exploring conceptual geodynamic models€¦ · introduction 1.1 preamble the mainstream role of...

Exploring Conceptual Geodynamic Models

Numerical Method and Application to Tectonics and Fluid Flow

Christopher P. Wijns, B.Sc., M.Sc.

Submitted for the degree of Doctor of Philosophy of Geophysicsat the University of Western Australia

School of Earth and Geographical Sciences

September 2004

This page is blank.

DECLARATION

This dissertation is my own composition. All co-authors involved in previous publications

have read and given their consent for inclusion of the relevant material. Any contributions from

co-authors are clearly detailed in section 1.5 of the Introduction.

Chris Wijns Date

Prof. David Groves (principal supervisor) Date

5

ABSTRACT

Geodynamic modelling, via computer simulations, offers an easily controllable method for

investigating the behaviour of an Earth system and providing feedback to conceptual models of

geological evolution. However, most available computer codes have been developed for engi-

neering or hydrological applications, where strains are small and post-failure deformation is not

studied. Such codes cannot simultaneously model large deformation and porous fluid flow. To

remedy this situation in the face of tectonic modelling, a numerical approach was developed to

incorporate porous fluid flow into an existing high-deformation code calledEllipsis. The resulting

software, with these twin capabilities, simulates the evolution of highly deformed tectonic regimes

where fluid flow is important, such as in mineral provinces.

A realistic description of deformation depends on the accurate characterisation of material

properties and the laws governing material behaviour. Aside from the development of appropriate

physics, it can be a difficult task to find a set of model parameters, including material properties and

initial geometries, that can reproduce some conceptual target. In this context, an interactive system

for the rapid exploration of model parameter space, and for the evaluation of all model results,

replaces the traditional but time-consuming approach of finding a result via trial and error. The

visualisation of all solutions in such a search of parameter space, through simple graphical tools,

adds a new degree of understanding to the effects of variations in the parameters, the importance

of each parameter in controlling a solution, and the degree of coverage of the parameter space.

Two final applications of the software code and interactive parameter search illustrate the power

of numerical modelling within the feedback loop to field observations. In the first example, vertical

rheological contrasts between the upper and lower crust, most easily related to thermal profiles and

mineralogy, exert a greater control over the mode of crustal extension than any other parameters.

A weak lower crust promotes large fault spacing with high displacements, often overriding initial

close fault spacing, to lead eventually to metamorphic core complex formation. In the second

case, specifically tied to the history of compressional orogenies in northern Nevada, exploration

of model parameters shows that the natural reactivation of early normal faults in the Proterozoic

basement, regardless of basement topography or rheological contrasts, would explain the subse-

quent elevation and gravitationally-induced thrusting of sedimentary layers over the Carlin gold

trend, providing pathways and ponding sites for mineral-bearing fluids.

CONTENTS

List of Figures v

List of Tables vii

Acknowledgements ix

1 INTRODUCTION 1

1.1 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Aims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4 Terminology and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.5 Published Content of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.6 Digital Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 PHYSICAL AND NUMERICAL MODEL 7

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.2 Code Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 Momentum Balance. . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.2 Constitutive Law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.3 Mass Conservation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.4 Yield Law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.5 Pore Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Permeability Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.1 Cubic Lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.2 Kozeny-Carmen Model. . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.3 Dilation Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.4 Permeability Tensor Rotation. . . . . . . . . . . . . . . . . . . . . . . 15

2.4 Physical Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5 Scaling Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.6 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.6.1 Pore Pressure Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . 17

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CONTENTS Wijns Ph.D. Thesis

2.6.2 Pore Pressure Gradient in Momentum Equation. . . . . . . . . . . . . . 19

2.6.3 Darcy Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.6.4 Interface Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.7 Analytic Solution Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.7.1 One-Dimensional Compaction of a Porous Halfspace. . . . . . . . . . . 21

2.7.2 One-Dimensional Consolidation of a Porous Halfspace. . . . . . . . . . 23

2.8 Miscellaneous Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.8.1 Deflection of Flow Field Due to Permeability Contrasts. . . . . . . . . . 25

2.8.2 Rotation of Anisotropic Permeability. . . . . . . . . . . . . . . . . . . 26

2.9 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 INTERACTIVE INVERSE MODELLING 27

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3.1 Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3.2 Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4 MODEL SPACE VISUALISATION 39

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2 Target, Model, and Inversion Results . . . . . . . . . . . . . . . . . . . . . . . . 40

4.3 Parallel Axis Display . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.3.2 Visualisation of the Inversion Data. . . . . . . . . . . . . . . . . . . . . 43

4.4 Multi-Dimensional Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.4.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.4.2 Projections of the Inversion Data. . . . . . . . . . . . . . . . . . . . . . 45

4.5 Self-Organising Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.5.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.5.2 Visualisation of the Inversion Data. . . . . . . . . . . . . . . . . . . . . 49

4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.7 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5 MODES OF CRUSTAL EXTENSION 55

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.3.1 Constant Viscosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.3.2 Temperature-Dependent Viscosity. . . . . . . . . . . . . . . . . . . . . 59

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Wijns Ph.D. Thesis CONTENTS

5.3.3 Field Validation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.4.1 Continuum Between Modes. . . . . . . . . . . . . . . . . . . . . . . . 62

5.4.2 Comparison with Other Modelling and with Nature. . . . . . . . . . . . 65

5.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6 CRUSTAL-SCALE FLUID FLOW: CARLIN SYSTEM 69

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.2 Geological Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.3 Numerical Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.3.1 Whole crust tectonics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.3.2 Upper crust tectonics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.3.3 Fluid flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.3.4 Local scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

7 CONCLUSION 79

REFERENCES 81

A Pore Pressure Approximation 91

B Mesh Dependence of Model Results 93

C Crustal Viscosity Profile 95

C.1 Viscosity at Brittle to Ductile Transition . . . . . . . . . . . . . . . . . . . . . . 95

C.2 Integrated Crustal Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

D Digital Appendix on CDROM

iii

LIST OF FIGURES

2.1 Strain weakening function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Permeability – strain relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Ellipsisflowchart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4 Interface boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.5 1D dry compaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.6 1D saturated compaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.7 1D saturated consolidation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.8 Flow deflection due to permeability contrasts . . . . . . . . . . . . . . . . . . . 25

2.9 Anisotropic permeability rotation . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.1 Inversion target and initial geometry . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Inversion generation 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32



4.1 Target, initial geometry, and inversion results . . . . . . . . . . . . . . . . . . . 41

4.2 Sample parallel axis display . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.3 Parallel axis display of inversion data . . . . . . . . . . . . . . . . . . . . . . . 43

4.4 Simulation rank versus generation . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.5 Variable parameters versus rank . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.6 2D crossplots with rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.7 4D projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.8 Sample SOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.9 SOM of inversion data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.10 Simulations mapped on to SOM . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.11 Individual parameter SOM plots . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.12 SOM vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.13 Six variations on the SOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.1 Initial extension model and schematic strength profile . . . . . . . . . . . . . . . 56

5.2 Constant viscosity extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.3 Metamorphic core complex mode with temperature dependent viscosity . . . . . 60

5.4 Metamorphic core complex mode with strong faults . . . . . . . . . . . . . . . . 60

v

LIST OF FIGURES Wijns Ph.D. Thesis

5.5 Cooling curves for exhumed lower crust . . . . . . . . . . . . . . . . . . . . . . 61

5.6 Endmember faulting behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.7 Number of faults versusrτ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.8 Fault spacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6.1 Tectonic timeline for western U.S. . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.2 Gravity and magnetic images over northern Nevada . . . . . . . . . . . . . . . . 71

6.3 Carlin conceptual model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.4 Carlin initial models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6.5 Effects of rheology and fault strength . . . . . . . . . . . . . . . . . . . . . . . . 74

6.6 Upper crust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.7 Upper crust with low basement step . . . . . . . . . . . . . . . . . . . . . . . . 75

6.8 Upper crust with low basement step . . . . . . . . . . . . . . . . . . . . . . . . 76

6.9 Local-scale folding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

A.1 Hydrostatic pore pressure approximation test . . . . . . . . . . . . . . . . . . . 91

B.1 Mesh and accuracy dependence of model results . . . . . . . . . . . . . . . . . . 93

vi

LIST OF TABLES

1.1 Mathematical notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3.1 Variable model parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.1 Variable model parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.1 Natural values for extension model parameters. . . . . . . . . . . . . . . . . . . 57

6.1 Natural values for Nevada model parameters. . . . . . . . . . . . . . . . . . . . 72

vii

ACKNOWLEDGEMENTS

Some of the work in this dissertation began during my tenure as an invited researcher at the

University of Western Australia (UWA) and the Commonwealth Scientific and Industrial Research

Organisation (CSIRO) of Australia. During this time, I was financially supported by an initial grant

from Prof. David Groves, and a living stipend from Dr. Alison Ord.

My doctoral work was supported by a scholarship funded jointly between UWA and CSIRO

(WACUP Scholarship), and, soon after my enrolment, I began to receive a top-up scholarship

from the Predictive Mineral Discovery Cooperative Research Centre (pmdCRC). Placer Dome

Asia Pacific Ltd. provided partial tuition funding during my Ph.D., and Mr. Greg Hall and Dr.

John Muntean were specifically involved in directing the modelling of the Carlin gold trend.

I received additional travel funding for conferences from the J.H. Lord Foundation, the Aus-

tralian Institute of Geoscientists, the European Union Young Scientists Fund, Placer Dome, and

the Society of Exploration Geologists.

My thanks go to my primary supervisors, Prof. David Groves and Dr. Alison Ord, for scientific

instruction, financial support, and, not least, for administrative support in securing me funding,

travel opportunities, and the attendant exposure to a world-wide community of geological science.

Valuable day-to-day scientific mentoring came from two sources: Dr. Louis Moresi, who left

CSIRO shortly after I began my Ph.D., and Dr. Roberto Weinberg at UWA, who also left halfway

through my studies. The time I did spend with Dr. Weinberg convinced me to quickly nominate

him as my third supervisor, and both he and Dr. Moresi continued their involvement in my research

efforts.

Drs. Boschetti and Klaus Gessner, at CSIRO, were my closest scientific collaborators during

the latter part of my Ph.D., for, respectively, interactive inversion and the extensional tectonic

modelling. Mr. Thomas Poulet, applied mathematician at CSIRO, helped me to learn much about

self-organising maps and several other data visualisation techniques.

My final and undoubtedly most pressing thanks go to my wife Jennifer, who patiently (most

of the time) put up with my irregular work hours and numerous missed dinners, while helping to

support my Ph.D.

ix

Chapter 1

INTRODUCTION

1.1 Preamble

The mainstream role of numerical modelling in mineral exploration today is forward and in-

verse modelling of potential field and electromagnetic data, and, to a much lesser degree, seismic

inversion. This type of modelling is relatively mature, but belongs to the realm of direct detection

of ore-bearing or ore-related geology. Exploration interest also encompasses the understanding of

physical and chemical processes in ore deposit formation, reflected in geological and laboratory

geochemical research. The recent entry of geophysics into this aspect of exploration is through

numerical modelling of physical and chemical processes involved in deformation and fluid flow.

The understanding of processes underpins direct geological, geochemical, and geophysical ob-

servation. It supplements empirical knowledge for the evaluation of mineral potential. Numerical

modelling of rock deformation, thermal effects, chemical reactions, and fluid flow contributes by

adding to this store of knowledge beyond what field and laboratory studies can deliver. Conceptual

models can be tested for feasibility, many variations can be modelled by altering input parameters,

specific questions can be answered in relation to the geological evolution of a model, and, most

importantly, results can be suggested that lie outside the envelope of the geoscientist’s initial ideas.

The focus in this thesis on the tectonic evolution of the crust is the culmination of numerical

code development (Chapter 2) linked to an interactive system for exploring model space (Chap-

ters 3 and 4). This “exploration system” for geodynamic modelling provides the environment for

research into extensional tectonics (Chapter 5), and the compressional evolution of the Carlin gold

trend in northern Nevada, U.S.A. (Chapter 6).

1.2 Aims

The purpose of developing and applying numerical process modelling to hypothetical and ac-

tual field situations is to predict the behaviour of an Earth system; in relation to mineral explo-

ration, it is to aid in targeting prospective areas. The scope of modelling in this dissertation is

tectonic deformation and associated fluid flow at crustal scales. In all cases, the aim is to deter-

mine, through intelligent exploration of model parameter space, the initial geometries and physical

properties that produce some hypothesised behaviour of the system.

The predictive capabilities of numerical process modelling are limited by two main factors:

“distance” from the initial to the final model state, and accuracy of the model description. The first

point refers to the simple fact that errors accumulate as calculations accumulate. Compounding

1

CHAPTER 1. INTRODUCTION Wijns Ph.D. Thesis

this, the evolution of a simulation is often deterministically chaotic, so that, with increasing time,

or other factors such as deformation, small differences in the initial state may lead to large differ-

ences in the final state. Many simulations should thus be run, with variations on model parameters,

before drawing conclusions about characteristic outcomes. Accuracy, the second factor, refers to

the level and consistency of input detail with respect to the desired outcome, in terms of geometry,

material properties, rheological descriptions, etc. An investigation of fluid flow through first- and

second-order faults, for example, depends on having an accurate representation of all faults with

influence on the chosen scale. In other words, the quality of output depends on the quality of

input. The input – for example, mapped faults – is often sparse, which may lead to output that,

although correct with respect to the model, is incorrect with respect to the actual field situation,

and therefore misleading for site-specific analysis such as in exploration.

It is crucial to understand the accuracy limit in numerical modelling. It decides the degree

to which a model is useful as a predictive tool in a field situation, versus an empirical tool for

understanding field observations. Returning to the above example of fluid flow through faults, a

model that includes all relevant second-order faults can be used to predict fluid flux in a mineral

district. A model that is incomplete can be used to predict fluid flux in relation to the geometries

and permeabilities of the faults in the model, that is, generic information can be compared to

similar field observations, and conclusions drawn in context. A model that relates fault spacing to

rock rheology (Chapter 5) is akin to chemical titration experiments that provide reaction products

based on chemical components. Such information does not locate drill holes in the case of mineral

exploration, but is used empirically by an explorationist in deciphering field observations.

High-deformation numerical modelling, of the kind undertaken in this dissertation, might never

be able to predict the full complexity of the real Earth, where the true heterogeneities are unknown.

Modelling should provide an envelope of behaviour, and this should be checked within the scope

of initial condition perturbations. In light of the above discussion, the numerical results in this

thesis should be considered as empirical aids, albeit based on sound physics, which can provide

new insight into the behaviour of Earth systems and their controlling parameters.

1.3 Methodology

Field observations provide a basis for constructing conceptual models of Earth systems. Nu-

merical models act as laboratories for testing the conceptual models, and, by comparing the results

to the original observations, the conceptual models can be validated or discredited. An iterative

process ensues whereby numerical results act as feedback to field interpretations, which can be

used to construct new hypotheses for numerical testing.

The modelling carried out for this dissertation can be considered the numerical equivalent of

sandbox (or sand and putty) modelling, calledanaloguemodelling. Numerical modelling is poised

to surpass analogue modelling as an investigative technique within the next decade. Analogue

modelling still has the advantage of resolution at fine scales, but, with increasing computer power,

this will disappear. Supercomputers today do allow equal detail, but they are not in common use

2

Wijns Ph.D. Thesis CHAPTER 1. INTRODUCTION

for geological modelling. Analogue models do not have the control over material behaviour that

numerical models have. Although some researchers take great pains in using well-documented ma-

terials and experimental conditions, the practice of mixing analogue materials may lead to proper-

ties that are poorly known outside strict limits of temperature, strain, and strain rate. Furthermore,

ambient conditions such as temperature are sometimes not controlled – for example, lighting over

the experimental setup – which leads to unconstrained material behaviour. Numerical models have

the distinct advantage of absolute control and the possibility of continuous observation at every

point. The weakness of present-day numerical modelling lies in the simplistic constitutive models

used to represent Earth materials. These do not allow the same wealth of deformation styles seen

in analogue models. This is the most serious impediment, as computer power for model resolution

will progress independently of the geosciences.

