combined 0.938 0.901 0.846 0.0910 0.115 0.120 1.00 1.00 1.00 -1.77 -1.46 -0.526 5.19 4.96 2.33 98 208 52


Epidote 0.890 0.109 0.970 -1.47 5.20 58

K-feldspar Plag. & qtz. Combined 0.932 0.919 0.959 0.0996 0.104 0.0824 1.00 1.00 1.00 -1.98 -1.79 -3.35 6.89 6.13 13.38 78 154 18


Epidote K-feldspar Plag. & qtz. Combined 0.891 0.853 0.860 0.860 0.118 0.168 0.148 0.135 0.955 1.00 1.00 1.00 -1.09 -1.35 -1.29 -1.02 2.70 3.94 4.02 3.25 17 66 152 69



0.05 0 0

0.1

0.2

0.3

0.4

0.5 ζ

0.6

0.7

0.8

0.9

1


N( ζ)/ Σ1ζ=0N

0.25 0.2 0.15 0.1 0.05 0 0

0.1

0.2

0.3

0.4

0.5 ζ

0.6

0.7

0.8

0.9

1


N( ζ)/ Σ1ζ=0N

0.25 0.2 0.15 0.1 0.05 0 0

0.1

0.2

0.3

0.4

0.5 ζ

0.6

0.7

0.8

0.9

1


N( ζ)/ Σ1ζ=0N

0.25 0.2 0.15 0.1 0.05 0 0

0.1

0.2 Epidote

0.3

0.4

0.5 ζ

K−feldspar

0.6

0.7

0.8

0.9

1


Figure 4.54: Convexity sample sandstone dike sample BRE1305A1. The shape descriptors are plotted as number of events are normalized by the total number of events (total number of P1 particles) in the histograms, N (i)/ i=0 N , to allow comparison between magnifications and different samples.

172


4.5. Sandstone dike

800x magnification

N( ζ)/ Σ1ζ=0N

0.25 0.2 0.15 0.1




Epidote K-feldspar Plag. & qtz. Combined 0.924 0.870 0.877 0.840 0.0897 0.132 0.153 0.132 0.955 1.00 0.940, 0.789, 0.970 0.910 -1.63 -1.72 -1.38 -1.01 4.85 4.87 4.45 3.67 19 19 78 40

0.05 0 0

0.1

0.2

0.3

0.4

0.5 ζ

0.6

0.7

0.8

0.9

1


N( ζ)/ Σ1ζ=0N

0.25 0.2 0.15


0.1 0.05 0 0

0.1

0.2

0.3

0.4

0.5 ζ

0.6

0.7

0.8

0.9

1


N( ζ)/ Σ1ζ=0N


Epidote K-feldspar Plag. & qtz. Combined 0.893 0.895 0.852 0.772 0.117 0.0923 0.133 0.147 0.955 0.970 1.00 0.714, 0.744, 0.910, 0.925 -1.06 -1.12 -0.893 -0.288 3.10 4.24 3.15 2.46 18 21 60 21




0.25 0.2 0.15 0.1 0.05 0 0

0.1

0.2

0.3

0.4

0.5 ζ

0.6

0.7

0.8

0.9

1


N( ζ)/ Σ1ζ=0N

0.25 0.2 0.15 0.1 0.05 0 0

0.1

0.2

0.3

Epidote

0.4

0.5 ζ

K−feldspar

0.6

0.7

0.8

0.9

1


Figure 4.54: (Cont.) Convexity sandstone dike sample BRE1305A1. The shape descriptors are plotted as number of events are normalized by the total number of events (total number of P1 particles) in the histograms, N (i)/ i=0 N , to allow comparison between magnifications and different samples.

173

4.5. Sandstone dike


Circularity The mean value of epidote and all phases combined generally decrease from ∼ 0.7 to ∼ 0.5 and ∼ 0.6, respectively, with increasing magnification. Similarly plagioclase & quartz decreases from ∼ 0.8 to ∼ 0.6. For K-feldspar the mean value varies between ∼ 0.5 and ∼ 0.7, but with no systematic trend. The peak values for plagioclase & quartz and all the phases combined are generally larger (∼ 0.8 − 0.9) than their corresponding peak values, except at high magnifications. For epidote and K-feldspar the peak values are both smaller and larger ∼ 0.5 − 0.9, respectively). For all the magnifications combined the peak values of K-feldspar and plagioclase & quartz (0.812 and 0.890, respectively) are close to the peak value for all the phases combined (0.890). There is no systematic trend with increasing magnification for either the individual phases or all of the phases combined. Epidote has small skewness, both negative and positive. The magnitude of the skewness is between ∼ 0.06 and ∼ 0.5. K-feldspar has similar signs and magnitudes (∼ 0.09 − 0.4, except for 200x magnification where the skewness is larger (−1.45). Plagioclase & quartz and all the phases have only negative skewness. The values generally compares to those of epidote and K-feldspar with small magnitudes of ∼ 0.1 − 0.5 and ∼ 0.2 − 0.5, respectively, except at low magnifications. At low magnification the skewness of plagioclase & quartz and all the phases combined are larger, ∼ 0.9 − 1.0 and ∼ 0.8, respectively. The kurtosis of epidote is between ∼ 2.1−2.7. The kurtosis for K-feldspar is ∼ 2.0−2.4, except at 200x magnification (4.96). Plagioclase & quartz and all the phases combined have a slightly larger range of kurtosis (∼ 1.8 − 3.3 and ∼ 2.0 − 2.9). At all magnifications combined the individual phases and all of the phases combined have similar location and spread of ∼ 0.6 − 0.7 ± 0.2. All the phases are skewered left. Epidote and plagioclase & quartz have longer left tails with skewness ∼ −0.3 and ∼ −0.6, respectively. While K-feldspar is centered around its mean value and has skewness of ∼ −0.1 Similarly K-feldspar has a relatively low kurtosis value of ∼ 2.1. While epidote and plagioclase & quartz have slightly larger values of ∼ 2.3 and 2.4, respectively.

174


4.5. Sandstone dike


N( ψ)/ Σ1ψ=0N

0.08 0.06 0.04 0.02


Epidote K-feldspar Plag. & qtz. Combined 0.607 0.791 0.658 0.708 0.170 0.158 0.188 0.187 0.523 0.851 0.832 0.851 0.0930 -0.902 -0.498 -0.487 1.98 2.88 2.33 2.23 52 98 58 208




Epidote K-feldspar Plag. & qtz. Combined 0.646 0.607 0.642 0.646 0.213 0.158 0.200 0.199 0.793, 0.542 0.832 0.832 0.832 -0.413 0.105 -0.358 -0.403 2.19 2.44 2.26 2.29 66 17 152 69

0 0

0.1

0.2

0.3

0.4

0.5 ψ

0.6

0.7

0.8

0.9

1


N( ψ)/ Σ1ψ=0N

0.08 0.06 0.04 0.02 0 0

0.1

0.2

0.3

0.4

0.5 ψ

0.6

0.7

0.8

0.9

1


N( ψ)/ Σ1ψ=0N

0.08 0.06 0.04


0.02 0 0

0.1

0.2

0.3

0.4

0.5 ψ

0.6

0.7

0.8

0.9

1


N( ψ)/ Σ1ψ=0N

0.08 0.06 0.04


0.02

Epidote K-feldspar Plag. & qtz. Combined 0.692 0.668 0.679 0.683 0.211 0.211 0.208 0.207 0.870 0.523, 0.870, 0.696, 0.870, 0.890 0.890 0.890 -0.312 -0.320 -0.335 -0.354 1.91 1.81 1.98 2.16 21 51 130 58

0 0

0.1

0.2

0.3

Epidote

0.4

0.5 ψ

K−feldspar

0.6

0.7

0.8

0.9

1


Figure 4.55: Circularity sandstone dike sample BRE1305A1. The shape descriptors are plotted as number of events are normalized by the total number of events (total number of P1 particles) in the histograms, N (i)/ i=0 N , to allow comparison between magnifications and different samples.

175

4.5. Sandstone dike



N( ψ)/ Σ1ψ=0N

0.08 0.06 0.04


0.02

Epidote K-feldspar Plag. & qtz. Combined 0.494 0.654 0.668 0.621 0.194 0.192 0.198 0.206 0.348, 0.890 0.600, 0.890 0.503 0.658 0.371 -0.485 -0.177 -0.187 2.19 2.42 2.05 2.06 39 70 57 166

0 0

0.1

0.2

0.3

0.4

0.5 ψ

0.6

0.7

0.8

0.9

1


N( ψ)/ Σ1ψ=0N

0.08 0.06 0.04


0.02

Epidote K-feldspar Plag. & qtz. Combined 0.664 0.600 0.617 0.603 0.170 0.189 0.171 0.164 0.638, 0.696 0.484, 0.484 0.851 0.600, 0.696 -0.285 -0.132 -0.107 -0.0639 2.22 2.41 2.57 2.37 19 78 19 40

0 0

0.1

0.2

0.3

0.4

0.5 ψ

0.6

0.7

0.8

0.9

1


N( ψ)/ Σ1ψ=0N


Epidote K-feldspar Plag. & qtz. Combined 0.588 0.605 0.575 0.534 0.157 0.147 0.152 0.151 0.465, 0.581 0.484 0.484 0.600, 0.696, 0.754 -0.111 -0.150 -0.0187 0.216 2.07 2.17 2.25 2.72 18 21 60 21




0.08 0.06 0.04 0.02 0 0

0.1

0.2

0.3

0.4

0.5 ψ

0.6

0.7

0.8

0.9

1


N( ψ)/ Σ1ψ=0N

0.08 0.06 0.04 0.02 0 0

0.1

0.2 Epidote

0.3

0.4

0.5 ψ

K−feldspar

0.6

0.7

0.8

0.9

1


Figure 4.55: (Cont.) Circularity sandstone dike sample BRE1305A1. The shape descriptors are plotted as number of events are normalized by the total number of events (total number of P1 particles) in the histograms, N (i)/ i=0 N , to allow comparison between magnifications and different samples.

176


4.6

4.6. Discussion

Discussion

Size and shape characteristics for fault gouge sample BRE1505-2 and sandstone dike sample BRE1305A1 have been quantified by image analysis and presented along with textural observations above. Now the observations will be compared and discussed in terms of tectonic environment.

4.6.1

Particle size analysis

For all the phases combined, sandstone dike sample BRE1305A1 has the highest Dvalue of ∼ 3.2, whereas fault gouge sample BRE1505-2 has a slightly lower value of ∼ 3.0. From the D-values for all the phases combined alone, it would look like the sandstone dike BRE1305A1 was a more evolved shear fracture than fault gouge sample BRE1505-2 (Marone and Scholz, 1989; Blenkinsop, 1991; An and Sammis, 1994; Blenkinsop and Fernandes, 2000; Storti et al., 2003). Also, when the shape distributions all of the phases combined are considered they appear comparable (figures 4.62 - 4.64). However, the two samples show distinct characteristics when it comes to field and textural observations, porphyroclast content, and importantly the grain size and shape distributions of the individual phases. fault gouge sample BRE1505-2 In fault gouge sample BRE1505-2, K-feldspar and plagioclase & quartz are the main constituents of the porphyroclasts (table 4.1). They generally show the same size and shape characteristics. Epidote, the minor porphyroclast constituent show different characteristics, but this is masked by the other phases when data for all the phases combined is evaluated. Generally plagioclase & quartz is the most abundant phase at all magnifications, while K-feldspar is intermediate and epidote is the least abundant. The porphyroclast proportion in the fault gouge sample increases from 35% to 61% with increasing magnification (table 4.1). K-feldspar and plagioclase & quartz have large mean (30 and 42 µm, respectively) and small peak values (3 − 4.4 µm), similar particle size ranges (2 − 200 µm), and D-values (table 4.1). Although, plagioclase & quartz, have some large survivor grains larger than ∼ 200 µm, the majority of the grains are smaller than ∼ 200 µm (figure 4.25). For both phases the porphyroclast proportion increase to a maximum at value intermediate magnification, before it decreases with further magnification (table 4.1). Epidote deviates from this trend. Epidote has a smaller mean value, close to the peak value (20 and 16 µm, respectively) (table 4.1). 177

4.6. Discussion


The particle size is limited to particles smaller than ∼ 100 µm and its porphyroclast proportion increases with increasing magnification. The D-values for fault gouge sample BRE1505-2 are generally large. The PSD of fault gouge sample BRE1505-2 can be fitted to a slope for the majority of the particle size range (D1 ) and a different slope near one end of the particle size range (D2 ) (figure 4.18). The smallest fraction of epidote can be fitted to slope with D2 value ∼ 1.8 which is smaller than the D1 -value of ∼ 3.3 . In contrast K-feldspar and plagioclase & quartz, and all of the phases combined which have D2 -values of 4.1, 3.3, and 3.7, respectively, for the largest fraction of grains. The D2 -values are smaller than their respective D1 values which are 3.0, 2.6 and, 3.2 for K-feldspar and plagioclase & quartz, and all of the phases combined, respectively. Although the D2 -value of smallest fraction of epidote is comparable to that reported from extensional fractures (Marone and Scholz, 1989; Blenkinsop, 1991; Blenkinsop and Fernandes, 2000), in this context it is unlikely that this small fraction is created by a tensile fragmentation process. The low D2 -value of epidote might be due to a over-sampling of larger grains (Blenkinsop, 1991) at high magnifications and the lower particle size might represent the grinding limit for epidote (An and Sammis, 1994). The higher D2 -values of the large fraction of K-feldspar, plagioclase & quartz, and all of the phases combined might due to the fact that although small particles have a greater fracturing strength than large particles (Marone and Scholz, 1989), it is difficult to remove the large particles completely from the fault gouge (Engelder, 1974; Ozkan and Ortoleva, 2000). Continued granulation will create more small grains that will cushion and protect large grains from further granulation. As the fraction of small particles increases and at the same time the number of large particles is reduced, the fracture probability of the large grains is reduced as predicted by the constrained comminution model (Sammis et al., 1987). The D1 -values in this study are higher than reported from theoretical models (Allegre et al., 1982; Turcotte, 1986; Sammis et al., 1987) and from faults in crystalline basement rocks which have D ∼ 2.6 (Sammis et al., 1986, 1987; Sammis and Biegel, 1989; An and Sammis, 1994), consistent with the constrained comminution model (Sammis et al., 1987). However, it is important to remember that D-values are affected by amount of shear displacement, number of fracturing events and confining pressure (Engelder, 1974; Sammis et al., 1986; Marone and Scholz, 1989; Blenkinsop, 1991; An and Sammis, 1994; Blenkinsop and Fernandes, 2000; Storti et al., 2003). The D1 -values comparable to that reported from shear localization in cataclastic rocks (Marone and Scholz, 1989; Blenkinsop, 1991; An and Sammis, 1994; Blenkinsop and Fernandes, 2000; Storti et al., 2003). Hence I suggest in situ granulation is consistent with my observations of grain 178


4.6. Discussion

size. Both field and textural observations also support in situ granulation. The apparent displacement observed in the field is relatively low, 13 cm (figure 4.4). The relatively high D1 -values might indicate high confining pressure and/or several faulting events, both sinistral and dextral movement. Repeated faulting events might be represented by the internal shear bands in the fault gouge (figure 4.15). Another possibility is that the apparent displacement we observe in the field is much smaller than the actual shear displacement. However, the shape and zonation of epidote both in the fault gouge and the faults in the wall-rock (figures 4.8 and 4.17) suggest repeated faulting. We also observe faulting with granulation in the wall-rock and fragmentation of wall rock material (figures 4.6, 4.7, 4.10, and 4.11) which suggest in situ granulation of wall-rock material.

