Numerical simulation of magma ascent by dykes and crust formation at spreading centres [Elektronische Ressource] / vorgelegt von Daniela Kerstin Kühn
178 Pages
English
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Numerical simulation of magma ascent by dykes and crust formation at spreading centres [Elektronische Ressource] / vorgelegt von Daniela Kerstin Kühn

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Learn all about the services we offer
178 Pages
English

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Numerical simulation of magma ascent bydykes and crust formation at spreadingcentresDissertationzur Erlangung des Doktorgradesder Naturwissenschaften im FachbereichGeowissenschaftender Universit¨at Hamburgvorgelegt vonDaniela Kerstin K¨uhnausBerlin SteglitzHamburg2005Als Dissertation angenommen vom Fachbereich Geowissenschaften der Universit¨atHamburgaufgrund der Gutachten von Prof. Dr. Torsten Dahmund Prof. Dr. Frank RothHamburg, den 10. Mai 2005Professor Dr. H. Schleicher(Dekan des Fachbereichs Geowissenschaften)ContentsList of Figures iiiNotation viiAbstract 11 Introduction 52 Elasticity and fracturing 132.1 Linear elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Fracturing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.1 Crack growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.2 Wholesale propagation of fractures . . . . . . . . . . . . . . . . . . . 243 Numerical method 273.1 Boundary element method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.1.1 The problem of a pressurised line crack . . . . . . . . . . . . . . . . 313.2 Combination of theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.3 Modifications to boundary element approach . . . . . . . . . . . . . . . . . 353.3.1 Fluid-filled fracture growth . . . . . . . . . . . . . . . . . . . . . . . 373.3.2 Fluid-filled fracture propagation . . . . . . . . . . .

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Published 01 January 2005
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Numerical simulation of magma ascent by
dykes and crust formation at spreading
centres
Dissertation
zur Erlangung des Doktorgrades
der Naturwissenschaften im Fachbereich
Geowissenschaften
der Universit¨at Hamburg
vorgelegt von
Daniela Kerstin K¨uhn
aus
Berlin Steglitz
Hamburg
2005Als Dissertation angenommen vom Fachbereich Geowissenschaften der Universit¨at
Hamburg
aufgrund der Gutachten von Prof. Dr. Torsten Dahm
und Prof. Dr. Frank Roth
Hamburg, den 10. Mai 2005
Professor Dr. H. Schleicher
(Dekan des Fachbereichs Geowissenschaften)Contents
List of Figures iii
Notation vii
Abstract 1
1 Introduction 5
2 Elasticity and fracturing 13
2.1 Linear elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Fracturing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.1 Crack growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.2 Wholesale propagation of fractures . . . . . . . . . . . . . . . . . . . 24
3 Numerical method 27
3.1 Boundary element method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1.1 The problem of a pressurised line crack . . . . . . . . . . . . . . . . 31
3.2 Combination of theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 Modifications to boundary element approach . . . . . . . . . . . . . . . . . 35
3.3.1 Fluid-filled fracture growth . . . . . . . . . . . . . . . . . . . . . . . 37
3.3.2 Fluid-filled fracture propagation . . . . . . . . . . . . . . . . . . . . 41
3.3.3 Simulation of dyke interaction . . . . . . . . . . . . . . . . . . . . . 43
4 Magma ascent in the mantle 47
4.1 Observations and constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2.1 Corner flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2.2 Dynamic pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2.3 Deviatoric stress field . . . . . . . . . . . . . . . . . . . . . . . . . . 58
iii CONTENTS
4.3 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3.1 Fracturing of the mantle . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3.2 Porous flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3.3 Focussed flow in dunite channels . . . . . . . . . . . . . . . . . . . . 72
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.4.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5 Modelling of oceanic crust formation 81
5.1 Thermodynamic and viscoelastic time ranges . . . . . . . . . . . . . . . . . 84
5.2 Dyke-dyke interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.3 Oceanic crust at different spreading rates . . . . . . . . . . . . . . . . . . . 90
5.3.1 Open questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.4 Numerical models of dyke interaction . . . . . . . . . . . . . . . . . . . . . . 97
5.4.1 Geological Interpretation . . . . . . . . . . . . . . . . . . . . . . . . 104
5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6 Conclusions 129
Bibliography 133
A Extended explanations 141
B Fortran subroutines and input file 157
Acknowledgments 163List of Figures
2.1 Plane strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 Plane stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Simple and pure shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Comparison of blocks undergoing simple and pure shear . . . . . . . . . . . 19