As indicated above, the strength and duty of numerical modelling is to investigate many vari-

ations on a hypothesis and to delimit the physical parameter space, and the resulting processes,

that control the behaviour of an Earth system. Variation of parameters by trial and error is the

traditional, but inefficient, method for exploring parameter space. Chapters 3 and 4 explain a new

method of interactive modelling that is an efficient system for exploring some target behaviour of

a system. Through various methods of visualising the resulting parameter space, a user can de-

termine limits on parameter values and draw conclusions about controlling factors. A unique and

powerful system for exploring conceptual geodynamic models results from coupling a computer

simulation code to the interactive technology.

1.4 Terminology and Notation

The wordmodelis used throughout this thesis to refer to a particular representation of nature

by mathematical means. This would include, for example, the underlying equations, initial ge-

ometries, material properties, and imposed deformations. The concept of a single model includes

variations of the above components, which are parameterised. Thus, one model gives rise to a

family of results,simulations, or solutions.

Mathematical notation follows conventional norms (e.g., Arfken, 1985). Table 1.1 contains a

list and explanation of mathematical symbols used throughout the text, including the subscript

notation known asindex notationor Einstein notation, which is a compact notation used for vector

and matrix operations.

1.5 Published Content of Thesis

Disregarding the editing required for continuity of the thesis, and revisions made in the course

of new learning, some of the following chapters have been published or submitted, in whole or

in part, as journal articles. Coauthors did not perform the modelling, nor write the articles: their

contributions are summarised explicitly for each article.

3

CHAPTER 1. INTRODUCTION Wijns Ph.D. Thesis

Symbol Explanation

b scalar

b vector

bi componenti of vector

b tensor (matrix)

bij componentij of tensor

xi unit vectors defining a coordinate system (e.g.,i=1,2,3 in 3D)ddx complete derivative with respect to variablex∂∂x partial derivative with respect to variablex

a · b scalar producta1b1 + a2b2 + ... + anbn (in nD)

∆b change in quantityb

∇ “del” operator (vector)[

∂∂x1

∂∂x2

... ∂∂xn

](in nD)

∇b gradient ofb

∇ · b divergence ofb

b,j∂b∂xj

for all values ofj (index notation)

bi,j∂bi∂xj

for all values ofi, j (index notation) (e.g.,bi,i = ∇ · b)

bij,j∂bij

∂xjfor all values ofi, j (index notation)

δij 1 if i=j, 0 otherwise (“Dirac delta”)

X ⇒ Y X impliesY

a → b a tends towardsb

a ≡ b a is equivalent tob

a ≈ b a is approximately equal tob

a ¿ b a is much less thanb

f(b) function ofb

|b| absolute value ofb∑ni=1 bi sum of allbi over valuesi=1 ton

Table 1.1: Mathematical symbols and notation used in the text.

Chapter 3 is published as:

Wijns, C., Boschetti, F., Moresi, L., 2003. Inverse modelling in geology by interactive evolution-

ary computation. J. Struct. Geol. 25 (10), 1615–1621.

F. Boschetti pioneered the use of this interactive method in the geosciences, and contributed to the

text of the Introduction and Method sections. L. Moresi was involved in discussions.

Chapter 4, apart from an expanded introduction and the sections on parallel axis display and multi-

dimensional projections, is published as:

Boschetti, F., Wijns, C., Moresi, L., 2003. Effective exploration and visualisation of geological

parameter space. Geochem. Geophys. Geosys. 4 (10), 1086.

4

Wijns Ph.D. Thesis CHAPTER 1. INTRODUCTION

Although he did not perform most of the work, F. Boschetti is first author because of his idea to

use a self-organising map (SOM) in this context. He produced only the raw graphical SOM output

(no figures) and portions of the text on SOM theory and visualisation.

Chapter 5 is submitted as:

Wijns, C., Weinberg, R., Gessner, K., Moresi, L. Mode of crustal extension determined by rheo-

logical layering. Earth Planet. Sci. Lett., submitted April 2004.

No coauthor contributed any text or figures, but all three coauthors participated in geological dis-

cussions of extensional tectonics, and provided reviews of the manuscript, with R. Weinberg most

heavily involved.

The results of Chapter 6 will be submitted as a paper to an economic geology journal. To date,

none of the eventual coauthors have provided anything more than geological discussion and ideas

for modelling.

1.6 Digital Appendix

A CDROM is included as a digital appendix to this disseration. It contains some code tests

from Chapter 2, the complete sequence of generations for the interactive inverse modelling of

Chapter 3, and selected animations of some simulations for Chapters 5 and 6. All publications and

abstracts related to the thesis research are also included.

5

Chapter 2

PHYSICAL AND NUMERICAL MODEL

2.1 Introduction

2.1.1 History

The predecessor to the present modelling code was a finite element computer code named

Citcom (California Institute ofTechnologyCOnvection in theMantle), developed by Dr. Louis

Moresi at the California Institute of Technology (Moresi, 1992). This code solved the equations

of viscous fluid dynamics, and, as the name implies, was geared towards modelling convection in

the Earth’s mantle. Moresi and Solomatov (1995) provide a detailed description of the multigrid

finite element algorithm. After this and during his time at CSIRO, Dr. Moresi modified the code to

include mobile integration points. The code became a particle-in-cell finite-element code named

CIItcom, which was able to track and evolve material properties on particles. Recent work at

CSIRO has resulted in a new generation of the software, namedEllipsis. The version developed

for this doctoral research incorporates porous fluid flow within the viscous matrix.

The decision to develop porous flow capabilities inEllipsis arose out of the need to fill a gap

in fluid flow modelling capabilities within the former Structural Controls on Mineralisation group

at CSIRO in Nedlands, Western Australia. Previous modelling was restricted to mechanical-fluid

flow models involving limited deformation, using finite-difference software (FLAC, Cundall and

Board, 1988), or large-deformation mechanical models using a particle code (PFC, Guest and

Cundall, 1994; Itasca Consulting Group, 2003) without a porous flow component. In many cases,

FLAC is sufficient for the problems at hand, but modelling of intense folding and thrusting, large

extension, and subduction, with the resultant fluid flow, is beyond the capabilities of this software

at the present time. The incorporation of pore fluid flow into a code allowing unlimited deforma-

tion (Ellipsis) provides a new modelling capability for research, with direct applications to mineral

and petroleum exploration.

2.1.2 Code Overview

Ellipsis is used to model crustal geodynamics by treating rocks as viscous fluids over long time

scales. This thesis does not describe the particulars of the various algorithms in the code, except

those that relate directly to the modification to porous flow. Details of the otherEllipsisalgorithms

are in Moresi et al. (2001, 2002).

The governing equations, solved by a finite-element scheme, are those for a coupled system of

mechanical deformation, porous flow, and thermal transport. The role of particles in the code is

7

CHAPTER 2. PHYSICAL AND NUMERICAL MODEL Wijns Ph.D. Thesis

that of integration points over an element, during the construction of the finite-element matrices.

Thus, the accuracy of the solution depends on the accuracy of this integration scheme and the

ability of the chosen shape functions to represent the variation of quantities across an element.

Bilinear shape functions are used over quadrilateral elements. Physical quantities are attached

to these integration points, or particles, such as material density, strain rate, and porosity. In

particular, the particles may carry time-integrated quantities such as accumulated strain, which

allows history-dependent material behaviour to be modelled. The fact that these time-integrated

quantities are not stored at nodal mesh points means that the underlying computational grid does

not have to deform with the material. There is no limit to the amount of deformation that can

be modelled, as the grid will not become tangled or otherwise distorted beyond the realm of

computational feasibility. An additional advantage is not having to remesh upon large deformation.

Particle positions are updated at the end of every time step based on the calculated nodal velocities.

2.2 Governing Equations

The system under study is that of a fluid-saturated porous matrix. Although the matrix repre-

sents a solid, such as the crust of the Earth, on the time scales involved, it behaves as a viscous

fluid. This assumption is justified by a comparison of typical viscous and elastic parameters for

the crust. The ratio of viscous to elastic shear modulus, which is the ratio of viscosityη to (ap-

proximately) Young’s modulusE, defines the Maxwell time

τMaxwell =η

E.

In problems with a timescale greater than the Maxwell time, the crust can be considered viscous.

For a (lower) crust viscosity of 1021 Pa·s, andE varying between 10 and 100 GPa (Goodman,

1989; Schultz, 1996), the Maxwell time varies from 300 to 3000 years. The scaled time steps in

the simulations of Chapters 5 and 6 are 7500 and 30,000 years, respectively. The Maxwell time

is thus better respected in Chapter 6. A viscoelastic model would be more appropriate for many

problems in geodynamics.

The viscous fluid that is the crust may also deform plastically, that is, it fails at sufficiently high

shear stresses. The pore fluid is single-phase, and considered inviscid except to the extent that the

viscosity controls the pore fluid velocity through Darcy’s law. The dynamics of the system are

governed by the conservation of momentum, mass, and energy. The effect of the pore pressure is

coupled into the stress state of the matrix, and the temperature feeds back into the rheology. In

the following development, the matrix is referred to as the solid, in order to distinguish it from the

fluid that fills the pores. As such, any variable with the superscripts is associated with the matrix,

and the superscriptp denotes the pore fluid.

The physics that describe the flow of a viscous fluid are generally known as the Navier-Stokes

equations, after the independent work of the French engineer Claude-Louis Navier, in 1822, and

the Irish mathematician George Stokes, in 1845 (Lamb, 1895, p. 515). The full Navier-Stokes

equations are difficult to solve, but can be considerably simplified for Earth systems that describe

the slow flow of geological materials. The addition of the pore component results in a two-phase

8

Wijns Ph.D. Thesis CHAPTER 2. PHYSICAL AND NUMERICAL MODEL

fluid dynamic system, described for geophysical applications by, among others, McKenzie (1984),

Scott and Stevenson (1984), and Bercovici et al. (2001).

2.2.1 Momentum Balance

The complete equation for the conservation of momentum of a representative elementary vol-

umeV of material with densityρ, as written by McKenzie (1984), is

d

dt

∫

VρvidV =

∫

VρgidV +

∫

VIidV +

∫

SσijnjdS −

∫

SρvivjnjdS .

In the order of the above terms, the time rate of change of momentum depends on body forces,

interaction with other materials, surface forces, and the advection of momentum. The velocity of

the volume isv, g is the acceleration due to gravity,I is an interaction force per unit volume,

andσ is the stress tensor over the surfaceS of the volume.S has a unit normaln. The material

velocities involved in this problem, as well as their accelerations, are sufficiently small for both

the time rate of change and the advection of momentum to be neglected. The above equation, for

the combined system of matrix and pore fluid, reduces to the simplified form

∂

∂xjσT

ij + ρbgi = 0 ,

where the divergence of the total stressσT is balanced by the body force due to the bulk density

ρb of the material. This bulk density depends on the porosityφ according to

ρb = (1− φ)ρs + φρp .

The total stress is a combination of the matrix and pore fluid stresses:

σTij = (1− φ)σs

ij + φσpij

= (1− φ)σsij − φppδij ,

wherepp is the pore pressure andδij is the Dirac delta function described in section 1.4. The

(inviscid) pore fluid does not support any shear stress.

In a porous medium, the pressure of the pore fluid modifies the stress regime. Biot (1941)

proposes the following relation to describe the effective stressσe due to the presence of a pore

pressure:

σeij = σs

ij + αppδij .

The parameterα is meant to account for grain compressibility. In the most general of cases,

this may be anisotropic andα would be a tensor. For the following development,α = 1, which

corresponds to incompressibility on a grain scale. Through this simplification, it is understood that

compressibility due to pore space dominates the system.

Since it is the effective stress that is related to the matrix strain rate (see below), the total stress

must be recast in terms of the known quantitiesσe andpp.

σTij = (1− φ)

(σe

ij − ppδij

)− φppδij

= (1− φ)σeij − ppδij .

9


The equation for the conservation of momentum is then

∂

∂xj

[(1− φ)σe

ij

]− ∂pp

∂xi+ [(1− φ)ρs + φρp] gi = 0 . (2.1)

2.2.2 Constitutive Law

The derivation of mass conservation, in the next section, depends on the materialconstitutive

law, the equation that describes the strain in response to applied stress. A porous material is

considered compressible even though the solid material itself may be incompressible, as in the

present case. Compaction or dilation of the pore space leads to a bulk volume change of the solid.

For this reason, a compressible formulation is required for the constitutive law of the Newtonian

viscous fluid that represents the solid. At high stresses, the solid deforms in a non-viscous manner,

described later, but at low stresses, there exists a linear relationship between stress and strain rate.

The bulk viscosityζ and the shear viscosityη of the solid matrix relate the effective stress to the

velocity according to (McKenzie, 1984)

σeij = ζ∇ · vsδij + η

(∂vs

i

∂xj+

∂vsj

∂xi− 2

3∇ · vsδij

)(2.2)

in three dimensions. In two dimensions the factor2/3 becomes unity.

The bulk viscosityζ depends on porosity as described in the next section. The shear viscosityη

incorporates the effect of the porosity throughη = (1− φ)ηm, whereηm is the true (non-porous)

material parameter. The temperature dependenceηm = ηoe−cT , whereT is temperature, is a

feature of the original code, and is used and explained more fully in section 5.2.

The stress decomposes into the deviatoric partτ and the isotropic partq, which incorporates

the solid pressureps.

σeij = τ e

ij − qδij = 2ηDij − qδij , (2.3)

whereD is the deformation tensor. In the incompressible case,

Dij =12

(∂vs

i

∂xj+

∂vsj

∂xi

)

has zero trace (the diagonal tensor components add to zero), andq becomes the true solid thermo-

dynamic pressure.

Compressible formulation 1. In order to adopt the same formulation for the compressible case,

D is replaced by

D∗ij = Dij − 1

3Dkkδij = Dij − 1

3∇ · vsδij

so thatD∗ has zero trace. In two dimensions, the factor1/3 becomes1/2. The stress decomposi-

tion

σeij = 2ηD∗

ij − psδij (2.4)

is now equally applicable to both compressible and incompressible cases, since in the limit of

incompressibility,

∇ · vs → 0 ⇒ D∗ → D .

10


Comparing the stress decomposition (2.4) with the constitutive equation (2.2), the solid pressure

is given by−ζ∇ · vs. In the incompressible case,ζ → ∞ as∇ · vs → 0, and this term becomes

the true thermodynamic pressure. This formulation meets with limited numerical success in the

code, and has therefore been superceded by the next case.

Compressible formulation 2. The following approach continues to use the incompressible form

of D. The compressible part, reflected in the divergence of the solid velocity, is contributed through

the parameterq that contains the remaining terms of the constitutive equation (2.2):

q = −(

ζ − 23η

)∇ · vs . (2.5)

This allows the original form (2.3) of the stress decomposition to be maintained. The finite element

stiffness matrix is constructed as for the incompressible case, and the effect of compressibility is

treated separately when satisfying the continuity, or mass conservation, equation.

The final form of the momentum equation (2.1), using the above stress decomposition (2.3), is

∂

∂xj[2(1− φ)ηDij ]− ∂

∂xi[(1− φ)q]− ∂pp

∂xi+ [(1− φ)ρs + φρp] gi = 0 .

2.2.3 Mass Conservation

The conservation of mass for a medium of densityρ, over the same representative elementary

volume used above, is∂

∂t

∫

VρdV = −

∫

SρvinidS +

∫

VΨdV ,

whereΨ is a volumetric source term. Disregarding any source term, the matrix of porosityφ obeys

∂

∂t[(1− φ)ρs] +

∂

∂xi[(1− φ)ρsvs

i ] = 0 .

In the present formulation, the density of the solid is constant except during the calculation of

buoyancy forces (Boussinesq approximation), so that

∂φ

∂t+ vs

i

∂φ

∂xi=

dφ

dt= (1− φ)

∂vsi

∂xi. (2.6)

The time derivatived/dt represents the material derivative, and the evolution of porosity is calcu-

lated on the moving particles.

The form of the continuity equation that is coupled to the momentum equation comes from the

constitutive law (2.2), and is apparent in the compressible formulation (2.5). Usingλ = ζ− 2/3η,

mass conservation is∂vs

i

∂xi+

q

λ= 0 . (2.7)

The only difference with the original non-porous version ofEllipsis is that the bulk viscosity now

depends on the porosity:

ζ = ζoφo

φ.