Sandstone dike sample BRE1305A1 From the field observations it is clear that the sandstone dike is apart of a pull-apart fracture (figure 4.31) and that the apparent extension is 10 cm (figure 4.33). Also the sandstone dikes are connected to sub-parallel fractures with granulated material. These fractures have the same orientation (160/70, 154/72) as the fractures and faults found elsewhere in the granodiorite and in the basin and it is a valid assumption that they are related to the basin development (see Chapter 2 Fieldwork on page 7). It seems that material in the pull-apart sandstone dike has three possible origins. Either it is (i) granulated material of the wall-rock that has been transported from the connecting fractures, (ii) material derived from the basin, or (iii) a combination of both. In sandstone dike sample BRE1305A1 the D-values are generally larger than those for fault gouge sample BRE1505-2 (table 4.2) and are larger than what is reported for extensional fractures (Marone and Scholz, 1989; Blenkinsop, 1991; Blenkinsop and Fernandes, 2000) and for faults in crystalline basement rocks (Sammis et al., 1986, 1987; Sammis and Biegel, 1989; An and Sammis, 1994), As with the fault gouge sample, the D-values of the sandstone dike material compare to that reported from shear localization in cataclastic rocks (Marone and Scholz, 1989; Blenkinsop, 1991; An and Sammis, 1994; Blenkinsop and Fernandes, 2000; Storti et al., 2003). If the fracture filling was derived from tensile fracturing of the wall-rock, one would expect the upper particle size bound to be equal or larger than that for fault gouge sample BRE15052. To the author’s knowledge there few detailed analysis of PSDs in undeformed sedimentary rocks. The PSD studies of sedimentary rocks mainly investigate the size 179

4.6. Discussion


distribution of pore space. There have also been studies of PSDs of unconsolidated sediments. The D2D -values range between 1.1 and 2.0 (Li et al., 2006; Kreina et al., 2003; Bryksina and Last, 2005). However, those values are not directly applicable to sandstone since the lithification involves compaction and growth of diagnetic minerals that will change both grain size and shape. In the sandstone dike, size and shape characteristics are distinct for K-feldspar and plagioclase & quartz. Here K-feldspar is the least abundant phase while epidote is much more abundant. Both epidote and plagioclase & quartz have a maximum particle size of ∼ 100 µm (table 4.2). Whereas all K-feldspar has grains are smaller than ∼ 65 µm. Epidote and K-feldspar behave similarly. They both have relatively large mean (∼ 18 µm) and small peak values (8 and 4 µm, respectively), and similar D-values (∼ 3.2 and ∼ 3.0, respectively). In contrast plagioclase & quartz have the largest mean value (∼ 23 µm) and the smallest D-value (∼ 2.7) of the individual phases. The peak value for plagioclase & quartz is larger than the mean value (∼ 37 µm and ∼ 23 µm respectively). Plagioclase & quartz dominates the porphyroclast assemblage at low magnifications, while epidote is the most abundant phase at high magnifications in the sandstone dike sample (table 4.1) (this contrasts to the fault gouge sample). K-feldspar, the least abundant phase, has a low porphyroclast proportion at all magnifications. The sandstone dike sample generally has higher porphyroclast proportion at high magnifications (table 4.1). Increasing from 28% to 80% with increasing magnification. In the field we observe both pull-apart opening of fractures and sedimentary flow structures (figures 4.31 - 4.33), but we also observe minor granulation in connecting fractures (figure 4.34). Minor granulation is also observed at the wall-rock in sandstone dike sample BRE1305A1 (figure 4.36 and 4.38). Furthermore this fracture material is in some ways texturally similar to that we observe in the fractures terminating at the sandstone dike and also in the sandstone dike (figures 4.39 and 4.42). What is also important is that in spite of the relatively large D-values, there are no survivor grains in the sandstone dike (figures 4.59, 4.60, and 4.61). Although few, they are as explained above, difficult to get completely rid of. The non-presence of survivor grains and the consequent low upper grain scale might indicate that the granulation is not in situ, but has happened in the connecting fractures and the material is transported to the pullaparts. Since the fractures and granulation zone are thin and/or under compressive stress, transport of relatively large grains to tensile fractures (low pressure zones) is inhibited. However, it should be noted that proximity to the basin margin and the presence of flow structures in the sandstone may imply infill of material from the basin. 180


4.6.2

4.6. Discussion

Comparison between fault gouge and sandstone dike

As mentioned above, the D-values found for both samples are relatively large and compareable. However, importantly the size characteristics of the different phases are distinct in the two samples. To sum up, it is clear that the fault gouge material is generated by in situ granulation. However, the sandstone dike appears to have a mixed signature, with evidence for both granulation and infill of basin material. In view of this, the size and shape characteristics of the different phases will be compared and discussed below. Combined shape and size analysis Epidote shape and size The epidote phase in fault gouge sample BRE1505-2 and sandstone dike BRE1305A1 have similar D-values (3.2) and particle size ranges (figure 4.59). In both samples the epidote grains becomes more smoother and spherical with decreasing particle size. However, fault gouge sample BRE1505-2 has a larger fraction of small epidote grains which have a rougher surface and are more non-circular (ζ < 0.4 and ψ < 0.2) than in sandstone dike sample BRE1305A1. When convexity and circularity of epidote is plotted against elongation and orientation, we observe that for all elongations and orientation the fault gouge sample has a fraction of more irregular and non-circular grains (figure 4.62(a) - (b), and 4.66 ). Importantly the fault gouge sample has a fraction of very rough surfaced and non-circular epidote grains that is not present in the sandstone dike (figure 4.62(c)). In the fault gouge sample the skewness and kurtosis of convexity are small (-0.432 and 2.08, respectively). Whereas for the sandstone dike the skewness and kurtosis are larger (-1.14 and 3.75, respectively), indicating a stronger influence of particles that are more irregular. Its clear that from the statistics and the scatter plots that the shape characteristics of epidote in the two samples are somewhat similar , except that the convexity and circularity of the epidote of fault gouge is more influenced small by irregular and non-circular grains. Epidote as comminution and fluid flow signature As mentioned above in both samples the epidote grains generally becomes more smoother and spherical with decreasing particle size (figure 4.59), but fault gouge sample BRE15052 has a fraction small epidote grains which are have a rougher surface and are more non-circular than the epidote particle in sandstone dike sample BRE1305A1. This might indicate that both preferential fragmentation of larger epidote grains during 181

4.6. Discussion


shear localization (Blenkinsop, 1991; Blenkinsop and Fernandes, 2000) and particle abrasion (Hattori and Yamamoto, 1999; Storti et al., 2003) have been active during faulting. Preferential fragmentation will create irregular and non-spherical grains, whereas particle abrasion will create smaller particles with smooth surfaces that will become progressively more spherical. In the sand stone dike the small grains have more smooth surfaces and spherical shapes indicative of preferential fragmentation of larger grains. In fault gouge sample BRE1505-2 the epidote grains are zoned with iron rich outer parts that display a complex shape shape, both in the wall-rock and in the fault gouge (figures 4.8 and 4.17). The shape is partly euhedral and and partly irregular suggesting that the outer parts must have grown after cataclasis and this will affect the particle size and shape distributions measured. However, this does at the same time document cataclastic deformation. In the wall rock the epidote is found as radiating needles. Fragmentation of the wall-rock material could potentially generate small epidote fragments that will act as nuclees for growth of new epidote. This implies several deformation events, otherwise there would be no need for the epidote to start to grow. Also, since it is connected with the fluid it is an indicator of fluid presence in the fault zone. The epidote in sandstone dike sample BRE1305A1 is comparable to that in the fault gouge sample. The epidote in the sandstone dike has a complex angular texture with black inclusions. However, they differ when it comes to concentric growth. The epidote in the fault gouge sample show iron-rich rims indicative of growth during cataclasis in both the wall-rock faults and in the fault gouge (figures 4.8 and 4.17), while the epidote in the sandstone dike do not (figure 4.42). This does not exclude epidote growth in the sandstone dike during cataclasis, but tells us that the iron-content in the fluid was the same before and after any cataclasis. It is not a unreasonable to assume that the proximity to the basin allows for substantial fluid infiltration that could buffer the iron-content in the fractures close to the basin margin. This would be unlikely in faults and fractures several hundred meters away from the basin margin. This difference might also be shown in the modal percentage of epidote in the two samples. In the fault gouge sample epidote is the least abundant phase and the relative porphyroclast proportion of epidote increases from 5% at low magnification to 20% at high magnification (table 4.1). In the sandstone dike epidote is much more abundant. The epidote porphyroclast proportion increases from 25 to 77% with increasing magnification in the sandstone dike (table 4.2). If the epidote is from the sandstone in the basin, you must disintegrate the sandstone and enrich the epidote relatively to 182


4.6. Discussion

quartz and transport it into the dike. Considering the high amount of epidote in the sandstone dike this is highly unlikely. Chlorite appears to be related to the faulting event since it increases in the wall-rock towards the fault zone. Sericite is also found in the fracture material in both samples. These observations suggest that in the fault gouge and sandstone dike samples there are two minerals that form during and/or after deformation. Both these minerals are hydrous and indicate fluid transport in the fault gouge and sandstone dike. The role of fluids is an important aspect which is often not considered in fault analysis and in experiments. The lack of irregular epidote grains in the sandstone dike might indicate that the deformation has not been a series of sequential faulting. Another aspect is the stress state of the system. It is possible that the difference in distance to the basin margin affects the stress state in such a way that there is a switch in deformation mechanism (i.e abrasion vs. fracturing along cleavage planes). K-feldspar shape and size The K-feldspar phase in fault gouge sample BRE1505-2 and sandstone dike BRE1305A1 have similar D-values (∼ 3.0) (tables 4.1 and 4.2). However, the upper particle size is limited to ∼ 65 µm for the sandstone dike sample, as opposed to ∼ 200 µm in the fault gouge sample(figure 4.60). As mentioned above, K-feldspar is a much more abundant in the fault gouge sample (13 - 40 % porphyroclasts) compared to the sandstone dike sample (6 - 22 % porphyroclasts). It is important to note that, K-feldspar and plagioclase & quartz require more processing than epidote to identify and separate the individual particles. Which has little affect on the particle size, however it may affect the shape determination. K-feldspar is more abundant in the fracture material than in the wall-rock, especially in fault gouge sample BRE1505-2, in terms of bulk mineralogy (porphyroclasts and matrix). This, in addition with the mesh texture might indicate that the majority of the K-feldspar in the fault gouge sample, and to some extent in the sandstone dike sample have been formed from fluids in the fractures and that the size and shape distributions may be somewhat biased by this complex texture. This is also apparent in fault gouge sample BRE1505-2. Figures 4.13 and 4.17 demonstrate that K-feldspar is mainly present as fracture material and is scarce in the wall-rock of the fault gouge sample. However, in fault gouge sample BRE1505-2 the size and shape characteristics compare well to that of plagioclase & quartz. Plagioclase & quartz do not show any signs a coupled reaction-deformation process, at least not to any great extent. This could be by coincidence, or it might indicate that the new -or on-growth of K-feldspar 183

4.6. Discussion


have had little affect on the shape and size characteristics. Elongation data The elongation distribution of both samples compare. They have similar mid-value mean values (0.389 and 0.449, respectively), low skewness and kurtosis (0.07 and 2.30, and 0.09, and 2.14, respectively) (figure 4.56). In fault gouge sample BRE15052, the large K-feldspar grains in fault gouge sample BRE1505-2 are semi-elongated, whereas the small K-feldspar grains both have elongated and non-elongated shapes (∼ 0.05 < ε N( εi))/ Σ1ε=0N

(N( ε) > N( εi))/ Σ1ε=0N

0.7

0.5 0.4 0.3

0.6 0.5 0.4 0.3

0.2 0.2 0.1 0.1 0 0

0.1

0.2

0.3

Epidote

0.4 ε K−feldspar

0.5

0.6

0.7

0.8

0 0

0.1

0.2

(a) All phases combined.

0.4 ε

0.5

0.6

0.7

0.8

0.6

0.7

0.8

(b) Epidote.

All magnifications

All magnifications

1

1

0.9

0.9

0.8

0.8

0.7

0.7 (N( ε) > N( εi))/ Σ1ε=0N

(N( ε) > N( εi))/ Σ1ε=0N

0.3


0.6 0.5 0.4

0.6 0.5 0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0 0

0.1

0.2

0.3

0.4 ε

0.5

(c) K-feldspar.

0.6

0.7

0.8

0

0.1

0.2

0.3

0.4 ε

0.5

(d) Plagioclase & quartz.

Figure 4.68: Accumulative distribution of grains with elongation ε larger than εi normalized against number of particles. We observe that in both phases for all phases approximately 50% and 20% of the grains have elongation larger than 0.4 and 0.5, respectively. This corresponds to aspect ratio of 1.7 and 2.0.