2.6 Griffith crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.7 Critical fracture length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.8 Fluid-filled fracture positioned at depth h within a plate . . . . . . . . . . . 25
2.9 Displacements of cracks with increasing half length a . . . . . . . . . . . . . 26
3.1 Difference between FD/FE method and boundary element method . . . . . 28
3.2 Interior and exterior problem . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3 Displacement discontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4 Segmentation of a fracture into N boundary elements . . . . . . . . . . . . 31
3.5 Segmentation of a fracture with arbitrary orientation . . . . . . . . . . . . . 32
3.6 Coordinate transformation from global to local coordinates . . . . . . . . . 34
3.7 Opening of fluid-filled fracture during ascent . . . . . . . . . . . . . . . . . . 36
3.8 Fracture discretised by segments of length δa . . . . . . . . . . . . . . . . . 36
3.9 Displacements at crack surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.10 Introducing lithostatic and hydrostatic pressure gradients . . . . . . . . . . 39
3.11 Displacement and stress field due to normal traction . . . . . . . . . . . . . 40
3.12 Displacement and stress field due to shear traction . . . . . . . . . . . . . . 41
3.13 Air-filled dyke in gelatine . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.14 Fracture propagation path . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.15 Program flow chart for static simulations . . . . . . . . . . . . . . . . . . . 44
3.16 Program flow chart for quasi-static simulations . . . . . . . . . . . . . . . . 45
3.17 Program flow chart for dyke interaction . . . . . . . . . . . . . . . . . . . . 46
iiiiv LIST OF FIGURES
4.1 Distinct layers of a typical oceanic crust . . . . . . . . . . . . . . . . . . . . 49
4.2 Schematic cross-section of the East Pacific Rise . . . . . . . . . . . . . . . . 50
4.3 Distinction between different flow regimes beneath mid-ocean ridges . . . . 52
4.4 Geometry of the corner flow model and resulting stream lines . . . . . . . . 56
4.5 Dynamic pressure contour lines . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.6 Deviatoric stress field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.7 Derivation of the principal stress field . . . . . . . . . . . . . . . . . . . . . 60
4.8 Melt extraction cycle by hydraulic fracturing . . . . . . . . . . . . . . . . . 62
4.9 Effect of deviatoric stress field on fracture propagation in contrast to the
effect of buoyancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.10 Impact of dynamic pressure gradient and deviatoric stress field . . . . . . . 66
4.11 Focussing of porous melt flow towards the mid-ocean ridge. . . . . . . . . . 69
4.12 Comparison of propagation paths of fluid-filled fractures and porous flow
lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.13 Simple shear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.14 Porosity fields in time resulting from the shear stress model . . . . . . . . . 72
4.15 Sketch of dunite conduits in mantle rock . . . . . . . . . . . . . . . . . . . . 73
4.16 Influence of a magma chamber . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.1 Regional dyke swarm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.2 Laboratory experiment of dyke interaction . . . . . . . . . . . . . . . . . . . 83
5.3 Cyclic Weertman model for formation of oceanic crust . . . . . . . . . . . . 83
5.4 Instantaneous heating of a semi-infinite half space . . . . . . . . . . . . . . 85
5.5 Solidification of a dyke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.6 Temperature profiles for solidifying dyke . . . . . . . . . . . . . . . . . . . . 87
5.7 Solidification times for dyke of width w . . . . . . . . . . . . . . . . . . . . 88
5.8 Dyke-dyke interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.9 Comparison of ridge topographies for different spreading rates . . . . . . . . 90
5.10 ”Infinite onion” model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.11 Model for oceanic crust accretion where dykes are fed by sills . . . . . . . . 95
5.12 Magma reservoir at crust-mantle boundary . . . . . . . . . . . . . . . . . . 96
5.13 Ascent of dykes from the same initial position . . . . . . . . . . . . . . . . . 98
5.14 Randomly distributed initial positions, regional stress field neglected . . . . 100
5.15 Randomly distributed initial positions, extensional stress field included . . . 101
5.16 Sector diagram for dyke ascent from randomly distributed initial positions . 102
5.17 Comparison of influence of initial positions . . . . . . . . . . . . . . . . . . 103LIST OF FIGURES v
5.18 Sector diagram for comparison of influence of initial positions . . . . . . . . 104
5.19 Study of length-dependent behaviour . . . . . . . . . . . . . . . . . . . . . . 105
5.20 Sector diagrams for study of length-dependent behaviour. . . . . . . . . . . 105