11


The system of solid plus pore fluid is a compressible system overall, although the solid itself is

considered incompressible. Thus, when all pore space has disappeared through compression of

the solid, the material becomes incompressible. The bulk viscosity, and henceλ, tends to infinity

under this condition, and the divergence of the solid velocity is zero through the above continuity

equation (2.7).

2.2.4 Yield Law

Plastic strain is the proxy for brittle failure inEllipsis. It is a feature of the original code,

but a description here is warranted because brittle failure, in the form of faulting, underpins the

results of all studies in this dissertation. Plastic strain occurs when the viscous stress (equation

2.2) reaches a specified yield stress. At such a point, the viscosity of the material is reduced in

order to respect the limiting stress. Zones of highly localised plastic strain represent faults. The

yield stressτyield may depend on pressure (depth) and previously accumulated plastic strainεp.

τyield = (co + cp p ) f(εp) , (2.8)

whereco is the cohesion, or yield stress at zero pressure, andcp is the pressure dependence of the

yield stress, equivalent to the friction coefficient in Byerlee’s law (Byerlee, 1968). The power law

functionf(εp) mimics the phenomenon ofstrain weakening, which is the progressive weakening

of a zone of plastic failure due to continued deformation.

f(εp) =

1− a (εp/εo)

n εp < εo

1− a εp ≥ εo

,

in whichεp is measured as the second invariant of the deviatoric plastic strain tensor. The roles of

the parametersa, εo, andn are illustrated graphically in Figure 2.1. The “saturation” strainεo is

the accumulated plastic strain beyond which no further weakening takes place, and, at this point,

the yield stress has been reduced by a proportiona. For example, there is no strain weakening

whena = 0. The exponentn describes the shape of the decay curve.

2.2.5 Pore Flow

The fluid filling the pores flows in response to pore pressure gradients according to Darcy’s

law (Darcy, 1856). Modified to include gravitational forces explicitly, the volumetric flow rate per

unit area of pore space, relative to the matrix, is

vpi − vs

i ≡ vpsi = −kij

φµ

(∂pp

∂xj− ρpgj

), (2.9)

in whichk is the permeability tensor andµ is the pore fluid viscosity. This is valid for laminar flow

and is thus restricted to relatively small flow velocities. Within the pore fluid mass conservation

equation∂

∂t(φρp) +

∂

∂xi(φρpvp

i ) = 0 ,

12


Figure 2.1: The strain weakening functionf(εp) is determined by the ratio1 − a of minimum to original

yield stress, the “saturation strain”εo after which there is no further weakening, and the exponentn, which

dictates the curvature of the weakening relation.

substitution of Darcy’s law in terms ofvpsi yields the following relation describing the pore pres-

sure:

− ∂

∂xi

[ρpkij

µ

(∂pp

∂xj− ρpgj

)]+ ρp∇ · vs = 0 . (2.10)

The second term represents compaction or dilation of the bulk matrix (i.e., changes in pore space),

and acts as a pressure source term. The density is not eliminated because the pore fluid cannot be

considered incompressible over the crustal scales involved in geological problems.

2.3 Permeability Models

The relationship between porosity and permeability is a field of study in itself. Permeability

models included inEllipsis have been either derived from basic geometry or else adopted from

published literature. Both porosity and strain may affect permeability.

2.3.1 Cubic Lattice

The following derived porosity-permeability law is an extension of an example by Turcotte and

Schubert (1982). A cubic lattice is composed of pores connected by cylindrical pipes of diameter

δ and lengthb. In physical terms,b would be the grain diameter andδ is a measure of the space

between grains. The scalar permeabilityk depends on flow through the connecting pipes only. The

pores at the vertices of the lattice act only to increase fluid storage capacity. In two dimensions,

the cylindrical porosityφk = 2δ/b for δ ¿ b. The 2D average channel (pipe) velocity

u = − δ2

12µ∇pp

gives the volumetrically averaged fluid flow through the lattice, or Darcy velocity

φkvps =

δu

b= − δ3

12bµ∇pp .

13


Comparing with the familiar form of Darcy’s law (2.9), the two-dimensional permeability

k2D =δ3

12b=

b2

96φ3

k .

The total porosity including the vertex pores is greater thanφk. Settingφk = αφ,

k2D =α3

96b2φ3 . (2.11)

A similar procedure in three dimensions yields

k3D =πδ4

128b2=

α2

72πb2φ2 .

2.3.2 Kozeny-Carmen Model

The Kozeny-Carmen relation (Carmen, 1956), modified by Mavko and Nur (1997), accounts

for a percolation threshold porosityφc, below which the porosity is effectively no longer connected

and the permeability becomes zero.

k =B

S2(φ− φc)

3 , (2.12)

in whichS is the specific surface area (pore surface area per volume of rock) andB is an empirical

geometric factor. For packed spheres of diameterb (Mavko and Nur, 1997),

S =3(1− φ)

2d.

This leads to

k = B(φ− φc)

3

(1 + φc − φ)2d2 ≈ B (φ− φc)

3 d2 ,

where the factor3/2 has been included inB. Two-dimensional packed spheres obeyS = 4(1 −φ)/d, which leads to the same relation as above with a different value forB. For packed cubes

(3D) with a small separation between faces,S = 6(1−φ)/d, and for squares (2D),S = 4(1−φ)/d.

2.3.3 Dilation Angle

A simple permeability increase with accumulated plastic strain mimics the effect of a dilation

angle in the material.

kij =

ko

ij (1 + ϕε) ε < εo

koij (1 + ϕεo) ε ≥ εo

, (2.13)

whereko is the initial permeability andϕ is the dilation factor (not the actual dilation angle). This

law may be modified by having the permeability begin to decrease after a specified strain has been

reached, as illustrated schematically in Figure 2.2. In this way, older faults may eventually become

less permeable than younger faults.

14


Figure 2.2: Schematic illustration of permeability change with increasing accumulated strain. The insert

shows the corresponding (schematic) dilation angleθd (not ϕ from equation 2.13). A dilation angle that

becomes negative (dotted line) results in a permeability decrease after some threshold.

2.3.4 Permeability Tensor Rotation

Anisotropic permeability due to rock fabric, for example, must follow any material rotation.

For this purpose, the orientation of a particle is tracked by means of a unit vectorn, called a

director, with some prescribed initial orientation. In 2D, after some rotation through an angleθ, n

becomesn′, with components

n′1 = − sin θ , n′2 = cos θ .

The rotation matrix

r =

(cos θ sin θ

− sin θ cos θ

)=

(n′2 −n′1n′1 n′2

)

transforms the initial permeability tensor through

k′ = rTkr .

Such a rotation is illustrated in Figure 2.9 in the test cases at the end of this chapter.

2.4 Physical Assumptions

Various assumptions, some of which are noted during the development of the relevant equa-

tions, render the mathematics more tractable. A connected pore network is assumed, such that

isolated pores that hold trapped fluid are not considered, in the sense that they do not affect ma-

terial properties. The system is fully saturated. The porosity is isotropic but the permeability is

not: it may depend on specified anisotropy in the material and/or become anisotropic as a result

of accumulated strain. Capillary pressures are ignored, as are thermal strains in the solid. There is

no individual grain compression in the solid.

15


Pore fluid density does vary with pressure and temperature, since water and other possible pore

fluids, over the scale of the crust, do not meet the incompressibility condition

|∆ρ|ρ

¿ 1 . (2.14)

Over a range of pressuresp, this implies

| ∫p ρβdp|ρ

= β|∆p| ¿ 1

in terms of the compressibilityβ, which is about10−9/Pa. For a depth of 20 km, the hydrostatic

pressure alone is 200 MPa, leading toβ∆p = 0.2. In terms of temperatureT , condition (2.14)

implies that| ∫T ραdT |

ρ= α|∆T | ¿ 1 .

For a thermal expansion coefficientα = 10−3/ C and a temperature of 750C at the bottom of

the crust,α∆T = 0.75. These incompressibility conditions are strictly valid for steady flow only.

There is no viscous energy dissipation accounted for in the solid since the Reynolds number is

negligible.

Re =UL

ν≈ 10−24

for velocitiesU varying by 1 cm/year over a length L = 1 km, and a (lower) crustal kinematic

viscosity ofν = 1015 m2/s. Shear heating in zones of plastic strain is treated empirically through

the strain softening equation (2.9).

2.5 Scaling Method

Modelling studies must accurately represent nature if they are to be useful tools. The issue

of scale is crucial to this accuracy. In analogue models, materials used to represent crustal com-

ponents will of necessity have properties, such as viscosity or density, that are not equal to their

corresponding natural properties. In numerical approaches, stability of the code may demand that

mathematical properties are also different than their natural counterparts.Dynamic similarityre-

quires that combinations of properties and dimensions produce similar measures between model

and nature.

A scale for a parameterx is defined by

x∗ =xmodel

xnature.

Dynamic similarity in terms of the equivalence of lithostatic versus viscous stresses demands that

σ∗ = ρ∗g∗L∗ = η∗ε∗ . (2.15)

Throughout this thesis, model densities are always equal to natural densities, and the gravitational

acceleration is also the true value, so that both the density scaleρ∗ and gravitational scaleg∗ are

16


equal to unity. The viscosity scaleη∗ is thus uniquely dependent on the length and strain rate

scales, and holds for the solid bulk viscosity as well.

In the yield law (2.8), the cohesionco scales as stress, that is,c∗o = σ∗, but the friction co-

efficient cp and the parameters in the weakening relationf(εp) are dimensionless and therefore

unscaled.

The Peclet number Pe= UL/κ, a measure of advective versus diffusive heat transport, must

be equal for both systems, leading to a thermal diffusivity scale

κ∗ = U∗L∗ = ε∗L∗2 (2.16)

in terms of the chosen independent length and strain rate scales.

TemperatureT can be scaled independently because it drops out of the heat equation:

κ∇2T = v · ∇T

κ∗T ∗

L∗2= U∗T

∗

L∗.

Parameters in a temperature-dependent viscosity law will scale in accordance with model temper-

atures. The thermal expansion coefficient has a scaleα∗ = 1/T ∗.The Darcy velocity must scale as the solid velocityU∗:

k∗

µ∗∇p∗ =

k∗

µ∗σ∗

L∗= U∗ ,

considering that solid and pore pressure scales (σ∗) must also be equal. The permeability scale

then follows:

k∗ =U∗µ∗

ρ∗g∗. (2.17)

Since the liquid viscosity is independent of the solid viscosity,µ∗ can be taken as unity, so that

k∗ = ε∗L∗.

2.6 Numerical Method

This section explains only the numerical solution of the equations relating to pore flow, devel-

oped in two dimensions. Figure 2.3 contains a flowchart for the code. The original multigrid loops

in the Ellipsis code, which solve for solid pressure and velocity, have been disrupted as little as

possible by consigning the pore pressure gradient (c.f. equation 2.1) to the right-hand-side force

vector, which is then updated iteratively within one time step.

There is greater detail in Hughes (1987) on the finite element formulations presented below,

and multigrid loops are explained in a tutorial by Briggs (1987).

2.6.1 Pore Pressure Solution

Following Hughes (1987), the computational domainΩ is bounded by surfacesΓh and Γd

having prescribed pore fluxesh and pore pressuresd, respectively. Ifqi ≡ φρpvpsi represents the

17


Figure 2.3: Solution flowchart for the porous flow version ofEllipsis.

Darcy mass flux, the boundary value problem can be stated as follows, using index notation:

qi,i = f ∈ Ω

pp = d onΓd

−qini = h onΓh .

Any source, or forcing terms, are represented byf , andni is the normal to the surface. Although

an exact solution to the problem is defined only at the nodes of the computational grid, the finite

element formulation begins by requiring that areally integrated quantities (overΩ) satisfy the fluid

mass conservation equation (2.10). This produces theweak form

−∫

Ωw,i

ρkij

µ

(pp

,j − ρpgj

)dΩ =

∫

Ωwρp∇ · vsdΩ−

∫

Γh

whdΓh ,

in which w is a weighting function and the source termf is given byρp∇ · vs, that is, the com-

paction or dilation rate of pore fluid mass.

Numerical integration over the domainΩ is made possible byshape functionsthat describe the

variation of quantities between nodes. Each nodeA in the grid has an associated shape function

NA, with the property thatNA is nonzero only within the elements that include nodeA. Ellipsis

uses bilinear shape functions on a rectangular grid of four-noded elements.

NA =4∑

e=1

14

(1− (xe − xA)) (1− (ze − zA)) ,

summed over the four elementse that touch nodeA at coordinates(xA, zA).

18


The solutionpp is composed of the unknown and prescribed partsp andd, respectively. When

the weighting function is taken equal to the element shape functionsNA (Galerkin method),

w =∑

A

NAcA , p =∑

A

NApA , d =∑

A

NAdA .

The functionsw, p, andd are linear combinations of the nodal shape functions with constant

coefficientscA, pA, anddA. Letting a permeability factork′ ≡ ρpk/µ , the weak form expands

as∑

A,B

∫

ΩNA,ik

′ijNB,jpBdΩ = −

∑

A,B

∫

ΩNAρp∇ · vsdΩ +

∑

A,B

∫

ΩNA,ik

′ijρ

pgjdΩ

−∑

A,B

∫

Γd

NA,ik′ijNB,jdBdΓg +

∑

A,B

∫

Γh

NAhdΓh . (2.18)

The material propertiesρp andk′ are calculated at the integration points, that is, the particle lo-

cations, and the solid divergence is calculated on an element basis, this being the quantity satisfied

by the solid continuity equation (2.7). In matrix form, equation (2.18) is represented by

Mp = f ,

whereM is the stiffness matrix,p is the vector of unknown nodal pore pressures, andf is the

force vector that holds all the known terms on the right-hand side of equation (2.18). In practice,

the stiffness matrix is built on an element by element basis. In a 2D elemente with coordinate

directions 1 and 2,

meab =

(Na,1 Na,2

)(k′11 k′12

k′21 k′22

)(Nb,1

Nb,2

)

is the element stiffness matrix entry for local node numbersa andb. The quantitiesk′ij are built

by averaging over all particle locations within the element. The entrymeab is added into a global

matrix according to the global numbersA andB of the four local nodes of the element. The force

vector entry

fea = −Naρ

p (∇ · vs)e +(

Na,1 Na,2

)(k′11 k′12

k′21 k′22

)(0

ρpg

)+ Nah

e

usesρp averaged over particles in the element. Solutions onΓd are left out because they are known

and solved separately. The fluxhe is included when prescribed.

2.6.2 Pore Pressure Gradient in Momentum Equation

The pore pressure gradient feeds back into the right-hand side of the governing momentum

equation (2.1) through the term ∑

A,B

∫

ΩNANB,ip

pBdΩ ,

where the nodal pore pressuresppB are known from the previous solution of pore pressure. The

element force contribution for directioni is

fea = NaNb,ip

pb .

19


Figure 2.4: The numerical interface approximates the particle interface according to the nodes on which

boundary conditions (BC) are applied.

2.6.3 Darcy Velocity

The pore fluid velocity is the end product of the final pore pressure field. Following a similar

procedure as above, each component of the Darcy flow is calculated independently using equation

(2.9), Darcy’s law.

∑

A,B

∫

ΩNANB (φvps

i )B dΩ = −∑

A,B

∫

ΩNA

kij

µ

(NB,jp

pB − ρpgj

)dΩ .

The element stiffness matrix for componentφvpsi is simply

meab = NaNb

and the force vector is

fea = −Na

1µ

(ki1 ki2

)(Nb,1p

pb

Nb,2ppb − ρpg

)

using known pore pressuresppb .

2.6.4 Interface Tracking

In the event that boundary conditions must be imposed on a material interface, the following

method locates and tracks any interface between two materials in the system. A nearest node algo-

rithm renders the interface only approximate and reliant on the density of the computational mesh

(Figure 2.4). This method is the only way of circumventing the need to mix both quadrilateral

and triangular elements if nodes were to be moved to interfaces. The alternative, which is to have

interfaces crossing through elements, is undesirable when integrating contrasting material proper-

ties within an element. Both pore pressure and temperature boundary conditions may be imposed.

20


Fluxes and stresses are far more complicated as boundary conditions because the problem arises

of actually defining the interface and its orientation, so as to integrate along it.