201

4.6. Discussion


Sandstone dike sample BRE1305A1 All magnifications 1

All magnifications 1

0.9 0.9 0.8 0.8 0.7 0.6

(N( ε) > N( εi))/ Σ1ε=0N

(N( ε) > N( εi))/ Σ1ε=0N

0.7

0.5 0.4 0.3

0.6 0.5 0.4 0.3

0.2 0.2 0.1 0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9 0

ε Epidote

0

K−feldspar

0.1

0.2

0.3

0.5

0.6

0.7

0.8

0.9

0.6

0.7

0.8

0.9

ε

(e) All phases combined.

(f) Epidote.

All magnifications

All magnifications

1

1

0.9

0.9

0.8

0.8

0.7

0.7 (N( ε) > N( εi))/ Σ1ε=0N

(N( ε) > N( εi))/ Σ1ε=0N

0.4


0.6 0.5 0.4

0.6 0.5 0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0

0.1

0.2

0.3

0.4

ε

(g) K-feldspar.

0.5 ε

(h) Plagioclase & quartz.

Figure 4.68: Accumulative distribution of grains with elongation ε larger than εi normalized against number of particles. We observe that in both phases for all phases approximately 50% and 20% of the grains have elongation larger than 0.4 and 0.5, respectively. This corresponds to aspect ratio of 1.7 and 2.0.

202


4.7

4.7. Conclusion

Conclusion

This chapter has documented qualitative and quantitative petrography and grain size and shape were analyzed. Fault gouge sample BRE1505-2 has large D-values indicative of shear fracturing. In the fault gouge sample the convexity and circularity of both epidote and K-feldspar show the same relationship as plagioclase & quartz and become smaller with increasing grain size (figures 4.59 - 4.61). This indicates that the large irregular and non-spherical grains of all three phases have been abraded into smaller smooth and spherical grains. This probably shows that any mineral growth has had only minor effect on K-feldspar in this sample. However, the epidote phase has a fraction of small irregular and non-spherical grains. I suggest this is related to the dynamics of the coupled reactiondeformation process. The shape and texture of epidote indicate repeated faulting events (figures 4.8 and 4.17). In summary, the data suggest that the material in the fault gouge is result of in situ granulation that happened during several repeated faulting events. In sandstone dike sample BRE1305A1 the size and shape characteristic of the particles appear to have a mixed signature. Large D-values that are comparable to those for the fault gouge sample, but show a deficit in large “survivor grains”. Transport of granulated material from fractures in the wall-rock or fractures terminating at the interface of the sandstone dike to the sandstone dike, may explain why relatively large survivor grains are missing. The epidote texture, shape (figures 4.42) and the high contents (table 4.2) suggest that it has been formed in situ. However the fluid infiltration from the basin have buffered the iron content during and after granulation. However, we do at the same time observe flow structures in the field and also in the thin section indicative of infill of basin material (figures 4.33 and 4.35). Also, we observe a interface between of a relatively thick zone of granulated material in connection with a lens of fine grained material with flow structures (figure 4.34). The field observation and textures we observe combined with grain size and shape characteristics indicates that this is most likely a result of a mixed process. Hence i suggest that the sandstone dike material is partly derived from granulation in connecting fractures that has been transported to the pull-apart fracture and partly of infill from basin material. We note that this process is complex and most likely happened during several stages and we do not understand the dynamics involved completely. The results also show that the fracture material is not spherical and it should 203

4.7. Conclusion


studied wether this affects the D-value obtained from a 2D-section since its assumed that a spherical geometry adequately describes the geometry. The results also show that phases recognition is important. Not only because of the complex and the poorly understood dynamics of coupled reaction-deformation processes and how this affects size and shape distributions. The size and shape distributions of the least abundant phase gets masked or the combined signal for all the phases will potentially not be representative for the phase behaviour. Also, we see that if only the bulk size and shape distribution were to be compared no difference between the samples would observed (figures 4.56 - 4.58). The results demonstrate the importance of incorporating both field and textural observation into the analysis and the advantage of combining grain size and shape analysis.

204

Chapter 5

Intrusion of material between rigid plates 5.1

Introduction

Near the vicinity of the basin margin there are fractures in the granodiorite filled with sandstone material. The majority of these fractures are oriented perpendicular to the margin and the orientation is comparable to fractures found elsewhere in the granodiorite and in the basin itself (see Chapter 2 Fieldwork on page 7). Furthermore, the fracture orientations are comparable to that which has been reported from faults found in the basin. These faults have been interpreted to have been active during the basin development (extension) (see Chapter 1 Geological Setting on page 3). In the previous chapter the petrography and size -and shape distribution of the fracture material was presented and discussed. For the sandstone dike located near the basin margin (∼ 20 m away) there were found mixed size and shape characteristics indicating that there was both granulation of wall-rock material and infill from the basin. Also presented was field evidence of granulation and faulting, and sedimentary flow structures in the sandstone dike near the basin margin. However, some places adjacent to the basin margin the fracturing is so extensive that a marginal breccia with no evidence of shear displacement have been formed (See 2.2.1 on page 23). The tip of the fractures oriented perpendicular to the basin margin extend to the basin margin, thus allowing for significant infill of basin material. To study the controlling factors on such processes I have used a numerical modeling approach. By using a simple setup with two rigid boudins separated by a gap (fracture) I have investigated how the pressure, velocity and infill flux of the gap material (fracture filling) is affected by different gap 205

5.2. Model setup

Chapter 5. Intrusion between rigid plates

widths and material properties.

5.2

Model setup

The analytical solution for linear flow between rigid plates have been described by Nadai (1963), Jaeger (1969), Dieter (1986), and others. Following Jaeger (1969), a brief outline of the analytical solution is given below. The symbols used are as follows Normal stress σij Shear stress

τij

Normal strain rate

ε˙ij

Shear strain rate

γ˙ij

Body forces

X and Y

Horizontal velocity

U

Vertical velocity

V

Spatial direction

x and y

Viscosity

µ

5.2.1

Incompressible flow in two dimensions

For incompressible fluids in two dimensions the equation of continuity reads:

εxx ˙ + εyy ˙ =

∂U ∂V + =0 ∂x ∂y

(5.1)

The constitutive relationship between the stress and the displacement velocities is given by

σxx = 2µε˙xx − P

(5.2)

σyy = 2µε˙yy − P µ ¶ ∂U ∂V τyx = µγ˙ = µ + ∂y ∂x

(5.3)

The force balance equations can be expressed in the terms of velocities: 206

(5.4)

Chapter 5. Intrusion between rigid plates ∂U ∂t ∂U ρ ∂t ∂U ρ ∂t ∂U ρ ∂t ∂U ρ ∂t

ρ

5.2. Model setup

∂σxx ∂τyx + + ρX ∂x ∂y ∂ ε˙xx ∂P ∂ γ˙ = 2µ − +µ +X ∂x ∂x ∂y µ ¶ ∂2U ∂P ∂ ∂U ∂V = 2µ 2 − +µ + +X ∂x ∂x ∂y ∂y ∂x ∂2U ∂P ∂2U ∂ ∂V = 2µ 2 − +µ 2 +µ +X ∂x ∂x ∂y ∂y ∂x ∂P ∂2U ∂ ∂V = µ∇2 U + µ 2 − +µ +X ∂x ∂x ∂y ∂x =

(5.5) (5.6) (5.7) (5.8) (5.9)

From the continuity equation (equation 5.1) we obtain ∂U ∂V + =0 ∂x ∂y ∂2U ∂ ∂V =0 ⇒ + 2 ∂x ∂x ∂y ∂2U ∂ ∂V =− 2 ∂x ∂x ∂y

(5.10) (5.11) (5.12)

Since U must be path independent Z Z Z Z ∂ ∂U ∂ ∂U U= dxdy = dydx ∂x ∂y ∂y ∂x

(5.13)

It follows that the integrands must be equal ∂ ∂U ∂ ∂U = ∂x ∂y ∂y ∂x Hence

∂2U ∂x2

(5.14)

∂ ∂V = − ∂y ∂x can be substituted into equation 5.9 and we get

µ ¶ ∂ ∂V ∂P ∂ ∂V ∂U 2 = µ∇ U + µ − − +µ +X ρ ∂t ∂x ∂y ∂x ∂y ∂x ∂U ∂P ρ = µ∇2 U − +X ∂t ∂x

(5.15) (5.16)

and similarly in the y-direction: ρ

∂V ∂P = µ∇2 V − +Y ∂t ∂y

Assuming that the motion is so slow that the accelerations

(5.17) ∂U ∂t

and ∂V ∂t are negligible,

equations 5.16 and 5.17 are reduced to 1 ∂P ρgx + =0 µ ∂x µ ρgy 1 ∂P ∇2 V − + =0 µ ∂y µ

∇2 U −

207

(5.18) (5.19)

5.2. Model setup


Differentiation of equations 5.18 and 5.19, with respect to x and y, respectively, and summing yields µ ¶ µ ¶ ρgy ∂ 1 ∂P ρgx ∂ 1 ∂P 2 2 ∇ U− + + ∇ V − + =0 ∂x µ ∂x µ ∂y µ ∂y µ ∂ ∂2U ∂ ∂2U 1 ∂2P ∂ ∂2V ∂ ∂2V 1 ∂2P + − + + − =0 ∂x ∂x2 ∂x ∂y 2 µ ∂x2 ∂y ∂x2 ∂y ∂y 2 µ ∂y 2

(5.20) (5.21)

Using the result from equations 5.10 - 5.14 the last expression reduces to −

5.2.2

1 ∂2P ∂ ∂2V ∂ ∂U ∂ ∂V ∂ ∂2U 1 ∂2P + − + − − =0 ∂x2 ∂y ∂x ∂y 2 µ ∂x2 ∂y ∂x2 ∂y 2 ∂x µ ∂y 2 ∂2P ∂2P + =0 ∂x2 ∂y 2

(5.22) (5.23)

Intrusion between rigid plates

The equations obtained in the previous section is now used for the specific problem. The numerical setup is presented in figure 5.1. We assume an incompressible fluid with viscosity µ between to parallel rigid plates of height 2h (figure 5.1(b)). The plates are pulled apart by two equal and oppositely directed forces parallel to the x-axis. We assume there is no slip at the interface and equations 5.1, 5.18, 5.19, and 5.23 are satisfied with the following boundary conditions: ) U = ∓U0 − h < y < h, when x = ±w V =0

(5.24)

where U and V are the flow velocity components of the gap material, U0 is the velocity of the plates, and w is the half width of the gap. In addition because of symmetry, V must be an odd function of y so that V (−y) = −V (y) and we seek a solution with ∂2V ∂y 2

= 0. A second degree polynomial 1 1 P = kx2 − ky 2 + C 2 2

(5.25)

where k and C are constants, satisfies equation 5.23. Hence equation 5.19 becomes 1 ∂P =0 µ ∂y ∂2V 1 ∂P + − =0 2 ∂y µ ∂y 1 ∂P = µ ∂y ky =− µ

∇2 V − ∂2V ∂x2 ∂2V ∂x2 ∂2V ∂x2

208

(5.26) (5.27) (5.28) (5.29)


5.2. Model setup

The solution of equation 5.29 is V =−

ky 2 x + f (y)x + g(y) 2µ

(5.30)

where f (y) and g(y) are unknown functions of y. The boundary conditions (equation 5.24) require that vertical velocity component is zero at the left and right interface V =−

ky 2 w + f (y)w + g(y) = 0 2µ

Which yields f (y) = 0 and g(y) =

ky 2 2µ w

V =

(5.31)

and equation 5.30 becomes

ky 2 (w − x2 ) 2µ

Inserting equation 5.32 into the continuity equation (equation 5.1) yields µ ¶ ∂U ∂ ky 2 k 2 =− (w − x ) = − (w2 − x2 ) ∂x ∂x 2µ 2µ Z kx k (w2 − x2 )dx = − (3w2 − x2 ) + F (y) U =− 2µ 6µ

(5.32)

(5.33) (5.34)

where F (y) is an unknown function of y. The first boundary condition (equation 5.24) gives F (y) = 0 and kw (3w2 − w2 ) 6µ 3U0 µ ⇒k= w3 Hence U , V and P (equations 5.34, 5.32, and 5.25, respectively) are U0 = −

U0 3 (x − 3w2 x) 2w3 3U0 2 V (x, y) = (w y − x2 y) 2w3 3µU0 2 P (x, y) = (x − y 2 ) + C 2w3 Equations 5.37 - 5.42 satisfy the boundary conditions (equation 5.24), but the U (x)

=

(5.35) (5.36)

(5.37) (5.38) (5.39) height

of the gap is not a parameter. We can assume that P = 0 at the corners of the gap (x = ±w and y = ±h). This gives in equation 5.42 3µU0 2 (w − h2 ) + C = 0 2w3 3µU0 2 ⇒C=− (w − h2 ) 2w3 and we obtain for the pressure P (w, h) =

P (x, y) =

3µU0 2 (x − y 2 − w2 + h2 ) 2w3 209

(5.40) (5.41)

(5.42)

5.2. Model setup

5.2.3


Characterization of analytical solution for spreading plates

In the analytical solution above the plates are approaching eachother. Since I am iteredtes int spreading of the plates, the equations are redefined so that U0 is positive for extension. Hence equations 5.37, 5.38, and 5.42 above become U0 (−x3 + 3w2 x) 2w3 3U0 V (x, y) = (−w2 y + x2 y) 2w3 3µU0 P (x, y) = (−x2 + y 2 + w2 − h2 ) 2w3

U (x)

=

(5.43) (5.44) (5.45)

The equations for velocities and pressure will now be evaluated. The horizontal velocity component U (equation 5.43) is a function of the coordinate x and half gap width w. It does not depend on the of vertical coordinate y (equation 5.43, figure 5.2(a)). The horizontal velocity component is a third order polynomial. However, for small gap ratios (h/w) ≤ 2.0 the first term in equation 5.43 is very small compared to the second term and the horizontal velocity is proportional with x (figure 5.2(b)). The vertical velocity component V (equation 5.44) is a function of x and ycoordinates and the half gap width w. V is symmetric around x = 0 figure 5.2(c). The magnitude of the vertical velocity component increases towards the top and bottom boundaries of the gap. V has a power-law relationship with x near the top and bottom boundaries and plots as a parabola curve with V (x, ±h) = V (−x, ±h) and a minimum/maximum value at x = 0 (figure 5.2(d)). Horizontally through the gap the vertical velocity is zero. It has a linear relationship with respect to y and the vertical velocity component is as mentioned above an odd function with respect to y (V (−y) = −V (y)) and is proportional with 1/y for x = 0 (figure 5.2(d)). At the right and left vertical boundaries (x = ±w) the vertical velocity is zero. The pressure P (equation 5.45) is a function of x and y-coordinates and the half gap width w. The pressure has a power-law relationship with the x and y-coordinates (equation 5.45). However, for small gap ratios the first term in equation 5.45 is very small compared to the second term. Hence the pressure variation is only minute with respect to the x-coordinate (figure 5.2(f)). The pressure varies quadratically with respect to the y-coordinate and plots as a parabola P (y) = P (−y)) with a minimum value of zero for y = 0 (figure 5.2(e) and (f)). The velocity field does not change their x and y dependence with increasing gap ratio (figure 5.3(a) - (h)). The velocities increase, but the relationship with the x -and 210


5.2. Model setup

y coordinates is the same. In contrast the above, the pressure the isobar changes from being horizontal to vertical with increasing gap ratio (figure 5.3(i) - (l)).