5.21 Madeira Rift Zone and Caldera de Taburiente, La Palma. . . . . . . . . . . 106
5.22 Transfer of geological settings into numerical model . . . . . . . . . . . . . . 106
5.23 Model of narrow dyke injection zone: different depths of initial positions . . 107
5.24 Ascent from reservoir at crust-mantle boundary . . . . . . . . . . . . . . . . 108
5.25 Schematic model for plate spreading . . . . . . . . . . . . . . . . . . . . . . 109
5.26 Dyke ascent from crustal magma chamber including plate drift . . . . . . . 109
5.27 Randomly distributed initial positions at depth of Moho . . . . . . . . . . . 110
5.29 Stress field resulting from randomly distributed initial positions . . . . . . . 112
5.30 Dyke cluster-induced melting . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.31 Modelling of regular spacing of volcanic edifices in central Iceland . . . . . . 115
5.32 Contemporaneous formation of sheeted dyke complex and seamounts . . . . 116
5.33 In situ basement drillholes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.34 Horizontal propagation of dykes along the spreading axis . . . . . . . . . . . 119
5.35 World stress map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.36 Interaction of magma injections, faulting and topography . . . . . . . . . . 122
5.37 Arrest of ascending dykes because of anomalous stress distribution . . . . . 123
5.38 Model for oceanic crust formation . . . . . . . . . . . . . . . . . . . . . . . 124
A.1 Stress components of the traction vectors . . . . . . . . . . . . . . . . . . . 142
A.2 Examples of stress conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 143
A.3 Elastic deformation of a specimen to illustrate elastic moduli . . . . . . . . 143
A.4 Coordinate transformation of displacement and stresses . . . . . . . . . . . 145
A.5 Errorfunction and complementary errorfunction . . . . . . . . . . . . . . . . 148
A.6 The Stefan problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
A.7 Transcendental function for the determination of λ . . . . . . . . . . . . . 1492
A.8 Influence of a large dyke or sill on succeeding, smaller dykes . . . . . . . . . 150
A.9 More realistic model of dyke ascent . . . . . . . . . . . . . . . . . . . . . . . 151
A.10 Influence of dyke length on ascent paths . . . . . . . . . . . . . . . . . . . . 151
A.11 Model runs with different sequences of dyke ascent . . . . . . . . . . . . . . 153
A.12 Influence of free ocean bottom interface on dyke propagation paths . . . . . 154
A.13 Influence of free ocean bottom interface on displacement field . . . . . . . . 155
A.14 Vertical displacement at ocean bottom . . . . . . . . . . . . . . . . . . . . . 155vii
Notation
1 2Variable Meaning Dimension
Latin
2A ,A fracture area m0 1
ij ij ij ij ijA ;A ,A ,A ,A boundary influence coeff. for stress Pa/mnn ss ns sn
ij ij ij ij ijB ;B ,B ,B ,B boundary influence coeff. for displacement nonenn ss ns sn
B dislocation density function none
a vertical fracture half length m
a critical fracture half length mc
a grain size mg
C boundary of region of interest m
c specific heat kJ/kg K
D displacement m
D ;D ,D ;D ,D displacement discontinuity mi x z n s
d cross section m
E Young’s modulus Pa
e~, e~, e~ unit vectors in Cartesian coordinate system nonex y z
~F,F body force Ni
f function of specified variable
3G Griffith force Pa m
2~g,g gravity acceleration m/s
h depth m
i,j counter
K bulk modulus Pa√
3K ,K ,K stress intensity factors N/ mI II III √
3K critical stress intensity factor N/ mc √
3K plane strain fracture toughness N/ mIc
2K permeability tensor m
2k permeability m
2k permeability at porosity φ m0 0
2k ,k principal permeabilities maa bb
1Names of variables are the same for all chapters. As far as possible, I tried to avoid using the same
symbol with different meanings; in the few cases I failed to do so, the meaning of the variable is clear from
the context.
2Please note that dimensions are given in 3-Dviii NOTATION
k thermal conductivity W/m Kt
L latent heat of fusion kJ/K
L characteristic length in porous melt flow models md
l length m
l horizontal fracture length m
N number of boundary elements none
~n unit vector normal to surface element none
P,p pressure Pa
P ,P lithostatic pressure caused by fluid/solid Paf s
p constant pressure Pa0
P dynamic pressure Pa
Q heat W
2q heat flux W/m
2R region of interest m
r,e~ radial distance (polar coordinates) mr
S surface energy J
T extensional stress Pa
T temperature K
T temperature at time t=0 K0
T temperature at dyke-rock boundary Kb
T melt temperature Km
T surface temperature Ks
t time s
t solidification time ss
~t traction Pa
U strain energy J
U work energy of external load Jex
U elastic energy Je
3V volume m
~u,u ;u ,u ,u ,u ,u displacement mi x y z n s
~u,u particle motion mi
~v,v velocity m/si
v Darcy velocity m/sD
v ,v velocity of fluid/solid m/sf s
v ,v velocity in direction of Cartesian coordinates x,y m/sx y
v ,v velocity in radial/angular direction m/sr θ
w fracture width m
w separation velocity between melt and matrix m/s0
~x,x ;x,y,z Cartesian coordinates mi
x time-dependent position of solidification boundary mm
z depth of asthenosphere m0