Nodes are deemed to belong to the material defined by the closest particle. The algorithm

works by running through nodes in each column of the mesh and noting boundaries. After the first

pass, a list is kept of nodes that are close to the interface, so that the search does not run through

the entire mesh during subsequent time steps. The list of searcheable nodes is determined by the

maximum displacement of the interface according to the computed time step.

2.7 Analytic Solution Benchmarks

Any numerical code must reproduce, to sufficient accuracy, analytic solutions for simple test

cases. The analytic solutions below have been either derived or taken from published literature.

All cases involve a viscous, porous matrix that is fully saturated with a pore fluid. The governing

equations of momentum and mass conservation assume appropriate simplified forms under the

assumptions of the following problems.

2.7.1 One-Dimensional Compaction of a Porous Halfspace

Compactionis defined as a bulk volume change of the matrix under the influence of applied

stresses, but in the absence of gravitational body forces. When gravitational forces are included,

the phenomenon isconsolidation. Compaction involves the simplified force balance equation

(1− φ)σeij − pp = σo , (2.19)

in which σo is the applied stress on a column of material resting against a rigid, impermeable

boundary.

2.7.1.1 Infinitely compressible pore fluid

In the case of an infinitely compressible pore fluid, for example, vacuum-filled pores, the pore

pressure remains zero. A constant compacting stress of magnitudeσo is applied to the matrix in

the vertical direction. The bulk viscosityζ changes as a result of changing porosityφ according to

ζ = ζoφo

φ,

whereφo is the initial porosity of the column,ζo = ζ(φo), andζ(φ → 0) → ∞. The shear

viscosityη remains constant. The momentum or force balance equation (2.19) reduces to

σo = (1− φ)σezz

= (1− φ)[(ζ − η)∇ · vs + 2η

∂vsz

∂z

]

= (1− φ)(

ζoφo

φ+ η

)∂vs

z

∂z.

21


0.0 0.2 0.4 0.6 0.8 1.0t

0.0

0.2

0.4

0.6

0.8

1.0

φ / φ

ο

φο = 0.05

0.020.01

0.20 analyticnumerical

Figure 2.5: 1D compaction of a dry porous column showing the evolution of porosity with time. Initial

porosities are 0.01, 0.02, 0.05, and 0.2.

Recalling that the porosity evolves in time according to

dφ

dt= (1− φ)

∂vsz

∂z,

the final force balance equation is

σo =(

ζoφo

φ+ η

)dφ

dt.

No spatial boundary conditions are required, since no spatial terms are involved in the equation.

Lettingφ′ = φ/φo and integrating,

∆t =[ζo lnφ′ + η(φ′ − 1)

] φo

σo. (2.20)

Computational results compare well with solutions to equation (2.20) for different initial porosities

(Figure 2.5). The relevant parameters areη = 102 Pa·s,ζo = 104 Pa·s, andσo = −103 Pa.

2.7.1.2 Incompressible pore fluid

Since there is no direct relationship for casting pore pressure in terms of porosity, it is difficult to

obtain a solution to the compaction equation (2.19) for porosity through time. Instead, Figure 2.6

contains solutions for the pore pressure with depth at fixed porosities. Parameters are identical to

those for dry compaction.

For vertical compaction, the pore pressure equation (2.10) simplifies to

∂2pp

∂z2=

µ

k

∂vsz

∂z

=µ

k

σo

ζ + η,

22


0.00 0.01 0.02 0.03 0.04 0.05 0.06

p p

0.0

0.2

0.4

0.6

0.8

1.0

z

φ = 0.01

0.10

0.05

analyticnumerical

Figure 2.6: 1D compaction of a saturated porous column showing pressure profiles at fixed porosities of

0.01, 0.05, and 0.1.

in which the constitutive equation (2.2) is used to replace the solid velocity gradient. Changes

in porosity change the bulk and shear viscosity values. At fixed porosity (or time), the above

pore pressure equation becomes a second-order ordinary differential equation with the quadratic

solution

p(z) =12

µ

k

σo

ζ + η

(z2 − h2

)

for a column of heighth and basez = 0. The boundary conditions arep(h) = 0, where the fluid

escapes freely out the top, and, since the fluid velocity is zero at the base,dp/dz = 0 atz = 0.

2.7.2 One-Dimensional Consolidation of a Porous Halfspace

2.7.2.1 Incompressible pore fluid

As above, there is no analytic solution for the variation of porosity with time in the case of

consolidation under gravitational forces. Ricard et al. (2001) use a finite difference solver to

calculate porosity versus depth at different times, for an infinite layer from0 ≤ z ≤ lo, and zero

matrix and fluid velocities at these boundaries. Their solution is for the case where the matrix

viscous term dominates in balancing the gravity term, that is, the matrix viscosity is much larger

than the fluid propertyµ/k, leading to a compaction length much greater thanL.

Figure 2.7 compares porosity profiles fromEllipsis with those from Ricard et al. (2001). Be-

cause of different formulations (Ricard et al. (2001) do not use a bulk viscosity), the shear and

bulk viscosity parameters inEllipsis were tuned so thatφ(z=lo) is equivalent att = 10 in both

approaches. Solutions appear to match very well at all times.

In the porous modelling in this dissertation, the effect of consolidation is negligible, because

23


0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8φ

0.0

0.2

0.4

0.6

0.8

1.0

z / l

o

t=0

t=2.5

t=5

t=10

t=20

a b

Figure 2.7: 1D consolidation of a saturated porous layer showing porosity profiles at different timest. The

initial porosity is 0.05, and all velocities are zero at the basez = 0 and the topz = lo. (a)Ellipsissolutions

are similar to (b) solutions from Fig. 13 of Ricard et al. (2001).

solid compression or dilation, which drives fluid flow, is dominated by boundary velocities that

are always much greater than the natural consolidation velocity.

2.7.2.2 Equal matrix and pore fluid densities

When the matrix and pore fluid are of equal density, the equilibrium momentum equation is

(1− φ)σeij,j − pp,i + ρgi = 0

at time zero when the porosity is constant. The divergence of the effective stress is balanced by

the gradient of the pore pressure and the body force due to gravity.

In the case of one-dimensional consolidation of a porous column on to an impermeable base,

the stress tensor is isotropic and diagonal, the components being equal to the solid pressure pa-

rameterq, which is only the thermodynamic pressure if the system is at rest:

(1− φ)q,i − pp,i + ρgi = 0 .

The pore pressure may be split into a hydrostatic part and a superhydrostatic partp′, in which case

the hydrostatic gradient is equal to the body force term, leaving

(1− φ)q,i = p′,i . (2.21)

McKenzie (1984) derives an analytic solution for the initial consolidation of a porous halfspace,

that is, at timet = 0 (constant porosity), on to an impermeable surface atz = 0. The solid vertical

velocity

vsz = −vp

oφ(1− e−z/δc

),

wherez is the height above the impermeable base,δc is the compaction length, andvpo , the initial

pore fluid velocity at largez, is a constant given by

vpo =

k

µ

(1− φ)φ

(ρs − ρp) g . (2.22)

24


Figure 2.8: Flow deflection due to permeability contrasts. A higher permeability center (a) causes the

flow field (black arrows with white tips) to deviate towards the center, whereas a lower permeability center

(b) causes the flow field to be deflected around the center. The profiles on top trace the pore pressure

horizontally through the center of the box.

From McKenzie’s inital pore fluid velocity (2.22),vpo = 0 when the matrix and fluid densities

are equal. Thus, the driving force for flowp′,i = 0 in Equation 2.21, and from this the solid

pressure gradient is zero also. Since the solid pressure is zero at the surface, it is zero everywhere

and the matrix is fully supported by the pore pressure. By the continuity equation,

vsi,i = − q

λ= 0 ,

and by the fact thatvs = 0 at the impermeable base, the matrix velocity is zero everywhere. This

zero velocity result is borne out by code tests.

2.8 Miscellaneous Test Cases

This section contains results for which analytic solutions have not been calculated, but which

may be compared with reasonable expectations. In all cases, the porous medium is fully saturated.

2.8.1 Deflection of Flow Field Due to Permeability Contrasts

Prescribed pore pressures applied to each side of a box maintain a flow field through a porous

material. The permeability is isotropic but there is a permeability contrast in the center of the

box. When the center is more permeable, the flow field, shown by arrows, is deviated towards the

center (Figure 2.8a). In the opposite case (Figure 2.8b), the flow field is deflected around the less

permeable center. The results are similar to those illustrated in Phillips (1991, p. 69).

25


Figure 2.9: Anisotropic permeability follows material rotation. The initial state (a) is rotated by approx-

imately 90 clockwise (b), as recorded by the stripes of marker particles. Fluid flow arrows (black with

white tips) from the high-pressure center demonstrate the permeability rotation.

2.8.2 Rotation of Anisotropic Permeability

A box is filled with a porous material having anisotropic permeability. Figure 2.9a illustrates

the initial configuration. The crossed stripes are for visualising the rotation. A constant pore

pressure in the center forces fluid to flow towards the edges of the box, which are held at zero pore

pressure. The moving walls of the box cause the material to rotate, and the anisotropic pore fluid

flow, shown by the arrows, follows the clockwise rotation (Figure 2.9b).

2.9 Chapter Summary

TheEllipsisfinite element code uses a fixed, rectangular computation grid, inside which parti-

cles move, representing different materials and carrying physical information such as porosity and

time-integrated quantities such as accumulated strain. The governing equations of momentum and

mass conservation describe a viscous fluid that represents a solid matrix, such as the crust, with

a porosity filled by an inviscid pore fluid that flows according to Darcy’s law. The solid matrix

can yield plastically, which is an approximation to brittle failure, and zones of concentrated failure

represent faults. Faults may continue to weaken with accumulated strain. Deformation causes the

porosity to evolve, and this, together with topographic differences or fluid sources, provides pore

pressure gradients that drive fluid flow. Simple tests against analytic solutions demonstrate the

accuracy of the code.

Numerical models represent nature when they are accurately scaled, which means that certain

parameter combinations must respect the values they have for real Earth systems. Parameters such

as length, strain rate, and permeability can then be translated into meaningful natural values in

order to assess the model results, and enter into the iterative process of refinement between field

observations and numerical tests.

26

Chapter 3

INTERACTIVE INVERSE MODELLING

3.1 Introduction

The sophisticated modelling of geological processes, made possible by a code such asEllip-

sis, is due to the recent advent of powerful computers. Tracking evolving material properties on

particles, as described in Chapter 2, is computationally intensive, but allows a degree of deforma-

tion not otherwise possible. Various codes in existence today can treat subduction, faulting and

folding, mantle convection, and fluid flow in a rigorous numerical fashion, much in the same way

as traditional geophysical applications such as seismic or potential field problems. As modelling

capabilities have increased, so has the number of parameters that enter into a model, and this has

led to the increasingly difficult task of discriminating the influence of these parameters. In the

non-linear geological systems under investigation today, predicting model behaviour through a

knowledge of individual parameter influences may often not be feasible.

In general,forward modellinganswers questions such as “What response should be expected

from this distribution of material properties under these initial conditions?” (e.g., “What faults will

be generated by this stress field in this particular material?”). The answer is obtained by providing

a computer code with certain input parameters and running the code for a number of time steps.

In many cases, the final result is a geological section or 3D representation.

Most real geological problems require an answer to a question that goes in the opposite di-

rection, that is, “What combination of material properties and initial conditions may result in this

geological response?” (e.g., “What was the stress field that generated these faults in this rock?”).

This is a more complicated problem, for which the answer must be found by iterative numerical

trial and error methods. This is computationally intensive, and the manner in which the search for

an answer is optimised is calledinverse theory, or simplyinversion(Tarantola, 1987).

Inversion is the natural step forward in geological modelling. Reconstructing initial configura-

tions from their geological responses is very much what geology is about. It is an implicit inverse

problem tackled on a daily basis by every geologist. The basic difficulties encountered in an in-

verse problem are: the lack of a guaranteed solution, or the probable existence of many solutions

giving the same answer (non-uniqueness); the efficiency with which a solution can be found that

matches a target within a given tolerance; the sensitivity of a given solution to changes in the initial

conditions. Parker (1977) and Tarantola (1987) explain the mathematical and technical aspects of

inversion.

To overcome the difficulties of inversion, geologists use their intuition and experience to focus

only on the “geologically reasonable” model results that explain the particular features they record.

27

CHAPTER 3. INTERACTIVE INVERSE MODELLING Wijns Ph.D. Thesis

This qualitative knowledge is often difficult to quantify, and the resulting conceptual geological

targets cannot be adequately described by numerical data. This is in contrast to, for example, the

inversion of a gravity profile, where the gravity data are the objective measure to be reproduced. To

compensate for the lack of a quantitative target, the method explained below combines the formal

methodology of mathematical inversion with the knowledge held in observational experience. This

inverse modelling technique can help every time a problem needs visual appraisal of the results or

a priori expert knowledge. The only requirement is a code that allows the user to forward-model

a process and view its result.

3.2 Method

At present, geodynamic modelling is often confined to the forward-modelling stage. Success

has been achieved in quantitative inversion of sedimentary basin models (e.g., Cross and Lessen-

ger, 1999; Bellingham and White, 2000), demonstrating cases where available quantitative data

(i.e., borehole logs and stratigraphic horizons from seismic interpretations) can be used for a direct

measure of misfit. Kaus and Podladchikov (2001) were able to invert a Rayleigh-Taylor instability

to restore initial conditions, but only for very restricted cases of initial geometry. In many applica-

tions of geological modelling, a forward solution is judged visually according to its resemblance

to patterns in the field, to the fact that it does not contradict basic geological principles, or simply

according to the modeller’s expectations.

If one accepts the fact that much of a geologist’s expertise is difficult to quantify, then it is

necessary to incorporate human interaction in directing the inversion process. Recently, research

in artificial intelligence has resulted in systems to support such human interaction in optimisation

problems (Takagi, 2001). They have been used in such diverse fields as graphic design, music

composition, and the engineering of hearing aids. These systems are known collectively under the

term interactive evolutionary computation(IEC).

This chapter extends the use of IEC to geological applications in which visual judgment is

necessary to evaluate model results in the absence of sufficient constraints. The system repre-

sents an advance on traditional, time consuming, trial and error approaches by providing a formal

role in the inversion process for geological experience that cannot be transformed into data. The

traditional numerical measure of data mismatch is replaced by the user’s subjective evaluation.

Humans find it hard to express subjective judgment with absolute values, while they generally find

it much easier to compare different instances of the same process and rank them according to cer-

tain criteria. Consequently, interactive inversion works by producing different possible solutions

and presenting them to the user for judgment and ranking.

Genetic algorithms (GAs) are one search method suitable for the inversion of non-linear func-

tions. Starting with a set of random solutions, these algorithms progressively modify the solution

set by imitating the evolutionary behavior of biological systems (selection, cross-over, and mu-

tation) until an acceptable result is achieved. GAs belong to a class of algorithms that work by

optimising multiple solutions, unlike other classes that optimise one single solution. They are

28

Wijns Ph.D. Thesis CHAPTER 3. INTERACTIVE INVERSE MODELLING

Figure 3.1: (a) Conceptual inversion target showing faults with a spacing on the order of the thickness of the

upper crust. (b) Initial geometry of the numerical model. The upper crust contains light-coloured marker

units with no material differences.

also suitable for handling ranked suites of solutions, which makes them an obvious choice as the

internal engine for interactive inversion applications.

GAs are an established technique today, with a wide range of applications to both theoretical

and industrial problems. Goldberg (1989) gives a basic description of GAs, and Boschetti et al.

(1996) provide a more detailed description of the specific GA implementation used in this work.

The IEC system works by linking a geological forward model (Ellipsis) to the GA. A geologist

runs the forward-modelling code with the aim of producing a geological simulation that matches

a conceptual target. A number of selected parameters is allowed to vary within given ranges. The

GA initially generates a suite of different simulations using randomly picked parameter values.

These simulations could be static geological images or animations showing time evolution. In

the example below, there is no automated method for discriminating between geologically ap-

propriate results, so the geologist ranks each of them according to criteria founded in his or her

experience and knowledge. A relative target misfit is now contained within these rankings. When

this stage is complete, the GA applies parameter swapping between highly ranked results to gen-

erate a new set of simulations that progressively converges towards the target geological section.