211

5.2. Model setup


b

0. 5

b’ a

0. 4 0. 3 0. 2 0. 1

2h

0 −0. 1 −0. 2 −0. 3 −0. 4 a’

−0. 5

2w −0. 2

0

0. 2

(a) Setup of rigid plates extrusion. Insert of

(b) h is the half height of the gap and w is the half

close up of close-up of gap region.

width. a − a0 is the center gap profile and b − b0 is the boundary flow profile.

Figure 5.1: Numerical setup of pure shear intrusion between rigid plates. The boudins are separated by gaps with aspect ratios (h/w) between 0.01 and 2.0. The boudins have aspect ratio 5 and the top and bottom boundaries are far away to avoid boundary effects. Pure shear velocities are specified at the sides and the bottom to obtain an extensional pure shear deformation with εxx = 0.5. The top boundary is a free surface which gives us zero normal pressure and the possibility to specify pressure. The boudin phase is blue (non texture), the gap matrix yellow, and the ambient matrix is red. (b) is a close-up of the gap region in (a).

212


5.2. Model setup

y −0.05 1

0

0.05 U(x,−0.05) U(x,0) U(−0.05,y) U(0,y)

0.8 0.6

0

−0.2

0.4 −0.4

−0.6 0

U(y) Anl

U(x) Anl

0.2

−0.8

−0.2 −0.4

−1 −0.6 −1.2 −0.8 −1 −0.05

(a) Horizontal velocity component contour.

0 x

0.05

(b) Horizontal velocity component profiles. y −0.5 15

0

0.5 15 V(x,−0.05) V(x,0) V(−0.05,y) V(0,y)

10

10

0

5

V(y) Anl

V(x) Anl

5

−5

−10

0 −0.05

(c) Vertical velocity component contour.

−15 0.05

0 x

(d) Vertical velocity component profiles y −0.5 500

0

0.5 0 P(x,−0.05) P(x,0) P(−0.05,y) P(0,y)

0

−500

−500 −1000

−1500

−1500

P(y) Anl

P(x) Anl

−1000

−2000 −2000 −2500 −2500 −3000 −3500 −0.05

(e) Pressure contour.

0 x

0.05

(f) Pressure profiles.

Figure 5.2: Evaluation of U, V and P (equations 5.44 -5.45) with respect to spatial directions. µ = 1, 2l = 1.0, 2w = 0.10 and U0 = −1.0 for all plots. The plots in (b), (d), and (f) are profiles along the top and left boundaries, and horizontal and vertical profiles though the center of the gap.

213

5.3. Results

Chapter 5. Intrusion between rigid plates Horizontal velocity U

(a) Gap ratio 0.1.

(b) Gap ratio 0.5.

(c) Gap ratio 1.0.

(d) Gap ratio 2.0.

Vertical velocity V

(e) Gap ratio 0.1

(f) Gap ratio 0.5.

(g) Gap ratio 1.0.

(h) Gap ratio 2.0.

Pressure P

(i) Gap ratio 0.1.

(j) Gap ratio 0.5.

(k) Gap ratio 1.0.

(l) Gap ratio 2.0.

Figure 5.3: Evaluation of U, V and P (equations 5.44 -5.45) with with respect to gap ratio. µ = 1, 2l = 1, and U0 = −1 for all plots.

5.3 5.3.1

Results Introduction

The numerical model used is a FEM code as described in Section 3.3 Numerical modeling on page 44. Several systematic runs where performed to study the validity of the analytical solution and how it compares to the numerical solution. We also wanted to extend from the analytical view of plates with a infinite lengths to plates with finite lengths and also study how it compares to non-linear viscous flow since many geological material can be described by such a rheology. In the numerical setup two boudins of height 2h = 0.98 with aspect ratio equal 214


5.3. Results

to 5 separated by a gap of width 2w were used (figure 5.1). The gap ratio (gap width/bodudin height = h/w) was varied systematically between 0.01 and 2.0. The matrix viscosity µmatrix was kept constant at 1, while power-law coefficient nmatrix was varied between 1, 3 and 10. The viscosity and power-law coefficient of the boudins were kept at constant at µboudin = 10e5 and nboudin = 1, respectively for all runs. The box was deformed during pure shear horizontal extension. The boundaries are geometrically placed far from the gap to avoid any boundary effects. Pure shear velocities was specified at the sides and the bottom to obtain an extensional pure shear deformation with εxx = 0.5. The top boundary is a free surface which gives us zero normal pressure and the possibility to specify pressure. Since we are only interested in the initial deformation stage only one time step was calculated. To assure accurate results convergence testing was performed with respect to number of elements in the gap, the convergence of power-law iterations, and compressibility. The results of the numerical simulation will now be compared to the derived analytical solution. The following parameters where compared to the analytical solution : The pressure deviation ∆P which is normalized against the maximum pressure value for the analytical solution (equation 5.49). A vertical pressure profile a-a’ through the center of gap is also measured and compared (figure 5.1(b)). The numerical pressure adjusted to coincide with the analytical pressure in the center of the gap to allow comparison. The vertical component of the flow velocity into the gap Vynorm (equation 5.50) a cross the profile b − b0 (figure 5.1(b)) was compared. The vertical component of the flow velocity is normalized against the calculated vertical background velocity V˜y . If we look at one quarter of the gap we see that when the gap wall is displaced during a time step this crates an area A1 (figure 5.4). Because of mass balance this area must be balanced by material flowing into the gap, area A2 . Since the boudins are relatively rigid this volume can be calculated by using the far-field strain rate. The area is then simply A1 = wε˙xx ∆th

(5.46)

where ε˙xx is the far-field strain rate. This must balance against A2 : A2 = V˜y ∆tw

(5.47)

When ∆t is small the deviation of this rectangular approximation is negligible and eqautions of A1 and A2 yields V˜y = ε˙xx h 215

(5.48)

5.3. Results


The deviation of the vertical flow velocity component into the gap ∆Vy is normalized against the analytical solution (equation 5.50). And the total flow velocity deviation of the gap material ∆V tot which is defined by equation 5.52. |P num − P anl | · 100% anl Pmax

∆P =

num Vy,norm =

∆Vy =

∆V

=

Vynum V˜y

(5.50)

|V num − V anl | · 100% V anl

q tot

(5.49)

2

(5.51)

2

(U num − U anl ) + (V num − V anl ) q · 100% anl 2 + V anl 2 Umax max

Vy∆t

A2

A1

h

y

ε

w

w

xx

∆t

x

Figure 5.4: Upper right quarter of the gap. Lateral displacement of the gap wall creates an Area A1 = wε˙xx ∆th. Because of mass balance this area must be balanced by material flowing into the gap with area A2 = V˜y ∆tw. This two areas must be equal and which yields V˜y = ε˙xx h.

216

(5.52)


5.3.2

5.3. Results

Pressure and pressure deviation in gap material

Vertical centerline profile a − a0 The pressure and pressure deviation across the vertical centerline profile a − a0 (figure 5.1(b)) of the gap are plotted. The pressure of the Newtonian fluid has a power-law relationship with y and generally plots as parabolas with a minimum value of 0 at y = 0 for all gap ratios (figure 5.5(a) and (b)). The pressure curve for gap ratios 0.01 and 0.02 cross with increasing vertical distance from the gap center. This is also observed for the analytical solution (figure 5.5(c) and (d)). The power-law fluids pressure profiles show the same symmetry, but have a more linear relationship with y (figure 5.5(e) - (h)). However, they have a parabolic section limited to a narrow area around y = 0. The parabolic section of the profile becomes generally larger with increasing gap ratio, especially for n = 3. The vertical pressure gradient becomes smaller with increasing gap ratio and power-law coefficient. In contrast to gap ratios ≤ 1.0, we the pressure increases towards the center of the gap gap ratio 2.0. For the Newtonian fluid the maximum pressure at the top/bottom boundary drops 4 orders of magnitude with increasing gap ratio (figure 5.5(a). For n = 3 and n = 10 the maximum pressure drop two and one magnitudes, respectively, with increasing gap ratio (figure 5.5(e) - (h)). For small gap ratios the maximum pressure for the power-law fluids are generally 2 - 3 magnitudes less than that for the Newtonian fluid. While at large gap ratios the maximum pressures are of the same magnitude. For the Newtonian fluid the pressure deviation a cross the profile a − a0 is generally less than 15% for all gap ratios (figure 5.6(a)). The pressure deviation and the vertical pressure deviation gradient generally increase with increasing gap ratio. However, the pressure deviation for gap ratio 0.01 deviates from this trend and has generally large pressure deviation towards the top/bottom boundaries, except near the gap center (−h/2 < y < h/2). Also we note that gap ratio pairs 0.05 and 0.01, and 1.0 and 2.0 have approximately the same pressure deviation for all values of y (figure 5.6(b)). However, this is also observed for the off-sequence pair gap ratio 0.02 and 0.5. For the power-law fluid the pressure deviation is generally large and plots as parabolas with a minimum at y = 0 (figures ?? and ??). The pressure deviation increases slightly with increasing power-law coefficient, but is generally comparable for both power-law fluids at all gap ratios. In contrast to the Newtonian fluid, the vertical pressure deviation gradient generally decreases with increasing gap ratio for the power-law fluids (figures ?? and ??). However, for gap ratio 2.0 the pressure deviation is largest. We also note that for gap ratio 2.0 the maximum pressure deviation is larger than 217

5.3. Results


100%, while for gap ratio smaller or equal to 1.0 it is less or equal to 100%. Similar to the Newtonian fluid, we observe and opposite is pressure profiles for gap ratio 2.0; the maximum pressure is found at the center of the gap for the power-law fluids and not at the top/bottom boundaries. Hence the large pressure deviations.

0.5

0.5 0.01 0.02 0.05 0.1 0.5 1.0 2.0

0.25

0

y

Gap Height, h

0.25

−0.25

0

−0.25

−0.5 0

0.5

1

1.5

2

−0.5

2.5

P

x 10

0

10

5

(a) Newtonian fluid.

20

30

P

(b) Close-up of center region around y = 0 in (a).

0.5

0.5 0.01, N = 1 0.02 0.05 0.1 0.5 1.0 2.0

0.25

0

y

y

0.25

−0.25

0

−0.25

−0.5 0

0.5

1

1.5

2

P

2.5 x 10

(c) Analytical solution for Newtonian fluid.

5

−0.5 0

10

20

30

P

(d) Close-up of center region around y = 0 in (c).

Figure 5.5: Vertical pressure profile a − a0 through the center of the gap as indicated in figure 5.1(b).

218


5.3. Results

0.5

0.5 0.01 0.02 0.05 0.1 0.5 1.0 2.0

0.25

0.4 0.3 0.2

y

y

0.1 0

0 −0.1 −0.2

−0.25

−0.3 −0.4 −0.5

−0.5 0

500

1000

1500

2000

2500

3000

3500

4000

0

4500

5

10

P

P

(e) Power-law fluid n = 3.

(f) Close-up of center region around y = 0 in (e). 0.5

0.5 0.01 0.02 0.05 0.1 0.5 1.0 2.0

0.25

0.4 0.3 0.2

0

y

y

0.1 0 −0.1 −0.2 −0.25

−0.3 −0.4 −0.5 0

50

100

150

200

250

300

−0.5 0

5

10

P

P

(g) Power-law fluid n = 10.

(h) Close-up of center region around y = 0 in (g).

Figure 5.5: (Cont.) Vertical pressure profile a − a0 through the center of the gap as indicated in figure 5.1(b).

219

5.3. Results


0.5

0.5 0.01 0.02 0.05 0.1 0.5 1.0 2.0

0

0.25

0

y

y

0.25

−0.25

−0.25

−0.5

−0.5 0

10

20

30

40

50

0

60

1 ∆P

∆P


2

(b) Close-up of center region around y = 0 in (a). 0.25

0.5 0.01 0.02 0.05 0.1 0.5 1.0 2.0

0.25

0.01

y

y

1.0 0

2.0

−0.25

0

−0.5 0

20

40

60 ∆P

80

100

0

120

(c) Power-law fluid n = 3.

5 ∆P

10

(d) Close-up of center region around y = 0 in (c). 0.25

0.5 0.01 0.02 0.05 0.1 0.5 1.0 2.0

0.25

y

y

1.0 0

0.01 2.0 −0.25

0

−0.5 0

20

40

60

80

100

∆P

(e) Power-law fluid n = 10.

120

140

0

5 ∆P

10

(f) Close-up of center region around y = 0 in (e).

Figure 5.6: Vertical pressure deviation profile a − a0 through the center of the gap as indicated in figure 5.1(b).

220


5.3. Results

Pressure deviation contours of gap material In figures 5.7 - 5.9 the pressure deviation of the gap matrix is contoured. For gap ratios ≤ 0.1 we observe that for all fluids the pressure deviation has a minimum value and is symmetric around y = 0 as descried above (figures 5.7(a) -(d), 5.8(a) -(d), and 5.9(a) -(d)). The pressure deviation does not generally change laterally in the gap for these gap ratios;

d∆P dx

≈ 0.