As in biological evolution, an element of randomness exists in the generation of new simulations,

so that unexpected results may suggest new possibilities outside the experience or expectation of

the geologist.

3.3 Application

3.3.1 Model

Fault spacing during crustal extension serves as an example to illustrate the inversion process.

The goal is to find a set of material parameters that gives rise to fault spacing on the order of the

thickness of the upper crust, while nowhere enabling the upper crust to be completely pulled apart.

The results of forward numerical simulations are ranked by comparing them to the simplified

line sketch of Figure 3.1(a). It is important to note that the method can proceed without any

29


Table 3.1: Variable model parameters.

Parameter Initial range Increment “Best” value Range

η l (×1.56× 1019Pa·s) 10 - 500 25 400 400

a 0.0 - 0.9 0.1 0.4 0.4

εo 0.0 - 1.0 0.1 0.7 0.7 - 0.9

n 0.4 - 2.0 0.2 1.0 0.6 - 1.0

Four model parameters, described in the text, are free to vary during the inversion, within the ranges and

by the increments indicated. The “best” values give rise to the top-ranked simulation of the final (sixth)

generation. The last column gives the range of parameter values for the five best simulations of the final

generation.

actual target image. The target is included as a guide, and the exact number, location, and dip of

the faults need not be reproduced. In fact, the location of faults is extremely sensitive to initial

perturbations, both in nature and in the numerical simulations, so that it is all the more appropriate

to look for general behaviour and relative spacing. Rheological controls on fault spacing are the

subject of many analogue and numerical modelling studies (e.g., Benes and Davy, 1996; Spadini

and Podladchikov, 1996; Thibaud et al., 1999; Bai and Pollard, 2000). The results generated below

are fairly simple and are intended to illustrate the method.

The model is composed of two initially homogeneous crustal layers. The upper layer behaves

as a visco-plastic material, yielding according to equation (2.8), and the bottom layer has a Newto-

nian viscosity. On top of these is a low-density, low-viscosity background material (“air”), which

does not interfere with the mechanics of the problem. The fact that the mantle is not included is

akin to specifying a strong mantle that does not appreciably deflect during extension. This initial

configuration is illustrated in Figure 3.1(b). Light-coloured horizons in the upper crust are simply

marker units with no physical differences. The upper crust has strain weakening properties (equa-

tion 2.9) that cause initial strain perturbations to localise. The fault geometry and successive fault

spacing arise naturally from the initial conditions of the problem.

The manner in which faults actually weaken is poorly understood, so the three parametersa, εo,

andn in the strain weakening law (2.9) are allowed to vary, in an attempt to gauge the influence of

weakening. The fourth parameter under study is the viscosity of the lower crust. These variables

are listed in Table 3.1, together with their ranges of investigation.

Numerous studies of the Basin and Range area in the western U.S.A. provide data on an envi-

ronment that most agree is extensional in nature, and so many of the parameter values are from the

literature for this region. The overall amount of extension is arguable, from a few tens of percent,

to greater than 100% (e.g., Jones et al., 1992) in certain regions. Total extension in the simulations

below is 50%. The initial3 × 1 model box of Figure 3.1(b) represents natural dimensions of 150

by 50 km. The upper and lower crust are each 20 km thick initially, and after extension the total

thickness of the crust is about 25 km, which is representative of large areas in the Basin and Range,

based on the regional crustal thickness model of Chulick and Mooney (2002).

30


The rate of extension is taken from data typical of the western U.S.A. Based on GPS data

through time, Thatcher et al. (1999) have determined velocities of between 2.8 and 6.5 mm/yr

for various domains of the Basin and Range, and Murray and Segall (2001) find similar rates of

between 2.3 and 3.6 mm/yr. Over the model length scale of 150 km, 5 mm/yr is equivalent to a

strain rateε =1.06×10−15/s.

Model densities for the upper and lower crust are 2700 and 3000 kg/m3. The choice of viscosity

ηu for the upper crust fulfills the maximum yield stress criterion at the interfacezu between upper

and lower crust (see Appendix C.1).

τyield(zu) = ηu ε

The rock cohesionco is 10 MPa (e.g., Suppe, 1985, p. 155), which is close to the value of zero,

relative to the stresses in the model, employed in Byerlee’s law for the upper 10 km of the crust,

but which avoids a cohesionless surface material that would allow plastic strain along the entire

air-rock interface. Numerous laboratory experiments by Byerlee (1968) resulted in a universal

friction coefficientcp of 0.6 – 0.85 for most rock types. However, this is for dry samples. As-

suming an average hydrostatic pore pressure, the solid pressure is reduced by more than one third,

so that an equivalent dry friction coefficient of 0.7 is reduced to 0.44 in the model. A test in

Appendix A shows this hydrostatic pore pressure approximation to be valid in terms of initial

brittle failure depth, although the relationship is certainly more complicated when deformation is

involved. Using these values ofco andcp , the maximum shear strength is about 100 MPa at the

brittle to ductile transition.

3.3.2 Inversion

Each step of the inversion involves running six forward simulations, and the four parameters

in Table 3.1 are allowed to vary. Initially, these parameter values are chosen randomly by the GA,

within the bounds specified. Extension proceeds by applying a uniform velocity to the right-hand

boundary. Figures 3.2, 3.3, and 3.4 illustrate the evolution of results using the IEC algorithm.

Bands of high localised plastic strain represent faults. Accumulated strain in the upper crust is

indicated by darkened material, and the degree of shading is a measure of the amount of strain.

The first generation (Figure 3.2) produces no satisfactory simulation. The image ranked first has

the most desirable fault spacing out of the choices presented, and also exhibits the most clearly

defined faults. The third and fourth simulations are ranked at the bottom because they have resulted

in complete dissection of the upper crust, which, as mentioned above, is an undesirable solution.

The GA now uses these rankings as a measure of relative misfit.

Figure 3.3 contains the second generation of the inversion. From this generation onward, results

are always compared with the best-ranked simulation of the previous generation, in order to assess

convergence towards the target. In this example, there are no second generation results that have

improved on the best first generation simulation. This is still a legitimate and useful result that

tells the GA that it has explored parameter space in the wrong direction. However, overall results

are better in the sense that there are no longer any simulations with a dissected upper crust. The

31


Figure 3.2: First generation of the inversion process. Dark areas in the upper crust represent accumulated

plastic strain. Model results are ranked according to their similarity to the concepts embodied in the target

image of Figure 3.1. (Faint white lines in the lower crust represent uneven particle distribution after de-

formation, and are not a visualisation of any physical process. They are interference patterns between the

deformed distribution of particles and the plotting routine that allocates colours to a regular array of pixels

based on nearby particles. Although particles reproduce in order to accomodate minimum requirements for

computation, they do not necessarily reproduce sufficiently for aesthetic purposes.)

simulation ranked third is so chosen in order to encourage denser fault spacing.

After iterating the ranking process a total of six times, five of the resulting images are qualita-

tively similar (Figure 3.4), and show no evidence of substantial improvement over the best-ranked

result of generation five. It is in fact difficult to assign different rankings to the first five images.

The simulations have a regular fault spacing that corresponds to the target concept. It is appar-

ent that, although the features of the final simulations are satisfactory in concept, they would not

32


Figure 3.3: Second generation of the inversion process. Results are compared with the best-ranked simula-

tion of the first generation.

match up well with the target image through any quantitative direct image comparison.

The outcome of this experiment is a set of the four chosen parameters that leads to the qual-

itative behaviour that was targeted. Table 3.1 lists the final (“best”) values that give rise to the

highest-ranked simulation of the sixth and final generation (Figure 3.4, second simulation). The

table also shows the range of each parameter for the five top-ranked simulations of the final gener-

ation, as a measure of variability within visually equivalent results. The final values permit some

physical conclusions about the system. The lower crust has an optimal viscosity that is about

33


Figure 3.4: Sixth and final generation of the inversion process. Almost all results are qualitatively similar,

with no substantial improvement over the best-ranked simulation of the previous generation.

6.2 × 1021 Pa·s. This is an order of magnitude greater than the upper limit determined by Pollitz

et al. (2001), based on geodetic measurements of post-seismic velocity fields after the 1999 Hec-

tor Mine earthquake in California. Flesch et al. (2000) determine a 100 km vertically averaged

crustal viscosity of between5× 1021 and5× 1022 Pa·s for most of the western U.S.A. Depending

on mantle viscosity and the method of vertical averaging, the inversion result may lie within this

range. The calculation of an effective viscosity for the upper crust influences the result for the

lower crust, so that rather than discuss absolute viscosity values, it would be fairer to state that the

34


optimal fault spacing occurs when the lower crust is about one fifteenth the strength of the upper

crust. Upon examination of all results, it is apparent that smaller viscosities for the lower crust

lead to the dissection of the upper crust (simulations 3 and 4 of generation 1) and exhumation of

the lower crust akin to metamorphic core complex formation. Higher viscosities lead to more even

stretching of the upper crust and probably finely spaced faulting that is below the model resolution

(e.g., simulation 3 of generation 2). The results for the strain weakening law settle on a value

of a = 0.4 that reflects a 40% drop in the yield strength after a total strain accumulationεo of

0.7 to 0.9. Major faults may undergo significantly more weakening than this, according to both

numerical experiments (e.g., Bird and Kong, 1994) and field-based heat flow measurements (e.g.,

Lachenbruch and Sass, 1980, 1992). The less pronounced weakening of the model faults in this

study may contribute to the phenomenon of distributed faulting, whereas greater weakening may

promote a higher degree of localisation on single faults. The last parametern varies between 0.6

and 1.0, indicating that the shape of the weakening curve is less important than the other parame-

ters in controlling fault spacing.

3.4 Discussion

The aim of the inversion in the example above is to arrive at some particular behaviour of the

crust during deformation: behaviour that cannot be sufficiently described by numerical measures.

Finding a suitable combination of parameters that gives rise to this behaviour would previously

have involved one of two more laborious approaches: the manual selection of parameters by trial

and error, or an exhaustive coverage of all parametric space. Trial and error may succeed with a

limited number of parameters, but depends on the user’s knowledge of the coupling and feedback

between parameters, which, in highly non-linear problems involving complex crustal rheologies,

may be impossible. A parametric sweep quickly becomes unfeasible due to the sheer number of

simulations that must be run as the number of parameters is increased. In this example, in excess

of 20 000 simulations would have to be run in order to cover all possible parameter combinations,

and each forward simulation takes a few hours to run on a 935 MHz desktop computer with 500

MB of RAM.

The IEC technique finds multiple solutions with only 36 simulations being run. This vast re-

duction in the number of individual simulations can be attributed to the fact that visual ranking

provides more information in this type of search than numerical misfit in a non-interactive inver-

sion. A result containing one or more features of paramount importance, but with a potentially

large numerical misfit because of, for example, spatial discrepancy in feature locations, is ranked

highly and provides a significant step forward in the search through parameter space. In fact, be-

cause of the combinatorial nature of the GA progression, two images that each contain a different

feature of importance can both be ranked highly in order to increase the likelihood of producing a

new result containing both desired features. Neither trial and error, nor a parametric sweep, takes

full advantage of the expert knowledge of a user, which in this case is the observational experi-

ence of a field geologist. Both the number of variable parameters and the number of individual

35


simulations in each generation is limited by available computer speed. The number of simulations

per generation would ultimately be limited by the capacity of a human to distinguish between, and

rank, the results.

Although the single GA search produces satisfactory results, it should be pointed out that, since

a GA depends on an element of randomness, the most correct procedure involves many GA trials.

In a statistical sense, the greater the number of trials, the more likely the optimal solution will

be found. In practice, the computational intensity of geodynamic modelling would not generally

allow this.

The power of inversion lies in demonstrating the range of non-uniqueness of a solution. A very

simple and quicka posterioriinvestigation highlights the sensitivity of the results to changes in the

variables. For example, referring to the five best simulations (Table 3.1), the viscosity of the lower

crust has settled on a unique value, which suggests that it exerts a strong influence on the outcome

of the problem. The same is true of the final amount of strain weakeninga. A parameter such

as the saturation strainεo has a weaker influence on the outcome, since it varies slightly without

affecting the final qualitative picture. The shape of the strain weakening function, determined

by n, has the least influence. With the inversion approach in general, most late-stage results are

close to the target in parameter space, and so a back-analysis through all generations is instructive

for looking at the sensitivity of solutions. However, such conclusions can only be drawn with

confidence on the strength of sufficient sampling of the model parameter space. To this end, and

to facilitate an understanding of the physical controls on the behaviour of the system, Chapter 4

concentrates on the effective visualisation of the whole multi-dimensional parameter space.

An important component of this interactive inversion technique is its ability to embrace un-

forseen results. An intuitive approach relies entirely on the experience of the modeller, and may

miss realistic targets that lie outside the realm of modelling space that is envisaged. The GA,

although converging to a specific area in parameter space, also provides for random solutions. If

ranked highly, such a random solution may open up an entirely different class of simulations that

also yield realistic results.

3.5 Chapter Summary

Conceptual models in geology are often difficult to describe numerically. They mostly reflect

the translation of qualitative field knowledge into verbal descriptions or geological cross-sections.

This usually precludes implementing the traditional notion of comparing a numerical simulation to

a target via one or more numbers. The interactive inversion system circumvents this difficulty by

using a visual comparison between forward model results and a target, in order to guide a genetic

algorithm to the parameters that generate a preferred solution. At each step, ranking between

interim results acts as a proxy for a numerical misfit.

The technique of interactive evolutionary computation considerably diminishes the effort re-

quired to explore parameter space during the inversion of conceptual models in geology. The

method is particularly geared towards cases of highly non-linear interactions between material

36


parameters, where the resultant behaviour is difficult to predict. This visual approach exploits

the unquantifiable aspects of an expert user’s knowledge in a situation where this is currently an

under-utilised resource.

37

Chapter 4

MODEL SPACE VISUALISATION

4.1 Introduction

As alluded to in Chapter 3, the end product of most inversion procedures is a single best solu-

tion. However, many solutions are generated during the course of the process. The focus of this

chapter is on maximising the information to be gained from an analysis of all solutions. Since the

user ranks each solution in the genetic algorithm search described in Chapter 3, this information

should be used to obtain some understanding of the parameter space, rather than be discarded at

the end of the inversion run. This, in turn, leads to a better understanding of the dynamical prob-

lem. From the statistical tools that are available to perform such an analysis, visual approaches

are preferable. The reason is twofold: first, visual displays are in keeping with the qualitative de-

scription and evaluation process for each simulation result, and second, this can provide a further

level of interactivity between the user and the inversion. If intermediate displays of the cumulative

solution space are prepared, the user not only judges the quality of each individual solution, but

follows the inversion progress, and, eventually, may control the direction of parameter exploration.

The three different techniques used below serve as complementary methods for visualising the

accumulated results of an inversion. These are aparallel axis display, multi-dimensional projec-

tions, and aself-organising map. In each case, a few simple plots reveal which combinations of

parameter values lead to “good” solutions, and what role individual parameters play in contributing

to “bad” solutions. The plots also offer straightforward assessments of the non-uniqueness of the

problem, and the extent to which the inversion has covered the specified ranges of investigation.

The example in this chapter is a modification of the previous crustal extension problem.

Before presenting the different visualisation techniques, it is imperative to caution on two

points. First, any reduction from a high-dimensional problem to a lower-dimensional visuali-

sation results in information loss. Of the three techniques presented, only the parallel axis display

does not involve reduction of dimensionality, that is, all data points are faithfully reproduced. Sec-

ond, any statistical method improves with the number of data points. The intensive computation

involved in geodynamic modelling results in relatively few simulations, and hence a paucity of

data for statistical manipulation. This cannot be avoided, and the challenge is to draw conclusions

from sparse data, keeping in mind that the addition of new data will usually change the working

view of parameter space. To reinforce an appreciation of the poor coverage of model space in a

typical study, the following problem has six variables per simulation and a total of 48 simulations.

The number of points needed to simply define a 6D hypercube is 26 = 64. The available data are

therefore equivalent to fewer than two points in 1D – sparse data by any stretch of the imagination.

39

CHAPTER 4. MODEL SPACE VISUALISATION Wijns Ph.D. Thesis

Table 4.1: Variable model parameters.