For gap ratios ≥ 0.5 the pressure deviation is also affected by the lateral xcoordinate (figures 5.7(e) -(g), 5.8(e) -(g), and 5.9(e) -(g)). This effect increases with increasing gap ratio and is most predominant for the power-law fluids. Also, boundary effects at the corners become more pronounced with increasing gap ratio. Although the maximum pressure deviation generally increases with increasing gap ratio, the pressure deviation for the Newtonian fluid is generally small for the majority of the gap at gap ratios ≥ 0.5 (figure 5.7(e) - (g)). Approximately 90% of the gap has pressure deviation ≤ 10% at gap ratios 0.5 and 1.0. For gap ratio 2.0 still 50% of the gap has pressure deviation ≤ 20%. The pressure deviation of both power-law fluids are comparable for all gap ratios (figure 5.8(a) - (g) and 5.9(a) - (g)). For the power-law fluids the pressure deviation for gap ratios ≥ 0.5 is characterized by relatively large deviations at the top and bottom boundaries and small deviations at the left and right boundaries (figure 5.8(e) - (g) and 5.9(e) - (g)). Diagonally from corner to corner there is a region of intermediate deviation. Although the pressure deviation increases with increasing gap ratio approximately 80% of the gap has pressure deviation ≤ 50% at gap ratios 0.5 and 1.0. For gap ratio 2.0 the pressure deviation is ≤ 50% for 50% of the gap.

221

5.3. Results

(a) Gap ratio 0.01.

(e) Gap ratio 0.5.


(b) Gap ratio 0.02. (c) Gap ratio 0.05.

(d) Gap ratio 0.10.

(f) Gap ratio 1.0.

(g) Gap ratio 2.0.

Figure 5.7: Newtonian fluid. Pressure deviation contours of the gap matrix (figure 5.1(b)) for gap ratios between 0.01 and 2.0. (a) - (d) not to scale.

222


(a) Gap ratio 0.01.

(e) Gap ratio 0.5.

(b) Gap ratio 0.02.

5.3. Results

(c) Gap ratio 0.05. (d) Gap ratio 0.10.

(f) Gap ratio 1.0.

(g) Gap ratio 2.0.

Figure 5.8: Power-law fluid n = 3. Pressure deviation contours of the gap matrix (figure 5.1(b)) for gap ratios between 0.01 and 2.0. (a) - (d) not to scale.

223

5.3. Results

(a) Gap ratio 0.01.


(b) Gap

ratio (c) Gap ratio 0.05.

(d) Gap ratio 0.10.

0.02.

(e) Gap ratio 0.5.

(f) Gap ratio 1.0.

(g) Gap ratio 2.0.

Figure 5.9: Power-law fluid n = 10. Pressure deviation contours of the gap matrix (figure 5.1(b)) for gap ratios between 0.01 and 2.0. (a) - (d) not to scale.

224


5.3.3

5.3. Results

Velocities and velocity deviation in gap material

Fluid flow across profile b − b0 num In figures 5.10 and 5.12 the the vertical velocity component Vy,norm (equation 5.50)

of the fluid flow across the b − b0 (figure 5.1(b)) into the gap is plotted and compared to the analytical solution (∆Vy , equation 5.51). The vertical velocity component has a parabolic shape with maximum velocity at the center of the gap and zero at the boundaries (figure 5.10). The the newtonian fluid has a smooth parabolic shape which is comparable to the analytical solution, whereas the power-law fluids have a “plugflow” shape at small gap ratios which becomes more pronounced with increasing powerlaw coefficient. The “plug-flow” effect is strong for gap ratios up to 1.0, but for gap ratio 2.0 the “plug-flow” has diminished (figure 5.10(g)). For the Newtonian fluid the vertical flow component increases with increasing gap ratio to a maximum at gap ratio 0.02, before it decreases with further increase in gap ratio (figure 5.10). While for the power-law fluid the vertical flow component decreases with increasing gap ratio. For small gap ratios (≤ 0.02) the vertical flow velocity component of the Newtonian fluid is much smaller than for the power-law fluids. (figure 5.10(a) and (b)). However, for ratio 0.05 the maximum velocities compare for all fluids (figure 5.10(c)) and for relatively large gap ratios (0.10 ≤ h/w ≤ 1.0) the maximum velocity of the Newtonian fluid is larger than for the power-law fluids (figure 5.10(d) and (f)). For gap ratio 2.0 the vertical flow velocity component for all fluids compare (figure. 5.10(e)). Also, the relationship between the the vertical flow velocity components of the power-law fluids remain approximately constant. The maximum vertical flow component is always larger for n = 3, except at gap ratio 2.0. We also note that the vertical velocities of the analytical solution for a Newtonian fluid calculated from the numerical velocities compare for large gap ratios, but not for gap ratios < 0.05 (figure 5.10). As mentioned above the vertical flow velocity generally decreases with increasing gap ratio, When the maximum vertical flow velocity is loglog-plotted against gap ratios it generally plots as a straight line (figure 5.11). The Newtonian fluid can be fitted to a straight line with slope equal to 0.9647 for gap ratios ≥ 0.05. The power law fluids with power-law coefficients 3 and 10 can be fitted to lines with slopes of 0.9876 and 0.9795, respectively over for gap ratios between 0.01 and 1.0. All the slope values compare to values calculated from the velocities of the gap material by using the analytical solution for a Newtonian fluid (0.9110, 0.9797, and 0.9809, respectively). The maximum velocity is over the given interval proportional to

225

5.3. Results


max log(Vy,norm ) = −a log(2w) + log(c)

(5.53)

max Vy,norm = (2w)−a c 1 max ⇒ Vy,norm ∝ (2w)a

(5.54) (5.55)

where a is the slope of the line. However, it is important to note that calling this a true power-law relationship is not strictly true since it only scales over one magnitude. Generally the maximum vertical flow velocity deviation is zero for all fluids at a distance from the gap center approximately equal to one quarter of the gap width (x ≈ ± 21 w) (figure 5.12). These points correspond to points where the vertical flow velocities of the analytical and numerical solution are equal (figure 5.10). However, for gap ratios 0.01 and 0.02 the vertical flow velocity deviation for the Newtonian fluid is parabola shaped across profile b − b0 with a maximum at the center of the gap (x = 0) and (x ≈ ± 12 w), respectively (figure 5.12(a) and (b)). For the Newtonian fluid the vertical flow velocity deviation is very large for the smallest gap ratio (0.01). As mentioned above it plots as parabola with a maximum at the center of the gap (figure 5.12(a)). However, for gap ratio 0.02 the vertical flow velocity deviation is less than approximately 10% for the entire gap width (figure 5.12(b)). For intermediate gap ratios (0.05 ≤ w/h ≤ 0.10) the vertical flow velocity deviation of the Newtonian fluid is even smaller at approximately 5% (figure 5.12(c) and (d)). With further increase of gap ratio the maximum vertical flow velocity deviation increases (figure 5.12(e) and (f)). However, at gap ratio 2.0 the maximum vertical flow velocity deviation is still less than 25% for the Newtonian fluid (figure 5.12(g)). The vertical flow velocity deviation is generally larger for the power-law fluids, except at gap ratio 0.01 (figure 5.12(a)). For the power law fluids with power-law coefficient 3 and 10 the maximum vertical flow velocity deviation oscillates between 20 and 30%, and 40 and 45%, respectively (figure 5.12(a) - (g)). However, we do note that the vertical velocity deviation of the power-fluids generally compare to the vertical flow velocity deviation of the Newtonian fluid at gap ratio 2.0 (figure 5.12(g).

226


5.3. Results

0

0 n=1

−200

n=1 −100

n=3 n = 10

−400

−300 V y,norm

V y,norm

n = 10

−200

−600

−800

−400

−1000

−500

−1200

−600

−1400 −5

n=3

−700 −4

−3

−2

0 −1 1 Gap width, w

2

3

5

4 x 10

−0.01 −0.008 −0.006 −0.004 −0.002

−3

(a) Gap ratio 0.01.

0 x

0.002

0.004

0.006

0.01

(b) Gap ratio 0.02.

0

0

−50

n=1

n=1

n=3

n=3

n = 10

n = 10

−100

−50 V y,norm

V y,norm

0.008

−150

−200

−100

−250

−300 −0.025 −0.02 −0.015 −0.01 −0.005

(c) Gap ratio 0.05.

0 x

0.005

0.01

0.015

0.02

−150 −0.05

0.025

−0.04 −0.03

0.01 −0.02 −0.01 0 Gap width, w

0.02

0.03

0.04

0.05

(d) Gap ratio 0.10.

Figure 5.10: Vertical flow velocity component into the gap across profile b−b0 (figure 5.1(b)). The numerical solution is plotted as whole drawn lines and the analytical solution for a Newtonian fluid calculated from the numerical velocities are plotted as dotted lines.

227

5.3. Results


0

0 n=1

n=1 −2

n=3

−5

n = 10

n=3 n = 10

−4

−10

V y,norm

V y,norm

−6 −15

−8 −10

−20

−12 −25 −14 −30 −0.25

−16 −0.2

−0.15

−0.1

−0.05 0 0.05 Gap width, w

0.1

0.15

0.2

−0.5

(e) Gap ratio 0.50.

−0.4

−0.3

−0.2

−0.1

0 x

0.1

0.2

0.3

0.4

0.5

(f) Gap ratio 1.0.

0 n=1

Figure 5.10: (Cont.) Vertical flow velocity

−1 n=3

V y,norm

−2

component into the gap across profile b − b0

n = 10

−3

(figure 5.1(b)).

−4

plotted as whole drawn lines and the analyt-

The numerical solution is

−5

ical solution for a Newtonian fluid calculated

−6 −7

from the numerical velocities are plotted as

−8

dotted lines.

−1

−0.8

−0.6

−0.4

−0.2

0 x

0.2

0.4

0.6

0.8

1

(g) Gap ratio 2.0.

max

V y,norm

10

n=1 anl. n=3 anl. n = 10 anl.

3

Figure 5.11: Loglog plot of gap ratio (2w) vs. maximum vertical flow velocity compomax ). The numerical nent into the gap (Vy,norm

solution is plotted as whole drawn lines and 10

2

the analytical solution for a Newtonian fluid calculated from the numerical velocities are plotted as dotted lines. For gap ratios ≥ 0.05

10

the loglog-plot for the Newtonian fluid can be

1

10

−2

−1

10 Gap ratio, 2w

10

0

fitted to a line with a slope equal to 0.9647. The loglog-plots for the power-law fluids can be fitted to lines with slope-values 0.9876 and 0.9795, respectively, for gap ratios between 0.01 and 1.0. All the slope values compare to values calculated from the velocities of the gap material by using the analytical solution for a Newtonian fluid (0.9110, 0.9797, and 0.9809, respectively).

228


5.3. Results

N=1 N=3 N = 10

120

N=1 N=3 N = 10

35

30

100

25

∆ Vy

∆ Vy

80

20

60

15 40

10

20

5 g −5

g’ −4

−3

−2

−1

0 Gap width

1

2

3

4

g 5

x 10

(a) Gap ratio 0.01.

g’ −0.008

−0.006

−0.004

−0.002

0 Gap width

0.002

0.004

0.006

0.008

0.01

(b) Gap ratio 0.02. N=1 N=3 N = 10

40

N=1 N=3 N = 10

40

35

35

30

30

25

25 ∆ Vy

∆ Vy

−0.01

−3

20

20

15

15

10

10

5

5 g

−0.025

g

g’ −0.02

−0.015

−0.01

−0.005

0 Gap width

0.005

(c) Gap ratio 0.05.

0.01

0.015

0.02

−0.05

0.025

g’ −0.04

−0.03

−0.02

−0.01

0 Gap width

0.01

0.02

0.03

0.04

0.05

(d) Gap ratio 0.10.

Figure 5.12: Vertical flow velocity component deviation across profile b − b0 (figure 5.1(b)) normalized to the gap width.

229

5.3. Results


40

45

N=1 N=3 N = 10

N=1 N=3 N = 10

40

35

35 30

30 25

∆ Vy

∆ Vy

25 20

20 15

15 10

10

5

5 g

−0.25

g

g’ −0.2

−0.15

−0.1

−0.05

0 Gap width

0.05

0.1

0.15

0.2

−0.5

0.25

(e) Gap ratio 0.50.

g’ −0.4

−0.3

−0.2

−0.1

0 Gap width

0.1

0.2

0.3

0.4

0.5

(f) Gap ratio 1.0. N=1 N=3 N = 10 25

∆ Vy

20

15

10

5

g −1

g’ −0.8

−0.6

−0.4

−0.2

0 Gap width

0.2

0.4

0.6

0.8

1

(g) Gap ratio 2.0.

Figure 5.12: (Cont.) Vertical flow velocity component error across profile b−b0 (figure 5.1(b)) normalized to the gap width.

230


5.3. Results

Total velocity deviation The total velocity deviation ∆V tot (equation 5.52) of the gap material is contoured in figures 5.13 - 5.15. It generally increases with increasing gap ratio and powerlaw coefficient. For the Newtonian fluid most of the gap has a small total velocity deviation (figure 5.13). The large deviation values are concentrated at the center of top and bottom boundaries. For gap ratios (w/h) ≤ 0.02 the total velocity deviation is large at the top and bottom, however the majority of the gap has a small total velocity deviation (figure 5.13(a) - 5.13(b)). For gap ratio 0.01 the majority of the gap has total velocity deviation smaller than 30%. Similarly for gap ratio 0.02 the majority of the gap has total velocity deviation smaller than 4%. For gap ratios ≥ 0.05 and ≤ 0.10 approximately 95% of the gap has total velocity deviation smaller than 2% (figure 5.13(c) - (d)). Although the total velocity deviation increases with increasing gap ratio, the total velocity deviation is small for the majority of the gap for gap ratios ≥ 0.5 (figure 5.13(e) - (g)). The total velocity deviation is less than 8% and 15% for the majority of the gap at gap ratios 0.5, 1.0, and 2.0, respectively. The total velocity deviation is also generally small for both power-law fluids at all gap ratios and compare, although it increases slightly with with increasing power-law coefficient (figures 5.14 and 5.15). For gap ratios ≤ 0.5 the total velocity deviation is less than 10% for the majority of the gap (figures 5.14(a) - (e) and 5.15(a) - (e)). For gap ratios 1.0 and 2.0 the total velocity deviation is less than 20%, for the majority of the gap (figures 5.14(f) and (g) and 5.15(f) and (g)).

231

5.3. Results

(a) Gap ratio 0.01.