Parameter Initial range Increment

viscosityηu (× 6.25×1018Pa·s) 5000 - 10000 1000

cohesionco (× 0.025 MPa) 0 – 2000 200

friction coefficientcp 0.1 – 1.0 0.1

tensile stress limitσc (× 0.025 MPa) 100 – 1000 100

strain weakeninga 0.1 - 0.9 0.1

saturation strainεo 0.1 - 1.0 0.1

Six model parameters, described in the text, are free to vary during the inversion, within the ranges and by

the increments indicated.

In spite of this, interactive inversion does produce good results, which may indicate that the non-

linearity of the problems is not great. “Non-linearity” in this case refers to a simple shape of the

solution space as defined by the rankings of the user, not the actual non-linearity of the physics.

This chapter illustrates methods of data visualisation that represent the best attempts in an area

of ongoing research. Wherever possible, different techniques, with different and complementary

strengths, should be used in parallel.

4.2 Target, Model, and Inversion Results

In this variation on the inversion in Chapter 3, the goal is to find one or more sets of material

parameters that give rise to fault spacingwider than the thickness of the upper crust, as sketched

in Figure 4.1a. The model and initial geometry (Figure 4.1b) are similar to the previous example.

The visco-plastic upper crust, with strain-weakening behaviour, is 15 km thick. This lies over

6 km of weaker, ductile lower crust. Above these is a highly compressible layer of low density,

low viscosity background material (“air”), which does not interfere with the mechanics of the

problem. The right-hand boundary extends with a uniform velocity equivalent to an average strain

rate of10−15/s. The thin lower crust would not in general be as representative of natural conditions

as the thicker model in Chapter 3. Although it may describe specific geological settings with a

very thinned lower crust, such as the extended area around Naxos in the Aegean (Jolivet, 2001),

or the crustal structure under parts of Antarctica (Mishra et al., 1999), there is no intent to link this

model to a particular field example.

The six model parameters that are allowed to vary are the viscosityηu of the upper crust, and

five yield parameters that describe the plastic failure of the upper crust (equations 2.8 and 2.9).

These are the cohesionco, the pressure dependence or friction coefficientcp, the strain weakening

parametera, the “saturation strain”εo, and a new parameterσc, the tensile limit of the crust. If the

solid pressure is negative (i.e., tensile stress) and has a magnitude greater thanσc, the viscosity is

reduced threefold to simulate tensile failure. Table 4.1 lists the variables together with their ranges

of investigation.

40

Wijns Ph.D. Thesis CHAPTER 4. MODEL SPACE VISUALISATION

Figure 4.1: (a) Conceptual target and (b) initial geometry of the crust. Panels (i) to (iii) hold the first two and

the last (sixth) generations of the inversion algorithm. Images are ranked within each generation according

to the spacing of faults (dark bands) in the upper crust. Unranked simulations have not extended because of

numerical non-convergence. The third image of panel (iii) shows the greatest fault spacing.

Six generations of simulations are run, with eight simulations per generation. After each gener-

ation, the user ranks the output images against the conceptual target (Figure 4.1a). As in Chapter

3, ranking is based on general behaviour and relative fault spacing, rather than trying to repro-

41


Figure 4.2: Schematic diagram of a parallel axis display for a three-parameter problem. Individual param-

eter values are joined to indicate the two solution vectorss1 ands2.

duce the image exactly. “Multi-criteria” human evaluation, where solutions are ranked according

to separate, but simultaneously evaluated criteria, is more effective than trying to devise a single

numerical target. The bottom panels of Figure 4.1 show the progression of the outputs and the

rankings accorded to each image for the first, second, and sixth (last) generations. By the sixth

generation, several images appear very similar, and represent “satisfactory” solutions. Unranked

solutions do not converge numerically, meaning that the computer code fails to calculate any result.

As discussed below, this is due to unrealistically high yield strengths.

At the end of the inversion, each solution is ranked in an absolute fashion, against all other

solutions, from 1 to 10, with 1 being the best. This ranking is distinct from the intra-generational

ranking that guides the genetic algorithm (GA), as displayed below each image in Figure 4.1, and

provides a classification fora posteriorianalysis of the inversion results.

4.3 Parallel Axis Display

4.3.1 Theory

The parallel axis display is a basic means of plotting all parameter values for all solutions. Each

parameter that makes up a component of the solution vector is assigned to a successive point on

the abscissa, and its normalised values are plotted above it on a common ordinate axis. Parameter

values belonging to a common solution vector are joined together (Figure 4.2). Unlike the next

two methods, the parallel axis display is quickly limited by the number of solutions that can be

plotted together before overcrowding makes the graph illegible.

One method for mitigating the inevitable obscuring of information that arises through the su-

perposition of similar solutions is to offset plot points, by either spreading them around each

parameter location (along the abscissa) or adding random noise to each parameter value. A com-

bination of both techniques proves useful for visualising the inversion data below.

42


Figure 4.3: Parallel axis display for all inversion solutions, ranked by colour from red (best) to purple

(worst). For easier visualisation of overlapping solution segments, points are spread slightly around param-

eter locations and some parameter values have small random noise added.

4.3.2 Visualisation of the Inversion Data

The inversion results consist of 48 points in a 6D space, the six dimensions corresponding to

the six model variablesηu, co, cp, σc, a, andεo. These data represent the extent to which the

inversion procedure has sampled the parameter space. The parallel axis display in Figure 4.3 was

produced with the free softwareggobi, available from http://www.ggobi.org. The best solutions

(red) are the result of a high viscosity, a low cohesion, friction coefficient, and tensile limit, and

pronounced fault weakening. The saturation strain varies.

It is apparent that the sampling has fallen short of the alotted range for the four parameters

co, cp, a, andεo, since points are not distributed over their full spectra. More simulations could

be run in order to fill in some of the missing data and increase the confidence that good results

have not been overlooked in unexplored space. In the most rudimentary manner, new parameter

combinations can be chosen manually by referring to blank areas in Figure 4.3. Based on the user

rankings of these new outputs, the GA would incorporate or cull this additional parameter space.

Highly ranked solutions, in orange and red (Figure 4.3), are in general quite clustered. They

occupy single values forco andcp, a small range of values forσc, and show greater variability for

the remaining three parameters. A possible conclusion is that model behaviour is strictly controlled

by co andcp, whereas good solutions exhibit some degree of non-uniqueness with respect to the

other parameters. The caveat is that a GA can possibly become locked in to particular values for

non-optimal reasons, including “the luck of the draw” in the first random distribution of inputs

to the model. One safeguard against this is to increase the frequency of mutations, leading to

more simulations that are not created according to the ranking. However, too high a mutation

rate may detract from the convergence to target behaviour. Increasing the number of individuals

per generation can also guard against locked values, but computation time and the challenge of

effectively ranking all solutions may preclude this option. In this example, the difference between

good (orange) solutions and the best (red) solutions lies in the parametersηu anda.

The worst-ranked solutions, in purple, including those simulations that failed to converge nu-

43


Figure 4.4: Simulation rank versus generation showing overall progress of the inversion towards better

results (1= best rank, 10= worst). Small random noise is added to each rank for easier visualisation of

close results. Eight simulations sit within each generational interval from 1 to 6.

merically, are widely spread in parameter space. This suggests that no single parameter is re-

sponsible for bad solutions. Highco andcp values characterise many of the numerically invalid

simulations. Fortunately, these high values translate into unrealistically high yield stresses, mean-

ing that this part of parameter space can be safely neglected.

4.4 Multi-Dimensional Projections

4.4.1 Theory

The simplest method for projecting data from high dimensions into 2D is to neglect all but two

parameters (dimensions) at a time. This is called anorthogonal projection, that is, all neglected

dimensions have their axes orthogonal to the viewing direction and therefore invisible. Such linear

projections are also known ascrossplots. Labels or colours can add a third dimension. As for the

parallel axis plot, small random noise is added to parameter values to avoid losing sight of multiple

points that are superimposed in 2D.

Perspective views of 3D data can be well represented in 2D, to which labels or colours can

again be added to achieve a fourth dimension. Beyond this it is necessary to use non-linear trans-

formations of the data to reflect, in 2D, relative distances between points innD. One suchmulti-

dimensional scaling, pioneered by Sammon (1969), is applied by Wijns et al. (in press) to evaluate

the effectiveness of an interactively guided 9D inversion.

44


Figure 4.5: Orthogonal projection of each variable parameter versus simulation rank (1= best rank). Small

random noise added to each parameter value and rank prevents strict superposition of points.

4.4.2 Projections of the Inversion Data

All projection plots in this section were produced with theggobisoftware used for the parallel

axis display. Figure 4.4 offers a simple way to judge the overall progress of the inversion towards

better-ranked simulations. With each successive generation, there are more solutions with good

ranks (low numbers), but the spread of rankings shows that the GA is still exploring far from the

well-ranked parameter space.

The series of six plots in Figure 4.5 illustrates the entire space explored by the GA as well as

any convergence to specific parameter values with respect to rank. The same conclusions hold as

were derived from the parallel axis display: highηu, low co, cp, andσc, high a, and variableεo

produce good results. The rapid convergence ofco (Figure 4.5b) is easy to visualise. Based on

a relatively complete coverage ofco space, a high cohesion invariably leads to bad results. Gaps

in the exploration of the parameter ranges are even more obvious than in the parallel axis plot.

The GA has missed a swath of intermediate values forcp, σc, and especiallyεo. Extremities have

not been sampled forcp, a, andεo. This type of analysis is essential for determining the strength

of any conclusions about parameter controls on the system. A highηu andco near zero can be

confidently concluded to lead to good simulations. It is unlikely that a highcp will produce good

solutions, but the true bounds between 0.0 and 0.4 have not been established (Figure 4.5c). The

range of good values forσc is similarly inconclusive (Figure 4.5d). The GA has not explored the

possibility of strong faults (lowa), but the evidence points to significant weakening as essential

45


Figure 4.6: 2D crossplots with a third dimension – rank – indicated by colour, from red (best) to purple

(worst). Small random noise added to each parameter value prevents strict superposition of points.

for good solutions (Figure 4.5e). Finally, no conclusions should be drawn about the control ofεo

(Figure 4.5f).

Colour represents the third dimension of rank in the selected crossplots of Figure 4.6. The

best-ranked results (red and orange) show combinations of high viscosity and low cohesion (Fig-

ure 4.6a), low cohesion and low friction coefficient (Figure 4.6b), and low tensile limit with gen-

erally large strain weakening (Figure 4.6c). The larger spread of good points in Figure 4.6a and c

shows that a range of high viscosity, and a range of large strain weakening, can produce acceptable

results. Referring only to the good points in Figure 4.6a, if the value ofco is fixed near zero, all the

variability is expressed throughηu, which is then known as theprincipal component(e.g., Jolliffe,

1986). A principal component rotation (PCR) seeks to express a maximum of the variation in data

through a minimum of axes. Like orthogonal projections, a PCR is a linear transformation that

may not always be appropriate for analysing non-linear data sets.

Figure 4.7 shows the result of manipulating four variables projected into 2D plus rank (i.e., 3D).

The four parameter axes are rotated into the position that provides the most distinct clustering of

similar solutions ranked in the top half. The nearly verticalσc axis indicates that this parameter is

extraneous in distinguishing between the better clusters. The displayed variability is orthogonally

resolved between thea axis and a new axis, for example,c′, that is a combination ofco andcp. All

mediocre (yellow) to good points are due to a lowc′ value, in keeping with previous conclusions

(e.g., Figure 4.6b). The three better clusters are aligned along thea axis, revealing that fault

weakening is responsible for differences between more highly ranked simulations.

4.5 Self-Organising Map

4.5.1 Theory

A self-organising map (SOM) is a transformation of high-dimensional (nD) data into a lower-

dimensional (usually 2D) plot (Kohonen, 2001). Broadly speaking, its aim is to spreadnD points

over a plane in such a way that topology is respected, that is, two points lying close to one an-

other in the higher-dimensional space should lie close in the 2D plot. In doing so, it acts as a

46


Figure 4.7: Projection of four variables plus rank into 3D. Rank is indicated by colour, from red (best) to

purple (worst). Small random noise added to each parameter value prevents strict superposition of points.

Axes have been rotated so as to isolate three groups of better simulations. The box contains normalised

projection distances for the four axes, along with total axis length in brackets. Theσc axis is almost vertical.

classification algorithm that assigns the input data to cells according to similarity.

The description and analysis of the mathematics underlying a SOM can be found both in the

neural network literature and in the framework of statistical clustering tools. The following de-

scription of the algorithm highlights the simplicity of the technique. Given the task of visualising

a set ofm data points innD:1. Generate a (2D) grid. This grid can have different shapes and sizes, and its nodes can also

have different shapes. The example in this section uses a grid with hexagonal nodes, which

allow contact with more nearest neighbours than rectangular nodes. The optimal number of

nodes to representm data points is determined via a heuristic calculation, in the absence of

any established theory.2. Initialise each node by assigning it a randomnD vector of the same dimensionality as the

data points. These are the SOM vectors.3. Pick one data point and find the SOM vector that is closest to the data point according to a

certain metric (usually a simple Euclidean distance after normalisation of the dataset). This

node is thebest matching unit(BMU). Modify the SOM vector belonging to the BMU by a

certain amount so that it is closer (more similar) to the data point.4. Choose a neighbourhood of the BMU over the grid. The SOM vectors belonging to this

neighbourhood are also modified to be closer to the data point, by an amount inversely pro-

portional to their grid distance to the BMU, that is, the farther the location of the vector on

the SOM grid, the less it is incremented.5. Repeat items 3 and 4 for each point in the dataset.6. Iterate items 3 to 5 several times (usually a few thousand) to ensure convergence.The result of this algorithm is that SOM vectors belonging to nearby grid locations will tend to

be similar, since they will tend to converge towards similar data points. Conversely, data points that

47


Figure 4.8: Illustration of nodes in a SOM. Two 3D SOM vectors(T1, P1, t1) and(T2, P2, t2), separated

by a distanced, belong to two SOM data nodes. The distance node that connects the data nodes is coloured

according to the magnitude ofd. Data nodes are also coloured, using an average of surrounding distance

nodes, for a more continuous display.

are close to one another will tend to fall in grid nodes close to one another, or in the same grid node.

This should achieve the dual purpose of clustering similar points and giving a best approximation

of the original topology in thenD space. In practice, all parameter values are normalised according

to their data extents before any calculations are made. Thus, each component is equally weighted

in the determination of distance. This can be both desirable or not. In the event that comparisons

are to be made between multiple GA trials, the data extents may be different. Computed distances

can therefore not be compared, leading to very different views of the parameter space. A better

approach, in this case, is to normalise according to a subset of the “best” results, which will provide

a more stable set of limits. This presupposes that at least two parameter values are available per

dimension, to provide a minimum and maximum. Very large and very small normalised values

will indicate outliers from the subset of best data points.

Clearly, what a SOM tries to achieve (mappingnD points into 2D) is impossible except for

trivial cases. Consequently, the final result is often non-optimal. The SOM depends on a number

of parameters that control the algorithm described above. These include the number of grid points,

the initialisation of the SOM vectors, the amount by which SOM vectors are updated towards the

data point, the size of the BMU neighbourhood, and the metric used to calculate distances. There

are no standard values for these parameters in the literature, and their choice is problem specific

and often left to the user’s discretion. Also, because the SOM grid is not tied to any specific refer-

ence location in thenD space, for example, the center of the SOM is not the center of thenD space,

the appearance of maps with different initialisations may be very different. This fact is demon-

strated in the next section, and the website http://davis.wpi.edu/∼matt/courses/soms/applet.html

also illustrates the effect of different parameter choices. Provided satisfactory convergence has

been achieved, the relationship between adjacent clusters should still be represented in a relatively

homogeneous way, and a trained user will recognise this.

The SOM display in Figure 4.8 employs two different types of nodes:data nodesanddistance

nodes. The data nodes represent the SOM vectors at the grid locations described above. The

distance nodes connect the data nodes, and indicate the relativenD distance between adjacent

SOM vectors. The distance nodes are coloured to show the magnitude of the distance between

adjacent data nodes. Data nodes are also coloured, according to an average of surrounding distance

48


Figure 4.9: SOM of the extensional faulting data. (a) Data nodes, indicated by superimposed dots, are

separated by distance nodes that indicate the difference between neighbouring SOM vectors. The colour bar

shows the distance scale. (b) Only data nodes are present, labelled with the absolute rank of the associated

simulations. Unlabelled nodes contain no input vectors. The dashed line encircles the domain of best model

results. Arrows illustrate three equivalent data nodes between the two displays.

nodes, to produce a more continuous map. Colours representing short distances indicate clusters

of close points, while colours representing large distances show cluster borders, which can also

be interpreted as steep ridges dividing valleys of similar points. This relationship between SOM

nodes is valid in a statistically averaged sense: over many maps of the same data, the colours will

reflect the distances between data points in a fair manner, but it is possible that, in a non-optimal

map, points that are close in data space are not close in the SOM. The only guaranteed relationship

in a SOM is that immediate neighbours represent points that are close in data space.

SOMs have been extensively employed in recent years in both scientific and engineering ap-

plications in order to visualise high-dimensional data and highlight data structure. The SOM plots

below have been produced with the use of the MatlabTM SOM Toolbox, written by Juha Vesanto.

More details about SOMs, and the specific implementation used in this work, can be obtained at

the SOM Toolbox website http://www.cis.hut.fi/projects/somtoolbox.

4.5.2 Visualisation of the Inversion Data

The input data for the SOM visualisation consist of 48 points in a 6D space, or 48 vectors of

six components each (η, co, cp, σc, a, εo). Figure 4.9a contains the main SOM plot, in which

thirty-five data nodes, indicated by superimposed dots, are considered optimal to classify the 48

input vectors. Depending on the convergence, each SOM data node may correspond to one or

more of the 48 output images from the simulations, or it may be “empty”, meaning that no data

point is close to it. A cluster in the SOM may be recognised as any collection of darker blue nodes

that is bounded by light blue to red nodes. Such an area should represent a domain of similarly

49


Figure 4.10: Mapping of model outputs on to the SOM grid of Figure 4.9a. Data nodes (every second node)

represent outputs from the inversion process, as illustrated by the mapped images. Images are labelled with

their absolute rankings (c.f. Figure 4.9b).

parameterised simulations.

The original data vectors can be represented on the SOM by plotting each point at the grid

location of its corresponding BMU (the grid node that is most similar to it). Each SOM node

is then labelled with the rank of its associated simulations (Figure 4.9b). Well ranked solutions

(closer to 1) are similar to the target. Unlabelled nodes are empty, as described above. It is clear

that the top left domain, encircled with a dashed line, contains the best cluster of model images.

Figure 4.10 illustrates the mapping of the output images on to the SOM. Both data nodes

and distance nodes are included, as in Figure 4.9a, and the output images are labelled with their

absolute rank. It is again clear that the area in parameter space at the top left corner contains the

solutions most closely resembling the target image. The simulation ranked 5.5 in Figure 4.10 is

far from the “best” corner, being separated by a ridge, but it nevertheless shares characteristics

of the best results. This may be indicative of either the non-uniqueness of the inversion problem

(simulations with very different parameter inputs may produce similar outputs) or of a non-optimal

SOM clustering. Further analysis will prove that this point is indeed far from the best simulations

in parameter space, and is thus well represented in the SOM.

In order to analyse the influence of each model parameter on the solution, the individual di-

mensions are mapped on to the SOM network of data nodes. The images in Figure 4.11 show

the (non-dimensional) magnitude of each model parameter at each node in the SOM. Viscosity

50


Figure 4.11: Individual parameter plots (data nodes only) showing the magnitude of each separate variable

at each data node in the SOM.

shows increasing values from the bottom right corner towards the top left (best corner). Similar

regular patterns occur for other parameters. When compared to the labelled SOM of Figure 4.9b,

these parameter plots show the component values associated with each solution obtained during

the inversion. There is a clear correlation, for example, between the best model images and a

high viscosity. However, because of the averaging effect of associating multiple input vectors

with one SOM node, some details are lost. The projection in Figure 4.5a shows that badly ranked

simulations exist with high viscosities, but this information is missing in the SOM plot forηu

(Figure 4.11). Such inconsistencies between visualisation methods illustrate the benefit of using a

diversity of approaches.

Figure 4.12 contains a combined representation of the individual parameter plots of Figure 4.11.

The height of each bar indicates the normalised magnitude of each component of the SOM vector.

The order of the bars at each node is the order of the dimensions in Figure 4.11. It is easy to

understand the distance between SOM data nodes (c.f. Figure 4.9a) by comparing bar charts. The

simulation ranked 5.5 in Figure 4.10, which shares characteristics of the best solutions, is the result

of a set of very different parameter values than those of the best solutions, confirming the SOM

representation of this point, and the non-uniqueness of the problem. The worst solutions (bottom)

are not dictated by any one component, for example, a high cohesionco, by itself, does not imply

a bad solution. An unrealistically high yield stress, through a combination of high cohesion and

pressure dependence, is responsible for the simulations that fail to converge numerically (labelled

“10” in Figure 4.10). Either of Figures 4.11 or 4.12 allows a quick scan of the parameter controls

on model outputs.

As specified earlier, different initialisations for the SOM will result in different final maps.

Figure 4.13 compares the SOM used above, with five variations, each labelled according to the

absolute ranking of solutions. In general, a greater number of isolated large-distance nodes (green

to red) reveals poor clustering, or a non-optimal SOM. The original map (Figure 4.13a) represents

all the better solutions, in the top half, with acceptable clustering, but the bottom half of the map

is not optimal. Figure 4.13e is the worst representation of the data, and Figure 4.13f is the best.

Labels are not consistent across all maps because, where more than one point occupies the same

SOM node, the label belongs to the last simulation added to that node. The solution ranked 5.5,

51


Figure 4.12: Visualisation of the normalised SOM vectors at each data node. The order of the bars refers to

ηu, co, cp, σc, a, andεo. The arrow indicates the best model output.

when labelled, is consistently far from the best corner, again reinforcing the non-uniqueness that

was signalled in earlier plots.

4.6 Discussion

Geoscientific problems, and geodynamic ones in particular, are among the most difficult inverse

problems. Their full solution is beyond not only current computational power, but also existing

mathematical tools. The combination of interactive inversion and visualisation is not a panacea,

but offers a number of advantages, some quite obvious, some less so.

The SOM is a successful method for displaying high-dimensional data, and provides a way to

deal with the many solutions that are generated by an inversion process. It produces relatively

simple plots that are useful for determining which parameters most affect the process under study.

Related to this is the determination of possible mechanical behaviour, given a certain modelling

scenario. The drawback of a SOM is that it cannot represent data that are nonexistent, meaning

that unsampled space is unrepresented. A parallel axis display or multi-dimensional projection

incorporates the full extent of the space under investigation, so data gaps can be easily detected.

The SOM is a clustering technique that is used to best advantage when a user is confident of

adequate sampling over the parameter space.

Keeping in mind the weaknesses in the coverage of parameter space that are highlighted by

the parallel axis display and multi-dimensional projections, all plots in combination could be sum-

marised as follows, converting to natural parameter values: the best matches to the target image

are achieved with a high viscosityηu > 5×1022 Pa·s, and a low cohesionco < 5 MPa. The low

52


Figure 4.13: (a) Original SOM from Figure 4.9a. (b) – (f) Five variations due to different random initial-

isations. Labels indicate simulation ranks. Labels are not consistent across all maps because, where more

than one point occupies the same SOM node, the label belongs to the last simulation added to that node.

friction coefficientcp (0.2 – 0.3), significant strain weakeninga (0.6 – 0.7), and fairly rapid rate of

weakening toεo (0.3 – 0.4) imply overall weak faults, as has been suggested for major structures

(Lachenbruch and Sass, 1980, 1992; Bird and Kong, 1994; Bird, 1995). Physically, this would

promote large fault spacing by ensuring that, once a fault is initiated, extensional stresses to either

side quickly drop to below yield values, so that new faults do not form nearby. The tensile limit

σc (2 – 6 MPa) may vary somewhat in the lower part of its range, again indicating weak rocks,

although this should only have an effect near the surface where tensile stresses are more likely.

Such a summary emerges quite naturally from the three sets of visualisation plots.

The approximate understanding of the search space can be used to further interact with the

inversion itself. One obvious way is to remove parameters once they have been “optimised”, or

parameters that are shown to have little impact on the results. This reduces the dimensionality

of the problem. Another approach is to focus the search into smaller domains, where the “good”

simulations lie. This can result in faster convergence as well as the opportunity to increase the

search resolution of certain parameters. A user with a good knowledge of the inversion algorithm

(a GA in this case) can also use the information of the current state of convergence in order to tune

certain parameters controlling the inversion itself. The opportunity to do this during the course

of an inversion can considerably improve performance. In the case of a GA, this may involve

changing the cross-over or mutation operator, or changing the number of individuals.

This inversion and visualisation system can be a vehicle to facilitate the transfer of technology

to applied modellers. The user-friendliness inherent in the approach (which removes the user from

53


the underlying mathematics and allows him orher to concentrate on the geological aspects of the

problem) has been very well received by industry representatives. It can be a crucial factor in the

acceptance of geological modelling as, for example, a mineral or oil exploration tool.

4.7 Chapter Summary

During any inverse modelling, it is in the interest of the modeller to capitalise on the availability

of accumulated solutions in order to understand the dynamics and non-uniqueness of the problem

as fully as possible. This is easy to do within the interactive inversion system, in contrast to most

inverse procedures that return a single best solution. Combined with the systematic ranking of

trial solutions, visualisation methods such as parallel axis displays, multi-dimensional projections,

and self-organised maps represent the high-dimensional parameter space in a clear and simple

2D visualisation environment. In the first instance, methods that represent the entire parameter

space, such as the parallel axis display and multi-dimensional projections, provide an overview of

the sampling by the inversion, and expose any gaps in coverage. The graphical summaries also

offer a platform from which to draw conclusions regarding the controlling physical factors and the

connections between them.

54

Chapter 5

MODES OF CRUSTAL EXTENSION

5.1 Introduction

Chapters 3 and 4 give some preliminary insights into rheological controls on fault spacing

during extension of the crust. This chapter builds on those results, using more sophisticated and

realistic representations of crustal strength profiles.

Continental lithosphere may be highly extended without entirely rifting to a new ocean basin.

For example, total Cenozoic strain estimates of up to 100% (β = 2) have been proposed for parts

of the Basin and Range in the western U.S.A., which have not been rifted (e.g., Lachenbruch

and Sass, 1978; Jones et al., 1992). A total of up to 80% extension is suggested on a very large

scale of 300–500 km (Wernicke et al., 1982), and Niemi et al. (1999) have even suggested 500%

extension in the Death Valley region, California, since the middle Miocene. Stretching of the

crust may then be accommodated by two contrasting phenomena: distributed, closely spaced, and

limited-slip normal faulting over a large area, or localised, large-strain normal faulting that can

result in the complete dissection of the upper crust and exhumation of the lower crust. Examples

of distributed faulting exist in various parts of the North Sea basin (e.g., Fossen and Rœrnes, 1996;

Viejo et al., 2002), where the faults are characterised by a relatively steep angle and small offset.

Normal faults that exhibit a low angle and very large displacement, juxtaposing exhumed high-

grade metamorphic rocks against near-surface rocks, are the hallmarks of a metamorphic core

complex (MCC), as in the Basin and Range (e.g., Axen et al., 1990; Duebendorfer et al., 1990;

Hodges et al., 1990; Spencer and Reynolds, 1991) or the Aegean (e.g., Gauthier and Brun, 1994;

Forster and Lister, 1999; Burchfiel et al., 2000; Gessner et al., 2001).

Why should extending lithosphere in places form a MCC rather than fail in a distributed man-

ner? One possibility is that major lateral discontinuities in the strength of either the upper crust

(e.g., pre-existing faults) or the lower crust (e.g., partial melt zones) focus stresses and localise

extension. Christiansen and Pollard (1997) document field observations where shear zones have

nucleated from pre-exisiting dikes that act as weaknesses, although these shear zones are not as-

sociated with a MCC. In their conceptual models of extensional tectonics, both Wernicke (1985)

and Wernicke and Axen (1988) assume an immediately active horizontal or low-angle detachment

surface through the upper crust, which then controls faulting and exhumation of the lower crust. In

some analogue modelling experiments, initial faults end up controlling fault spacing and the mode

of extension (e.g., Koyi and Skelton, 2001). Brun et al. (1994) conclude that a weak rheological

zone at the base of the upper crust, such as a magma body, is required to trigger a MCC. Other ana-

logue modellers (e.g., Mulugeta and Ghebreab, 2001; Corti et al., 2001) employ a pre-weakened

55

CHAPTER 5. MODES OF CRUSTAL EXTENSION Wijns Ph.D. Thesis

Figure 5.1: (a) Initial geometry and boundary conditions for the numerical model. Stripes in the upper

and lower crust are marker materials for visualising deformation. (b) Representative maximum shear stress

profile (solid line) through the crust for a given strain rateε. Neglecting localised strain weakening, strength

increases with pressurep from co at the surface to a maximum value at the basezu of the brittle upper

crust, at which point the yield curve intersects the viscous, temperature-dependent flow law. To avoid any

extremely low viscosity, the value at some depthzc is adopted as a minimum value and is constant to the

basezl of the crust.

lower lithosphere to localise stresses and influence the resulting extension, and Corti et al. (2001)

add an imposed basal velocity discontinuity to this. Some numerical modelling experiments also

assume initial weaknesses, and proceed to investigate related factors that vary the results of ex-

tension (e.g., Dunbar and Sawyer, 1989). Although such weak features do serve to focus stresses,

in the absence of lateral heterogeneities, vertical contrasts in rheology dictate whether one or the

other mode of extension will result. In fact, small lateral heterogeneities are often insufficient to

trigger a MCC mode when the rheological layering promotes distributed faulting.

The mode of extension is thus determined by the combined ability of both lateral and vertical

rheological structure to focus stresses in the brittle upper crust. The numerical modelling in this

chapter concentrates on the role of vertical rheological contrast in dictating the spacing between

fault zones and the accompanying mode of distributed faulting or MCC formation, building on

the results of the inversion in Chapter 3. The 2D vertical sections are readily compared to much

published analogue modelling, including that of Brun et al. (1994).

5.2 Model

As in previous models, the upper crust behaves viscoplastically, which gives rise to the local-

isation of deformation and necking of layers. However, the transition from brittle behaviour to

ductile flow in the lower crust is now governed by temperature, which creates the possibility of

ductile material cooling and entering the brittle (plastic) field. The initial geometry of the model

(Figure 5.1) corresponds to 20 km of upper crust above 40 km of lower crust, along a length of

160 km. Horizontal and vertical stripes are simply marker materials to enhance visualisation (as in

Brun et al., 1994). Above these two crustal layers is the “air” layer that does not influence the me-

chanics of the problem. All walls of the closed bounding box are free-slip, and vertical walls have

zero heat flux. Extension proceeds by applying a uniform velocity to the right-hand boundary,

equivalent to 100% strain in 5 Ma (Gessner et al., 2001), or 6.3×10−15/s. This boundary velocity

56

Wijns Ph.D. Thesis CHAPTER 5. MODES OF CRUSTAL EXTENSION

Table 5.1: Natural values for extension model parameters.

Parameter Value

depthzu, upper crust 20 km

depthzl, lower crust 60 km

velocityU , boundary 3.1 cm/yr

strain rateε, initial 6.3×10−15/s

densityρu, upper crust 2700 kg/m3

densityρl, lower crust 3000 kg/m3

gravityg 10 m/s2

cohesionco 16 MPa

friction coefficientcp 0.44

strain weakeninga 0.8 (0.2, simulation D)

“saturation” strainεo 0.5 (1.0, simulation D)

thermal diffusivityκ 10−6 m2/s

temperatureTs, surface 0oC

is low enough to not create tensional stresses, so that gravity, in effect, drives the deformation.

The yield law (2.8) for brittle crust uses a rock cohesionco = 16 MPa and a pressure depen-

dencecp = 0.44, similar to the model in Chapter 3 The strain weakening parameters,a andεo

(equation 2.9), vary in order to investigate the influence of fault weakening on the mode of ex-

tension, but the exponent is fixed atn = 1, based on the previous conclusion that it is the least

important parameter for controlling the outcome (Chapter 3).