(e) Gap ratio 0.5.


(b) Gap ratio 0.02.

(c) Gap ratio 0.05.

(d) Gap ratio 0.10.

(f) Gap ratio 1.0.

(g) Gap ratio 2.0.

Figure 5.13: Newtonian fluid. Total velocity error of the gap matrix (figure 5.1(b)) for gap ratios between 0.01 and 2.0. (a) - (d) not to scale.

232


(a) Gap ratio 0.01.

(e) Gap ratio 0.5.

(b) Gap ratio 0.02.

5.3. Results

(c) Gap ratio 0.05.

(d) Gap ratio 0.10.

(f) Gap ratio 1.0.

(g) Gap ratio 2.0.

Figure 5.14: Power-law fluid n = 3. Total velocity deviation of the gap matrix (figure 5.1(b)) for gap ratios between 0.01 and 2.0. (a) - (d) not to scale.

233

5.3. Results

(a) Gap ratio 0.01.

(e) Gap ratio 0.5.


(b) Gap ratio 0.02.

(c) Gap ratio 0.05.

(d) Gap ratio 0.10.

(f) Gap ratio 1.0.

(g) Gap ratio 2.0.

Figure 5.15: Power-law fluid n = 10. Total velocity deviation of the gap matrix (figure 5.1(b)) for gap ratios between 0.01 and 2.0. (a) - (d) not to scale.

234


5.4

5.4. Discussion

Discussion

The effects of velocities, pressure, and gap ratio have been studied for flow of material between rigid plates using a numerical approach. Simulations for Newtonian and power-law fluid fracture material have been presented. For the Newtonian fluid, gap ratios ≤ 0.02 have large pressure and velocity deviations at the top and bottom boundaries of the gap. The small gap ratios also deviates from the general trend which is observed with increasing gap ratios. This is interesting since the analytical solution for flow between rigid plates for a Newtonian fluid has been derived for the case of approaching plates where w ¿ h (Jaeger, 1969). However, this is not observed for the power-law fluids. In figure 5.16 the normalized lateral velocity along the right gap boundary is plotted for the different fluids. We observe that for gap ratios ≤ 0.02 the lateral velocity of the Newtonian is not constant along the vertical gap boundaries, but plots as a parabola with a minimum velocity in the center of the gap (figure 5.16(a)). The lateral velocity at the center of the vertical boundaries are only 20% and 80% of the maximum velocity at the top and bottom of the boundary for gap ratio 0.01 and 0.02, respectively. This also explains why there is an overlap in the vertical pressure profile a − a0 for the Newtonian fluid at gap ratios 0.01 and 0.02. In contrast the power-law fluids have approximately constant lateral velocities along the vertical boundaries of the gap (figure 5.16(b) - (c)). The analytical solution uses U0 to calculate velocities and pressure (equations 5.43, 5.44, and 5.45). In my setup I have used the velocity at the center point of the vertical boundaries as representative of U0 : U0 = − U

num (w,0)−U num (−w,0)

2

1 ).

This would explain why

for gap ratios smaller the < 0.05 the Newtonian fluid show large pressure and velocity deviation near the top an bottom boundary. But more importantly, gap ratio 0.05 seems to represent a lower limit where the rigid boudin material changes from having a uniform lateral displacement which conserves the original rectangular shape to a lateral displacement promoting a “dumb-bell” shaped geometry. This could mean that because of this it is becomes increasingly difficult to fill the gap with material from the ambient matrix for gap ratios < 0.05, at least for the initial deformation stage. This is also what is observed for the Newtonian fluid in figure 5.11. With decreasing gap ratios the maximum flow velocity into the gap increases to a maximum value for gap ratio 0.05, before it decreases with further decrease in gap ratio. As opposed to the power-law fluids where the maximum flow velocity into the gap increases with decreasing gap ratio and the maximum flow velocity into the gap scales to the gap ratio as 1

Negative value for extension.

235

5.4. Discussion max ∝ Vy,norm

1 (2w)a

Chapter 5. Intrusion between rigid plates max is not observed for for gap ratios ≤ 1.0. Since, this lowering of Vy,norm

the power-law fluids at gap ratios < 0.05 it could indicate that the localization effects and larger velocities generated in power-law fluids with n > 1 are strong enough to allow for more significant infill at very small gap ratios. The strain localization in the power-law fluids is also observed as a strong “plug flow” which increases with decreasing gap ratios. We also note that this is in itself interesting since it is also observed for fluids with free-slip boundary conditions at the vertical boundaries. However, for gap ratios > 0.05 the relationship between the maximum flow velocity into the gap and the gap ratio of the Newtonian fluid is comparable to the power-law fluids. For all fluids the exponential coefficient is approximately equal to 1 and the maximum flow velocity into the gap scales to the max ∝ gap ratio as Vy,norm

1 (2w) .

This is also what is expected from the analytical solution.

From equation 5.43 we get V (0, h) =

3U0 2w h

for vertical flow velocity at the center

of the profile b − b0 , which is comparable to the measured relationship between the maximum flow velocity into the gap and the gap ratio. The horizontal spreading velocity of the vertical gap boundaries generally increases for all fluids with increasing gap ratio (figure 5.17). For large gap ratios the horizontal spreading velocity is similar for all fluids. However, for gap ratios < 0.10 the horizontal spreading velocity of the Newtonian Fluid is much smaller than for the power-law fluids (figure 5.17(b)). The horizontal spreading velocity for the Newtonian Fluid decreases dramatically, whereas the spreading velocity for the power-law fluids increase to a local maximum value at gap ratio 0.02. This explains why the vertical velocities of the analytical solution for a Newtonian fluid calculated from the numerical velocities are comparable for large gap ratios, but not for gap ratios < 0.05 (figure 5.10). To sum up, for power-law fluids with n > 1 and very small gap ratios the initial opening of a fracture creates large vertical velocities at the fracture tip and thus allowing for material from the ambient matrix to be sucked into the fracture. For Newtonian fluids there is a lower gap ratio boundary of 0.05 of which it gets harder for material from the ambient matrix to be sucked into the gap. Since the power-law fluids cause much smaller stresses/pressures, it is easier to separate the plates if there is a power-law fluid in between, because less force is needed. So for the fractures adjacent to the basin margin this means that if the material can be described by a power-law rheology, the initial opening will generate suction of basin material into the fractures. In order to demonstrate this one would have to measure the integrated normal stress at the lateral boundaries, ie. the force, and compare it for the constant strain rate driven expansion of rigid plates. 236


5.5. Conclusion

The analytical solution is said to hold for relatively small gap ratios (w ¿ h) (Jaeger, 1969), however we observe that for the relatively large gap ratios the pressure deviation and velocity deviation is relatively small. For gap ratios ≤ 1.0 the maximum pressure and total velocity deviation is less than 10% for the majority of the gap. Still for gap ratio 2.0 the pressure deviation is less than 20% for approximately half of the gap material and the total velocity deviation is less than 15% for the majority of the gap. Also interesting is that the velocity deviation of the power-law fluids are relatively small for all gap ratios, even though the analytical solution is derived for a Newtonian fluid. For all gap ratios ≤ 0.5 the majority of the gap has total velocity deviation less than 10%. Still for gap ratios 1.0 and 2.0 the total velocity deviation is less than 20% Also the noticeable is that the total velocity deviation and the pressure deviation, although relatively large, are unaffected by power-law coefficient.

5.5

Conclusion

I have used a numerical modeling approach to investigate the main factors affecting flow of material between rigid plates. This might give insight into processes which may generate fractures near the basin margin of the Hornelen basin. From the results presented and discussed above it is clear that the analytical solutions for flow between rigid plates also hold for relatively large gap ratios. Power-law fluids do not deviate significantly from the analytical solution for a Newtonian fluid for all gap ratios when in terms of flow velocities, but do differ in terms of pressure. The pressure and velocity deviation of the power-law fluids are unaffected by power-law coefficient. The power-law fluids are characterized by a strong a “plug flow” at small gap ratios and the “plug flow” effect increases with power-law coefficient.

237

5.5. Conclusion


0.5

0.5 0.01 0.02 0.05 0.1 0.5 1.0 2.0

0.4 0.3 0.2

0.3 0.2

0.1

0.1 0

y

0

y

0.01 0.02 0.05 0.1 0.5 1.0 2.0

0.4

−0.1

−0.1

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0.3

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1

0.9984 0.9986 0.9988

U num/Unum max

0.999 0.9992 0.9994 0.9996 0.9998

1

U num/Unum max


(b) Power-law fluid n = 3.

0.5 0.01 0.02 0.05 0.1 0.5 1.0 2.0

0.4 0.3 0.2

Figure 5.16:

Lateral velocity normalized

against the maximum lateral velocity along the right right boundary of the gap. Notice

0.1

that for the Newtonian fluid (a) there is a rela-

−0.1

tively large difference in lateral velocity along

−0.2

the vertical boundary at small gap ratios. For

y

0

the power-law fluids (b and c) the difference

−0.3 −0.4

in velocity is minute along the vertical gap

−0.5 0.99988

0.9999

0.99992

0.99994

0.99996

0.99998

1

boundary for all gap ratios.

U num/Unum max

(c) Power-law fluid n = 10. 3 2.5

n=1 n=3

2.5

2

n = 10 2

Vx

Vx

1.5

1.5

1

1 0.5

0.5 0

0 0

0.5

1 Gap Ratio, 2w

1.5

2

(a) Spreading velocity for gaps ratios 0.01 ≤ 2w ≤ 2.0.

0

0.05 Gap Ratio, 2w

0.1

(b) Close up of small gap ratios in (a).

Figure 5.17: The spreading velocity of the gap. The maximum velocities are plotted as whole drawn lines and the minimum velocity is plotted as lines with diamonds. The spreading velocity of the Newtonian fluid is much lower at gap ratio < 0.05. At large gap ratios the spreading velocity is equal for all fluids. The spreading velocity generally increase with increasing gap ratio. In contrast the spreading velocity of the power-law fluids increases to a local maximum for gap ratio 0.02 before decreasing with decreasing gap ratio.

238


5.5. Conclusion

The vertical flow velocity into the gap generally increases with a power-law relationship with decreasing gap ratio for both Newtonian and power-law fluids with n > 1 max and scales as Vy,norm ∝

1 (2w)

where 2w is the gap width. For power-law fluids with

n > 1 the initial opening of a fracture with very small gap ratio creates large vertical velocities at the fracture tip and a large lateral spreading velocity of the fracture, thus allowing for material from the ambient matrix to be sucked into the fracture. For Newtonian fluids there is a lower gap ratio boundary of 0.05 of which it gets harder for material from the ambient matrix to be sucked into the gap. Below this limit the forces become very large for small linear viscous gaps, as it causes deformation of the “rigid” boudins and therefore deviations from the analytical solution. However, the analytical solution is generally, and especially if only velocities are analyzed, a good approximation for plates of finite length, ie gap ratios, and also non-linear materials. Since the power-law fluids cause much smaller stresses/pressures, it is easier to separate the plates if there is a power-law fluid in between, because less force is needed. So for the fractures observed in the marginal breccia adjacent to the basin margin this means that if the material can be described by a power-law rheology, the initial tensile opening will generate suction of basin material into the fractures. This has also implication to other features than boudinage. Subduction channel flow, channel flow in orogenesis and other deformation features involving flow between rigid plates.

239

Chapter 6

Synthesis and conclusions This thesis demonstrate the use of combination of techniques; field work, textural and quantitative image analysis, and numerical modeling to better understand the deformation processes that occur near the basin margin The results presented show much more extensive deformation that has not been recognized earlier. Three domains of deformation have been recognized with respect to the basin margin: • 0 - 5 m away from the basin – Extensional fracturing and development of a marginal breccia. ∗ This is supported by numerical modeling of flow between rigid plates. When an initial fracture is formed non-linear viscous fracture material cause much smaller stresses/pressures than linear viscous fracture material. Hence it is easier to separate the plates if there is a power-law fluid in between, because less force is needed. Since there fractures terminate at the interface with the basin it indicates that this is an earlier deformation that happened before the sand was complectly lithified. • 20 m away from the basin margin – Sandstone dikes (pull-apart fractures) ∗ A mixture of material partly from granulation in nearby connecting fractures and partly from basin material. Both field and textural analysis show evidence of granulation of wall-rock material. Quantitative image analysis show a mixed signature. Large D-values supporting 241

6.1. Outlook

Chapter 6. Synthesis and conclusions

granulation, but the lack of “survivor grains” indicate transport of granulated material to the fracture. However, flow structures indicate infill of basin material. • 200 m away from the basin margin – Fault gouge ∗ Both field evidence, textural analysis and quantitative image analysis show evidence of cataclasis. Large D-values and suggest a high amount of deformation. Shape and texture of epidote suggests repeated faulting events. We observe an ductile deformation in the granodiorite in the form of foliation and shear zones that predates the brittle deformation. The foliation and shear zones are generally orientated W-E and dips to the south, Other brittle deformation structures are found up to 200 - 400 m a way from the basin. These include faults, fault zones, fractures and breccias. The orientation of these are generally comparable and might be related to the basin development

6.1

Outlook

The results presented in this study show that the Northern margin of the Hornelen basin is a much more tectonized contact than previously recognized. However, to fully understand the implications of these deformation feautures, more work is needed. It should be investigated if the deformation features are only locally, or if similar deformation features are found at the Southern margin of the Hornelen basin, and if similar deformation features are found in other basins. The observation of kink bands that break up in fractures near the basin is a very interesting feature. Kink bands form at relatively large pressures and if the kinks were not formed from an preexisting foliation, it would imply that at least the some of the observed deformation happened during relatively high pressures. Also the shear zones display a connectivity with brittle deformation. Since the cataclastic cores are only found locally in the shear zones it implies that it formed simultaneously with the rest of the shear zones.

242

Appendix A

Geological maps

243

Appendix A. Geological maps

244

g g

g

Rylandsvatnet g

g

g

g

g

g

g

Sildefjellet

s Q g

g g

N N

N

350

g

N g g

g N

N

g

g g

N

g

g q g g

300

n

n n

Skipperdalsnes

n

250 400

Geitanipa Q g

200

s

150

Q

Djupedalen

g

100

g

g

Bukta

q

g g

g g

g

50

Q

g

g

g

g

g

350

Frøysjøen

g g

g g

g

g

g

q

g

g

g

g

g

g

300 g

q

g g

150 g

g

P7

g

g

Q g

g

Q g g

g

Q

g

g

Q

g

g

Q n

g

P1

g g

150

q

g

100

q

g n

g

g q

200

g

g

g

50

Vågane

g gq

g

g

g

g

g g

250

g Q

g

g

BRE59-05

g

g

100

g

g g

BRE60-05

g

g

g

g g

200

g

g g

Rylandsholmen

g

g

250

g

q n n

n

n

N g g

g

g

g q

q

g g

g gq

Smørhamn-Vågane area 1 : 20 000

g

g g

g

g

g

0

g Q

q

Q

Q

q g

g

Q Q

Smørhamn

Rota

1200

Equidistance: 25 m

g

q

800

400

Q

g

g

200

Q

Q Q g

Legend Granodiorite/tonalite (BGC) Conglomerate, sedimentary breccia (HB) Melangê (KM) Lithological boundary/Lithological boundary uncertain Sandstone dike, marginal breccia, sedimentary breccia Crush breccia, cataclasite Pseudotachylite Foliation with dip indicated (RHR) Shear zone with sense of shear indicated Mylonite/mylonite zone

Bedding with dip indicated (RHR)

g q

Q

n

N

Transform fault with relative movement indicated (RHR) Normal fault, tick points towards dip (RHR) Major joint, possible fault Granulated material Quartz fractures (< 10 cm/< 10cm) Quartz netvains Rivers and lakes Boulders Roads

Appendix B

Image analysis In the following chapter the image analysis procedure is explained in detail. The program “Gray scale image analysis” (Bjørk, 2006) is written in MatLab and uses MatLab’s Image Processing Toolbox (The MathWorks, 2001). The code is open-source and allows for individual user alterations if needed. The program requires a minimum amount of user input and is able to semi-automatically produce reliable results for a large amount of images. The program code is given in Appendix C Program code.

B.1

Phase separation and grain identification

The “Gray scale image analysis program” (Bjørk, 2006) has two main methods of identifying and separating particles. Both methods use grays scale bit-map images, but any color image can be used and will be converted into gray scale format. The important factor determination good phase differentiation and particle identification is sufficient contrast in the image. The program offers two different methods: (i) gray scale thresholding and (ii) “watershed” method. The gray scale thresholding generally works best on material where you want to differentiate between phases and identifying their particles. The “watershed” method works best when you only want to differentiate between particles and matrix. High resolution images give best results.

B.1.1

Common user input

The program requires little user input. However the following needs to be specified in the USER INPUT-section: output: Output file name. Mat-file with the processed image. 245

B.1. Phase separation and grain identification

Appendix B. Image analysis

magnification: Magnification to display image (%). convert2grayscale: Convert colour image to 8-bit gray scale image (true/false). contrast: Increase image contrast. Saturate low a high intensity values (true/false). stretch_scale: Stretch and adjust intensity values to full length of gray scale (true/false). manual: Manual removal of particles (true/false). min_area: Matrix threshold. I.e. limit for the minimum particle size detectable. se: Set shape and size of structuring element for morphological opening and closing(Shape: square, disk, diamond etc. Size: pixels). manual: Manual removal of particles (true/false). method_choice: Method selection (1: thresholding and 2: watershed). Colour images are converted into 8-bit gray scale images using the NTSC standard for luminance (Gray scale intensity = 0.2989R + 0.5870G + 0.1140B). The contrast can be improved by (i) saturating the bottom 1% and the top 1% of the intensity values using the imadjust-function, and/or (ii) stretching and adjusting the intensity values of the gray scale image so that it uses the entire gray scale range with the histeq-function. A matrix threshold, that is a limit for the minimum particle size detectable, needs to be specified (min_area). For high resolution images of 2500x2000 I suggest that all grains with area less than 200 pixels are removed. This is a reasonable limit, smaller matrix threshold would add error to calculations of particle shape (see below). Some particle may need to be removed manually. This is done using a graphical interface (see below). Common for both methods is the reading and display of the image data. The image is read from the graphics file and displayed by using MatLabs imread- and imshow\verb-functions (fig. B.1). A duplicate matrix1 of the image is made for further processing. The original image data is kept for evaluation of the result (see below). 1

For clarity matrix is typeset matrix when referring to a mathematical matrix and matrix when

referring to matrix material (image background).

246


B.2. Gray scale thresholding

Figure B.1: Original BSE-image.

B.2

Gray scale thresholding

B.2.1

Input

The gray scale thresholding method also requires choices for processing type and morphological opening an closing: lower_boundary: Lower boundary for thresholding. upper_boundary: Upper boundary for thresholding. heavy: Processing type (light/heavy) open_factor: Number of times to perform morphological opening (erode + dilate) to separate and identify particles (used if heavy = true). close_factor: Number of times to perform morphological closing (dilate + erode) to make the particle boundaries more smooth. The gray scale image has a gray scale range of 256, where black equals 0 and white equals 255. A histogram of the gray scale intensity is plotted using the imhist-function to identify the different intensity ranges of the phases (fig. B.2). It is also possible to use the data cursor as a guide to finding the appropriate gray scale ranges (fig. B.3). When the data cursor mode is enabled you can click on the grains in the image and 247

A.2. Gray scale thresholding

Appendix A. Image analysis

data values of the x− and y-coordinates, and gray scale intensity value are displayed. Once the appropriate gray scale range is found the upper and lower boundaries are needs to be specified. The pixels in within the range is set to 1 (white), while the background is set to 0 (black). The result is displayed to show the result (fig A.4).

Figure A.2: Histogram of gray-scale intensity.

Light processing (heavy = false)is sufficient when the phase of interest consists of isolated grains or grains with nearly touching boundaries. If the particles share relatively long boundaries (heavy = true), the grain separation becomes more advanced and requires more processing. It is important to note, that every morphological operation affects the size and shape of the particle slightly and therefore should be kept at a minimum (se below). It is therefore advised that any particles that share any long boundaries that will not be separated without applying numerous repetitions of morphological processing, should be manually separated using a standard image-editing software (e.g. if a dark phase requires manually separation of particles. Use imageediting software and separate the particles in question by drawing thin white lines at the boundaries. Save the image as a copy and use it for further processing of that phase).

248



Figure A.3: With the data cursor mode enabled you can click on the grains in the image and data values of the x− and ycoordinates, and gray scale intensity value are displayed.

Figure A.4: Thresholded B&W-image of phase of interest.

249



Light processing Several morphological operations using MatLabs bwmorph-function are performed on the image to identify the individual grains. All morphological operations inspect the pixels in a 3x3 pixel-environment. First image noise is reduced by removing isolated white pixels with the clean-option (fig. B.2.1). That is, individual 1’s that are surrounded by 0’s. To separate the grains H-bridges and spurs are removed with the hbreak and spur options, respectively. An H-bridge pixels is the pixels that has two opposite 0 and 1 nearest neighbours pairs (fig. B.2.1). A spur is a pixel with only one 1 next nearest neighbor and seven 0 neighbours (fig. B.2.1). Finally inclusions in the grains are filled with the imfill-function using the holes option. A inclusion defined as a set of background pixels (0’s) that cannot be reached by filling in the background from the edge of the image. This is done last to avoid that any grains touching at two places before morphological processing becomes one single particle. The removal of inclusion will potentially add particles a different phase to that grains. But if that happens, it will be visible int the graphical evaluation of the end result (see below). To remove any remaining matrix and to set a limit for the minimum grain size detectable on each magnification image, all grains with area less than 200 pixels are removed. Note, this is a reasonable limit, smaller matrix threshold would add error to calculations of particle shape (see below) . The image is then labeled labeled using the bwlabel-function. The connecting white pixels (i.e. the particles) are specified by a positive integer value while the background remain set to 0 (fig. B.8). The connectivity of the pixels is chosen to be nearest neighbour only. That is, in a 3x3-environment the the neighbours at the faces if the center pixels are considered neighbours (fig. B.2.1). The next nearest neighbours at the vertices are not. This is done to avoid that grains which are in contact with only one next nearest neighbours remain regarded as individual particles. If high resolution images are used 4-connectivity instead of 8-connectivity does not affect the size and 0

0

0

0

1

0

0

0

0

−→

0

0

0

0

0

0

0

0

0

Figure B.5: Removal of isolated pixels. Isolated piexls are individual 1’s that are surrounded by 0’s.

250



1

0

1

1

1

1

1

0

1

−→

1

0

1

1

0

1

1

0

1

Figure B.6: Removal of H-connected grain. The H-bridge pixels has two opposite 0 and 1 nearest neighbours pairs.

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

−→

Figure B.7: Removal of spur. The spur has only 1 next nearest neighbor and seven 0 neighbours.

shape of the final particle. To remove any matrix noise left and to set the limit for the minimum size detectable all areas with an area smaller than the matrix threshold value (min_area) are removed with the bwareaopen-function. This function produces a binary image that needs to be re-labeled (fig. B.10). Grains that are located at the edges of the image are removed since we do not know their entire size and shape (fig. B.11). This is done simply by finding the unique nonzero integer values at the edges of the labeled image and setting pixels with those values to zero. The area of the edge-grain is calculated using the regioprops-function and subtracted from the total area of the processed image. The edges of the separated particles are usually much rougher than the edges of the real particle in the original image (Figure B.12(a)) because image noise and the morphological operations used to separate the particles. To remove this artifact the particles are eroded and dilated one or several times (morphological closing) (fig. B.12(b)). This is done with the imdilate -and imerode-functions. The morphological closing is constrained with the option ’same’. This option keeps the particle numbering unchanged so that if any grains touch after dilation, the particle do not merge to one particle. Excessive morphological closing should be avoided, because it imposes the structure of the structuring element onto the particles (see below).

251



Figure B.8: The particles are labeled so that each particle is specified with an positive integer. The matrix is set to zero. Image shows part of particle number 140 and the matrix.

1

1

1

1

1

1

1

1

1

Figure B.9: 4-connectivity of pixels in a 3x3 pixel-environment. The center pixel (1) is connected to its nearest neighbours (1’s),but not its next nearest neighbours (1’s).

252



Figure A.10: Labeled image after morphological processing. Individual grains are randomly coloured to highlight their separation from particle neighbours.

Figure A.11: Labeled-image with edge grains removed.

253



(a) Before.

(b) After.

Figure B.12: Morphological closing to make grain boundaries smoother. The grain boundaries are superimposed onto the original image for comparison.

254



Heavy processing For phases with long adjacent particle boundaries the processed image does not display a reasonable result. The individual particles need to be separated by more advanced processing. That is done by eroding and applying the morphological operation mentioned above a number of times. After each loop any grains smaller than 1/3 of the specified minimum particle size is removed. It is reasonable that any particle smaller than 1/3 of the matrix threshold will be removed in the end and this removal speeds up the dilating procedure later on. Then the grains need to be dilated back to their original size and shape while allowing particles to touch at their boundaries. This could be done by constraining the labeling with the imdilate-function, but this does not produce reasonable particle boundaries for repeated morphological opening operations. Any overlap is given to one of the particles. Instead the particles are dilated separately and compared to its neighbours in a for-loop. If any overlap exists the overlap is divided between the grains on the basis of distance from particle center.

That is, the half of the overlap which is

furthest away from the particle A’s centroid is removed from particle A and similarly the other half of the overlap which is furthest away from the particle B’s centroid r is removed from particle B. The distance from the centroid is calculated as ³ ´2 particle P article )2 . If there is only on pixel overXoverlap − Xmean + (Yoverlap − Ymean D= lap or odd number of pixels overlapping the last pixels is randomly deleted from one of the grains. If any particle should be smaller than the matrix threshold they are removed, else the particles are added up in a dummy matrix that will replace the previous one. When the procces is finished the edge particle are removed and the matrix is re-labeled in a similar fashion of that of for light processing option. The process is time consuming, but produces reliable results (fig. B.13). Although as mentioned above the morphological processes will have an effect on the particle boundaries and the number of morphological should be kept to a minimum.

255

B.3. Watershed method


Figure B.13: Because the particles share long particle boundaries light processing is not sufficient to separate and recognize the particles. However after more processing a reliable result is obtained.

B.3

Watershed method

The “watershed” method is also based on difference is gray scale intensity. But rather than specifying a specific gray scale range to isolate the particles the “watershed” method uses segmentation to divide the image into separate particles. Simplified this can be thought of as viewing the image as a surface (figs. B.14 & B.19(a)). When “water” is pored on to the image it will “flow” from the “mountains” (high intensity values) to the “catchment basins” (low intensity values). The particles acts like the “basins” and are separated by the “valleys”. The parts of “watershed” method is based on the watershed demo found in The MathWorks (2001, chap. 9).

B.3.1

Input

The “watershed” method also requires choices for image enhancements: inverse: Inverse image (particles need to be dark to function as catchment basins) (true/false). exaggerate: Exaggerate particle bounadries (true/false). 256



ex_number: Number of times to add top-hat and subtract bot-hat image from original image exaggerate: Threshold catchment basins. Detect catchment basins “deeper” than a specified threshold value (true/false). basin_min: Set threshold value. oversegmentation: 1. If exaggerate = true: Remove catchment basins with upper boundaries below the specified threshold value. 2. If exaggerate = false: Remove any catchment basin shallower than the specified threshold value. overseg_removal: Set threshold value. To properly identify all particles the boundaries needs to be exaggerated (fig. B.15). This is done by adding the top-hat-image to the original image, and then subtracting the bottom-hat image from the new image. Top-hat filtering is the equivalent of subtracting the result of performing a morphological opening operation on the original image from the original image itself. Similarly bottom-hat filtering is the equivalent of subtracting the result of performing a morphological closing operation on the original image from the original image itself. This result of the filtering is that top-hat image contains the “peaks” and the bottom-hat image contains the “valleys”. The adding and subtracting of images is done by using the imadd -and imsubtract-functions. The imadd -and imsubtract-functions adds and subtracts, respectively, the pixels values of two images together, but truncates the values to the maximum (255) and the minimum (0) limits, respectively, if they are exceeded. This increases the contrast between the particles and their boundaries. Since the watershed-function identifies catchment basins, the image need to be inverted if the particles have higher intensity values than their boundaries. This is done with the imcomplement-function (fig B.16). The pixel values are subtracted from the maximum pixel value (255) and the difference is used as the pixel value in the output image. In the output image, dark areas become lighter and light areas become darker. The image is then thresholded to detect all catchment basin deeper than the specified threshold. This is done by using the imextenedmin-function that return regional minima that are less than their neighbours by the speficied threshold (figs. B.19(a) & B.19(b)). That is, a marker image with the locations of the catchment basins ar specified. 257



Then the image is modified with using the imimpose-function to only contain the catchment basins that have been specified. The imimpose-function sets the pixel values of the catchment basin returned by the imextenedmin-function to zero. It also changes the values of all the other pixels in the image to eliminate the other minima (fig. B.19(c)). The result might be over-segmentated due to too many catchment basins in the particles. This is can be corrected for by removing any catchment basins that are shallower than a specified threshold with the imhmin-function (fig. B.19(d)). This will now remove catchment basins that originally was on the the low end of the intensity scale event though their relative depth was equal of that catchment basins found in the other end of the intensity scale. It is reasonable to assume that the the matrix will have highest intensity values and that catchment basins found at lower intensity values represents original image noise and/or intra-particle fractures in the particles. Also it is clear from fig. B.19(a) that if imhmin-function was applied on the original image, it would not be possible to differentiate between the two deep catchment basins. However, this will not yield the appropriate result if the intra and inter-particle fractures have the same absolute gray scale ranges. Of course it possible not to use the imextenedmin -and imimposemin-functions and only use the imhmin-function, and vise versa. The watershed-function a returns a labeled matrix identifying the watershed regions (i.e. the particle ) in the image (fig. B.20). The particles are identified by a positive integer while the background (fracture space or matrix) is set to zero. Similarly like for gray scale thresholding method particles can be manually removed and edge particle are removed automatically and their area subtracted from the total area. To compare the result of the processing the labeled matrix is superimposed ont to the original image (fig. B.21). The end result is not too bad as a first hand approximation (fig. B.21). The original image shows a set of perpendicular fractures in peridotite (fig. B.14). The problem areas are the eroded areas, and the dark areas with strong shadows. The eroded areas created lot of over-segmentation and in the dark areas the contrast is insufficient to differentiate between all the particles. However, as a first hand approximation the result is not bad and with some image enhancements using standard image-editing software are reliable result should be possible.

258


A.3. Watershed method

(a) Original image.

(b) Image converted to gray scale. Out

Figure A.14: Image showing perpendicular fracturing in peridotite (image courtesy of K.H. Iyer).

259



Figure A.15: The gaps are exaggerated by adding the tophat-image to the original image, and then subtracting the bottom-hat image from the new image.

Figure

A.16:

The

image

is

inverted

using

the

imcomplement-function. The pixel values are subtracted from the maximum pixel value (255) and the difference is used as the pixel value in the output image. In the output image, dark areas become lighter and light areas become darker.

260



Figure A.17: The imextenedmin-function return regional minima that are less than their neighbours by the specified threshold.

Figure A.18: The imimpose-function sets the pixel values of the catchment basin returned by the imextenedmin-function to zero.

261



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(a) Original gray scale image. There are two “deep” catchment basins of equal depth (30) and one “shallow” catchment basin (20).

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(b) Extended minima image returned by the imextended-min-function. The image is binary and contains catchment basins “deeper” than the specified threshold value of 20 (1’s).

Figure B.19: Development of image matrix during image enhacement. The pixels are plotted as intensity values.

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(c) Imposed minima image returned with imimpose-function. Note, the imimposefunction also changes the values of all the other pixels in the image to eliminate the other minima.

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(d) All catchment basins below the specified threshold value of 41 are suppressed with the imhmin-function. Note that the depth of catchment basins that originally was deeper than the specified valuer is now set to that depth.

Figure B.19: Development of image matrix during image enhacement. The pixels are plotted as intensity values.

263



Figure B.20: Labeled image after morphological processing. Individual grains are randomly coloured to highlight their separation from particle neighbours.

264



(a) The particles are superimposed on top of the original image for comparison.

(b) Close up of region to the left of the pen cap in (a). The eroded areas are over segmented and the some particles are not identified properly.

Figure A.21: Graphical evaluation of processing result.

265

Appendix C

Program code: Gray Scale Image Analysis In this chapter the program code of “Gray scale image analysis” (Bjørk, 2006) is given. The program is written in MatLab and uses MatLab’s Image Processing Toolbox (The MathWorks, 2001). Image_analysis.m is the main file where to user-input is set. The program is has three function files which handles the processing of the image depending on the user-input: thresholding_light.m, thresholding_heavy.m, and watershedding.m. Image_post_processing is the post-processor which handles the graphical plotting.

C.1

Image_analysis.m

1

% = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = %

2

% GRAY SCALE IMAGE ANALYSIS

3

%

4

% PHASE RECOGNITION AND PARTICLE IDENTIFICATION

5

%

6

% Torbjørn Bjørk, November 2005

7

% Updated June 2006

8

% = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = %

9 10

clear; close all; clc;

11 12

warning off;

13 14

% = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = %

15

% USER INPUT

16

% = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = %

17 18

% OUTPUT FILE NAME

267

C.1. Image_analysis.m

19

Appendix C. Program code

output = ’epidote’;

20 21

% READ IMAGE GRAPHICS FROM FILE

22

%image

23

image

= imread(’BRE0204B2_5_1_pq.tif’); = imread(’pan_3.jpg’);

24 25

% MAGNIFICATION TO DISPLAY IMAGE GRAPHICS

26

mag = 50;

% of original size

27 28

% CONVERTION TO GRAYSCALE

29

convert2grayscale = true;

30 31

% INCREASE CONTRAST

32 33

% 1. Saturate low and high intensity values

34

adjust = false;

35 36

% 2. Stretch and adjust intensity values to full lenght of gray scale

37

stretch_scale = false;

38 39

% STRUCTURING ELEMENT

40

% Set shape and size of structuring element for morpohlogical opening and closing.

41

se = strel(’square’,2);

% See reference page in Help browser for other shapes

42 43 44

% SET CONNECTIVITY OF PARTICLES (3x3 ENVIRONMENT)

45

connectivity = 4;

% 4: Only nearest neighbours. % 8: Nearest and next-nearest neighbours.

46 47 48

% MANUAL REMOVAL OF PRATICLES

49

% Manual removal of particles using a graphical interface

50

manually = true;

51 52

% MINIMUM DETECTABLE PARTICLE AREA

53

% Set minimum particle area (pixels)

54

min_area = 100; % 200

55 56

% POST-PROCESSING

57

% Plotting of particles

58

boundaries = false;

% true: super-impose particle boundaries

59

% (Note: Touching particles will be plotted

60

% with same colour regardless of particle labeling).

61

% false: Super-impose transparent particles.

62 63

% METHOD SELECTION

64

METHOD = {’Thresholding’, ’Watershed’}; method_choice = 2;

65 66

% METHOD INPUT

67

switch METHOD{method_choice}

68 69

case ’Thresholding’

268



70 71

% Gray scale range (0 - 255) for thresholding

72

lower_boundary = 0;

73

upper_boundary = 90;

74 75

% If particle particle density is high and more processing is

76

% needed to identify particles

77

heavy = false;

78 79

% Number of times to perform morphological closing (dilate + erode)

80

% to make the particle boundaries more smooth

81

close_factor = 1;

82 83

% Number of times to perform morphological opening (erode + dilate)

84

% to separate and identify particles (used if heavy = true)

85

open_factor = 1;

86

case ’Watershed’

87 88 89

%Inverse image (particles need to be dark to function as catchment basins)

90

inverse = true;

91 92

% Exagerate particle bounadries

93

exaggerate = true;

94

ex_number = 1; % Number of times to add tophat and subtract bothat from original image

95 96

% Detect intennsity valleys lower than specified threshold

97

threshold = true;

98

basin_min =70;

99 100

% Oversegmentation

101

oversegmentation = true;

102

overseg_removal = 90; % Remove any catchment basin shallower than overseg_removal

103

end

104 105

% = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = %

106 107

% MAKE OUTPUT DIRECTORY

108

[SUCCESS,MESSAGE,MESSAGEID] = mkdir(’OUTPUT’);

109 110

%ADD PATH TO ADDITIONAL FUNCTIONS

111

addpath([pwd, dir_div, ’FUNCTIONS’]);

112 113

disp(’% = = = = = = = = = = = = = = = = = = = = = = = %’)

114

disp(’% GRAY SCALE IMAGE ANALYSIS

%’)

115

disp(’%

%’)

116

disp(’% PHASE SEPARATION AND PARTICLE IDENTIFICATION

%’)

117

disp(’% = = = = = = = = = = = = = = = = = = = = = = = %’)

118

disp(’ ’)

119 120

if convert2grayscale | stretch_scale | adjust

269



121

disp(’% = = = = = = = = = = = = = = = = = = = = = = = %’)

122

disp(’PRE-PROCESSING’)

123

disp(’% = = = = = = = = = = = = = = = = = = = = = = = %’) disp(’ ’)

124 125

end

126 127

% = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = %

128

% DISPLAY IMAGE

129

% = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = %

130 131

% Convert to grayscale using the NTSC standard for luminance

132

if convert2grayscale

133

disp(’Converting to gray scale’)

134

image = .2989*image(:,:,1)...

135

+.5870*image(:,:,2)... +.1140*image(:,:,3);

136 137

end

138 139

image = image(1:500,1:500);

140 141

% Stretch and adjust intensity values to full lenght of gray scale

142

if stretch_scale disp(’Stretching and adjusting intensity values to full lenght of gray scale’)

143 144

image = histeq(image); end

145 146

% Saturate at low and high intensity values

147

if adjust disp(’Saturating low and high intensity values’)

148

image = imadjust(image);

149 150

end

151 152

% ORIGINAL TOTAL SIZE OF IMAGE

153

original_total_area = size(image,1)*size(image,2);

154 155

% MAKE IMAGE DUPLICATE FOR PROCESSING

156 157

label = image;

158 159

% Display image

160

figure(1) clf imshow(image,’InitialMagnification’,mag)

161

title(’Original Image’)

162 163

% = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = %

164

% METHOD SWITCH

165

% = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = %

166 167

disp(’ ’)

168

disp(’% = = = = = = = = = = = = = = = = = = = = = = = %’)

169

disp(’PROCESSING’)

170

disp(’% = = = = = = = = = = = = = = = = = = = = = = = %’)

171

disp(’ ’)

270



172 173

switch METHOD{method_choice}

174

case ’Thresholding’

175

fprintf(1, ’ Method: Thresholding,

176

’);

177

if heavy == false

178 179

fprintf(1,’Option: Light processing’)

180

disp(’ ’)

181

[label,num,area_particles,total_area,label_edges,area_edge_grains,area_matrix] = ...

182

threshold_light(label,image,mag,lower_boundary,upper_boundary,min_area,... original_total_area,close_factor,se,manually,boundaries);

183

else

184 185

fprintf(1,’Option: Heavy processing’)

186

disp(’ ’)

187


188

threshold_heavy(label,image,mag,lower_boundary,upper_boundary,min_area,... original_total_area,close_factor,open_factor,se,manually,boundaries);

189

end

190 191

case ’Watershed’

192 193

disp(’Method: Watershed’)

194

disp(’ ’)

195


196

watershedding(label,image,mag,min_area,original_total_area,se,oversegmentation,... overseg_removal,basin_min,exaggerate,ex_number,threshold,inverse,manually,boundaries);

197 198

end

199 200

% = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = %

201

% SAVE DATA

202

% = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = %

203

disp(’% = = = = = = = = = = = = = = = = = = = = = = = %’)

204

disp(’POST-PROCESSING’)

205

disp(’% = = = = = = = = = = = = = = = = = = = = = = = %’)

206

disp(’ ’)

207

save([’OUTPUT’,dir_div,output],’label’,’num’,’area_particles’,’total_area’,’label_edges’,’area_edge_grains’,’area_matr

208

disp([’Data saved as ’,pwd,dir_div,’OUTPUT’,dir_div,output,’.mat’])

209

disp(’ ’) disp([output,’.mat contains the following matrices:’])

210

disp(’ ’) disp(’label:

211

disp(’num:

Number of particle’)

212

disp(’area_particles:

Area of particles’) disp(’area_matrix:

213

Area of matrix’) disp(’total_area:

214

particles + area of matrix)’) disp(’label_edges:

215

matrix of removed edge particles’) disp(’area_edge_grains:

216

of edgde particles’) disp(’original_total_area:

217

image (toal area + area of edge particles’) disp(’ ’) disp(’*The

218

"regionprops"-function can be used for calculation’) disp(’of size

219

and shape properties of the particles’) disp(’ ’)

220

disp(’% = = = = = = = = = = = = = = = = = = = = = = = %’)

Labeled matrix of particles*’)

The total area (area of Labeled

221 222

Area

Area of original

close(warndlg)

271



223 224

close(warndlg(’FLOOD & FILL - REMOVE PARTICLES MANUALLY’))

225

\end{verbatim}

226 227 228

\normalsize

229

\subsection{thresholding\_light.m}

230

\label{sssec:thresholding_light.m}

231 232

\begin{Verbatim}[fontsize=\scriptsize,,numbers=left]

233 234

% = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = %

235

% GRAY SCALE THRESHOLDING - LOW PROCESSING

236

%

237

% Torbjørn Bjørk, November 2005

238

% Updated June 2006

239

% = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = %

240 241

function

242

[label,num,area_particles,total_area,label_edges,area_edge_grains,area_matrix]=...

243

threshold_light(label,image,mag,lower_boundary,upper_boundary,min_area,....

244

original_total_area,close_factor,se,manually,boundaries,connectivity)

245 246

% = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = %

247

% DISPLAY GRAY SCALE INENISTY HISTOGRAM (WHITE = 255 & BLACK = 0)

248

% = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = %

249 250

figure(2) clf imhist(label,256) title(’Gray scale intensity

251

histogram’) ylabel(’Number of pixels’)

252 253

% = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = %

254

% THRESHOLDING

255

% = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = %

256 257

% Find pixels whit values within specified range

258

white=find(label>=lower_boundary & label=lower_boundary & label

Quantification and Modeling of Deforamtion Processes ... - heim.ifi.uio.no

Quantification and Modeling of Deforamtion Processes ... - heim.ifi.uio.no

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Modeling and quantification of security attributes of ... - CiteSeerX

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