The boundary between upper and lower crust is defined at all times by a fixed density contrast.

Initially, this corresponds to the depth at which the temperature dictates a change from brittle to

ductile behaviour. With increasing extension, the initially Newtonian lower crust may be exhumed

into the brittle field and undergo faulting. It behaves mechanically as the upper crust does, but

retains its original density (i.e., metamorphic grade and mineralogy). There is no thermal expan-

sion, which provides a minor buoyancy effect compared to compositional density contrasts; nor is

melting modelled.

Viscosity η varies with temperatureT according to the Frank-Kamenetskii relation (Frank-

Kamenetskii, 1969, Chapter 6, p. 340–41)

η(T ) = ηo e−cT . (5.1)

This is a simplification of an Arrhenius rheology, where the constantsηo andc are chosen such

that the viscosity at the interfacezu between upper and lower crust, which is the initial brittle to

ductile transition, satisfies the maximum yield stress

τyield(zu) = η (T (zu)) ε ,

that is, the maximum shear stress profile through the crust is continuous. Appendix C.1 contains

a full development of this relation. To avoid an extremely low viscosity at the base of the crust,

57


η(T ) > η(Tc) for some maximum temperatureTc. The constants used in equation (5.1) then result

in a maximum viscosity variation of two orders of magnitude from the top of the lower crust to the

point of minimum viscosity. In a constant velocity extension model, the crustal strength profile

(Figure 5.1b) evolves with a changing strain rate, but provides a convenient starting point from

which to characterise the crust.

The surface of the upper crust is maintained at 0C. Since the temperature scale can be chosen

independently of other variables, it is fixed only for comparison with field data in a later section.

For reference, an initial temperature gradient of 17.5C/km results in a brittle to ductile transition

at 350 C, in the range discussed by Brace and Kohlstedt (1980) and references therein, and

McKenzie and Fairhead (1997). This transition temperature also corresponds to theoretical flow

laws derived by Handy et al. (1999) for various crustal components.

Table 5.1 contains parameter values for the natural system, which apply to all simulations

unless explicitly stated otherwise. Values for the viscosity law (5.1) are not listed because they

change with every simulation, but the discussion section offers some natural viscosity equivalents.

5.3 Results

As in previous simulations, bands of localised plastic strain are proxies for fault zones in the

continuum code. Unless otherwise noted, a maximum of 80% strain weakening (a = 0.8) occurs

after an accumulated plastic strainεo = 0.5. These values reflect evidence, from both numerical

experiments (e.g., Bird and Kong, 1994) and field-based heat-flow measurements (e.g., Lachen-

bruch and Sass, 1980, 1992), that major faults may undergo significant weakening.

The different behaviours of the crust under extension are partly parameterised by the ratiorτ

of integrated upper to lower crustal strength. Appendix C.2 contains the calculation of integrated

strengths, based on the maximum sustainable shear stress at a given strain rate.

5.3.1 Constant Viscosity

The two contrasting modes of crustal extension are first illustrated for a constant viscosity

lower crust (Figure 5.2), for comparison with the results of Brun et al. (1994). When the system

is characterised by a smallrτ = 0.53 (i.e., relatively high-viscosity lower crust), the result is

distributed faulting (simulation A, Figure 5.2i). The upper crust develops many closely-spaced

steep faults, each of which accomodates limited strain, and the interface between upper and lower

crust remains relatively flat. This result is very similar to the tilted block mode of Brun et al.

(1994), although, according to the information supplied,rτ ≈ 1.8 for their analogue model. This

discrepancy is probably due, in large part, to differences in material properties for the experiments,

and is addressed below in the discussion. Even with more than 80% extension (β > 1.8), the upper

crust in simulation A is never completely dissected. New steep faults form soon after older faults

have accommodated a small amount of shear strain, breaking the upper crust into a series of small

blocks that undergo limited rotation.

Although Brun et al. (1994) propose that a weakness is needed in the lower crust in order to

58


i ii

Figure 5.2: Simulations with constant viscosity lower crust. (i) Simulation A: evolution of distributed

faulting mode withrτ = 0.53. (ii) Simulation B: evolution of MCC mode caused by a uniformly weak

lower crust andrτ = 2.52. Total extension of (a) 25% (1.2 Ma), (b) 53% (2.7 Ma), and (c) 83% (4.2 Ma).

localise stresses and trigger a MCC mode of extension, simulation B (Figure 5.2ii) shows that

a uniformly weak lower crust (highrτ = 2.52) is sufficient to achieve this behaviour. Defor-

mation is accommodated by only a few normal fault zones. Displacement is large within each

fault zone, which, through fault weakening, remains active even after rotating to a shallow dip.

The lower crust flows easily to isostatically compensate for localised thinning of the upper crust,

which enhances block rotation and continued strain on low-angle faults. Despite the Newtonian

rheology of the lower crust, strain is localised as a result of kinematic interaction with the plastic

(non-Newtonian) upper crust. The lower crust is first exhumed atβ = 1.6.

5.3.2 Temperature-Dependent Viscosity

The main limitation in using a constant viscosity for the lower crust, and therefore a limitation

of the analogue modelling, is that lower crustal material that rises towards the surface remains

very weak, although it should cool and strengthen, even becoming brittle. This phenomenon does

not affect the case of distributed faulting, where the lower crust remains buried below a laterally

stable brittle to ductile transistion (Figure 5.2i). In MCC mode, however, exhumation and cooling

will influence the tectonics by changing the rheological behaviour of lower crust material.

A weak lower crust with a temperature-dependent rheology, in a laterally uniform system,

gives rise to the MCC mode in Figure 5.3. The rising lower crust enters the brittle domain, but

cooling is slow enough, relative to exhumation, that brittle deformation of the lower crust does not

penetrate to great depth. The propagation of faults from the upper crust into the exhuming lower

crust allows continued strain along the original structures, until they become very gently dipping

or even flat-lying detachment surfaces accomodating large displacements. There is evidence in the

vertical lower crust markers that these faults are spatially connected to diffuse, high-shear zones

59


Figure 5.3: Simulation C: evolution of MCC mode (rτ = 2.15) with both a viscosity and a brittle to ductile

transition that are temperature dependent. Total extension of (a) 16% (0.8 Ma), (b) 53% (2.6 Ma), and (c)

82% (4.1 Ma). Faults continue to propagate through the newly brittle lower crust as it nears the surface, and

are connected to diffuse, high-shear zones in the ductile region, indicated with pairs of solid lines in (c).

Figure 5.4: Simulation D: MCC mode withrτ = 2.15, but a maximum fault weakening of 20%, rather than

80% as in simulation C. Total extension of 80% (4.0 Ma).

that extend to the base of the entire crust. These deep shear zones are traced in Figure 5.3c, and

show much greater localisation than in the case of constant viscosity (Figure 5.2ii). Such brittle

fault to ductile shear continuity has been suggested in an intraplate setting by Zoback et al. (1985).

Despite a decrease inrτ relative to the constant viscosity MCC of simulation B, fewer fault

zones are generated in simulation C. The greater tendency to MCC mode is due to the variable

viscosity profile that, although higher in an integrated sense, allows much lower viscosity material

near the base of the model to dominate crustal flow and tectonic expression. This is apparent when

comparing the strain visualisation markers in the lower crust (Figures 5.2(i)c and 5.3c).

In order to determine the importance of fault weakness with respect to MCC formation, simu-

lation D (Figure 5.4) has a maximum strain weakening of only 20% (a = 0.2) instead of 80% as in

simulation C, and this weakening accumulates more slowly throughεo = 1.0 instead of 0.5. The

resulting extension still produces a MCC mode, although the characteristics are different. Fault

zones are more numerous and more diffuse, and the upper crust tends to neck rather than produce

planar shear zones. Although these stronger fault zones initially evolve as for the case of simula-

tion C, they are less effective in accomodating continued strain, so that higher inter-fault stresses

60


Figure 5.5: (a) Temperature-time paths for an exhumed footwall of a MCC from fission-track ther-

mochronology and Ar40/ Ar39 ages (from Gessner et al., 2001) and from numerical simulation C. The

numerical data are scaled to match the 300C initial field temperature. (b) Schematic cross section of

Central Menderes MCC from Gessner et al. (2001). The asterisk denotes the location, in high-grade meta-

morphic material, of the laboratory analyses. (c) Part of simulation C with an asterisk showing the final

location of the numerical temperature sampling point.

develop, causing new faults to form.

5.3.3 Field Validation

A test of the physical validity of MCC simulation C is the comparison of numerical results

with cooling data from the Kuzey detachment in the Central Menderes. After 100% extension,

the crust is approximately 30 km thick, in agreement with a seismic interpretation of western

Turkey by Saunders et al. (1998). Both field and numerical temperature-time curves for exhumed

footwall material are similar (Figure 5.5a). The field data, from Hetzel et al. (1995) and Gessner

et al. (2001), reflect measurements from apatite and zircon fission-track thermochronology, and40Ar/ 39Ar ages for the higher temperatures. Temperature in the numerical model is scaled to

match the initial 300C of the exhumed field sample. The numerical curve does not reach the

surface temperature of 0C because, due to the continuum nature of the code, the upper crust is

never completely removed from above the exhumed lower crust, so that the lower crust remains

marginally buried and begins to stabilise at approximately 75C.

The match between cooling rates for the numerical and field data suggests that the model for-

mulation provides an adequate physical description for the investigation of extensional tectonics.

Parsons et al. (2001) provide observations that could further argue the case that weak lower crust

61


Figure 5.6: (a) Endmember of MCC mode (rτ → ∞) represented by the brittle upper crust over a free-

slip lower boundary. Total extension of 3% (0.15 Ma). The profile above is the normalised basal velocity.

(b) Same simulation as (a) at a total extension of 38% (1.9 Ma). Because of the continuum nature of the

code, there is some stress transmitted across the graben. At this stage, gravitational slumping also begins to

affect the simulation because there is no lower crust isostatic compensation. (c) Endmember of distributed

faulting mode (rτ → 0) represented by the brittle upper crust over a no-slip boundary. Total extension of

40% (2.0 Ma).

promotes MCC formation. A low seismic-velocity middle crust (hotter and/or weaker) is present

under the Buckskin-Rawhide core complex in Arizona, U.S.A. Further west, the middle crust

shows high velocities (colder and/or stronger) under the steep, normal faults of the Salton Trough.

5.4 Discussion

5.4.1 Continuum Between Modes

Pre-existing faults or thermal and rheological heterogeneities are not required to produce a

MCC mode of extension. A large ratio of upper to lower crustal strength is necessary, often

overcoming distributed initial faults. Increasing the ratiorτ eventually leads to a state where a

single fault is nucleated and remains the only zone of failure. This is illustrated by the endmember

caserτ → ∞ of a single brittle layer extended over a free-slip lower boundary (Figure 5.6a and

b), which is the limit of very weak lower crust that effectively decouples the upper crust from the

tectonics below. Once a fault or symmetric graben is nucleated, boundary stresses can no longer

be communicated laterally through the layer, and the velocity gradient approaches zero away from

the fault, as apparent in the basal velocity profile plotted above the section. If the fault was truly

a discontinuity (impossible with theEllipsis continuum code), the velocity gradient would be

62


2

3

76

3

6

6

88 8 9

3

4 4

5 6

77

0 1 2 3 4 5strength ratio rτ

0

2

4

6

8

10

12

num

ber

of fa

ults

alo

ng 1

60 k

m

η = cstr

h = 0.5

rh

= 0.4

rh

= 1.0

rρ = 1.0

wf = 0.1

Figure 5.7: Number of major fault zones as a function ofrτ . Data labels indicate a subjective evaluation

of the model result between the distributed endmember (0) and the MCC endmember (10). Unless other-

wise indicated, the standard simulations (filled circles) have a temperature-dependent lower crust viscosity,

thickness ratiorh = 0.5, density ratiorρ = 0.9, and fault weakness factorwf = 0.53. Whilerτ controls

the overall trend, changes in the other variables, especiallyrh, move results vertically for equalrτ , and

horizontally for equal mode or number of faults.

zero everywhere except at the discontinuity, unlike the profile in Figure 5.6b, where some stress

continues to be transmitted. The lower crust, where present, fulfills the role of stress distributor

and ensures non-zero traction at the base of the brittle crust. For the hypothetical case of a lower

crust that is so strong as to maintain a constant stress at this interface,rτ → 0 and a zero-slip basal

condition exists (Figure 5.6c). The resulting uniform yield of the upper crust is the endmember of

distributed faulting.

Therτ → ∞ case in Figure 5.6b will not necessarily result in MCC formation. The presence

of a volume of mobile lower crust may be essential for block rotation, shallowing of fault dips, and

lower crust exhumation. Fault zone localisation and significant block rotation are complementary

expressions of the weak lower-crust endmember. These two phenomena might not be separable in

nature.

Simulations A to C, together with the endmember cases in Figure 5.6, suggest that the number

of major fault zones that develop to accomodate extension correlates inversely with the tendency

towards MCC mode. This is borne out in Figure 5.7, in which the number of faults and mode of

extension are plotted against the strength ratio of the crustrτ . Faults are counted if they transect

the upper crust, and a graben is counted as one fault zone. To illustrate, in Figure 5.6b, ther

is only one fault zone. The trend is that fewer fault zones develop with increasingrτ . More

simulations have been run than are shown in the previous section, in order to explore the phase

space of extension modes. Results are labelled with a subjective rank from 0 to 10, corresponding

to position between the endmembers of, respectively, distributed faulting and MCC mode. In

63


34

3

6

6

8

4

8

5 6

8 9

0 1 2 3 4 5 6 7 8 9rτ / rh

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

(∆f

β /z

l )

Figure 5.8: Spacing∆f of major fault zones (normalised by final crustal thicknesszl/β) as a function of

rτ/rh for temperature-dependent simulations withrρ = 0.9 andwf = 0.53. Data labels indicate mode as

in Figure 5.7. The trend should pass through the origin (c.f. Figure 5.6c forrτ/rh → 0) and towards infinite

spacing forrτ/rh → ∞, although these cases would not be reached for natural parameter ranges for the

crust.

addition torτ , the following parameters simplify the classification of results: the ratiorh of upper

to lower crust thickness, the ratiorρ of upper to lower crust density, and the fault weakness factor

wf = a/(1 + εo). A largerwf implies greater and/or more rapid fault weakening. According to

Figure 5.7, variations inrτ exert the greatest control on the mode of extension. Of secondary but

significant importance isrh, while rρ andwf have a minor influence. The extension rate in these

simulations may be too high for density contrasts to appreciably affect exhumation of the lower

crust. Buoyancy forces will be more evident for lower extension rates, at some point dictating a

transition to diapirism forrρ > 1.

Numerical work by Bai and Pollard (2000) may explain why strength contrasts control the

mode of extension. They relate fracture spacing in layered rocks to the stress distribution between

fractures, which depends on the physical properties of neighbouring layers. The observations on

the numerical simulations in this chapter are that:

1. at the onset of extension, many faults are nucleated in both modes;

2. whenrτ is small, many of the nascent faults zones are activated;

3. whenrτ is large, few fault zones are activated.

For a fractured layer sandwiched between two unfractured layers, Bai and Pollard (2000) find

that there is a critical fracture spacing to layer thickness below which the interfracture stress is

everywhere compressive and new fractures will not form. At greater fracture spacings, the stress

becomes tensile, and eventually overcomes the tensile strength of the layer, leading to new in-

64


fill fractures. This critical spacing to thickness ratio increases with increasing ratio of Young’s

modulus between the fractured and neighbouring layers. Although the Bai and Pollard (2000)

analysis is for linear elastic media, there is an exact analogue in the results for normal faulting of

viscoplastic layers. As the strength ratiorτ increases, the spacing between active faults increases.

Thus, the magnitude of lateral stress transfer from the unfaulted lower crust to the upper crust is

key to the mode of extension, where the mode is specifically controlled by the number of active

fault zones. A relatively strong lower crust provides greater traction at the base of the upper crust,

which results in the yield stress being reached at shorter spacings between faults. The existence

of few fault zones in the presence of a weak lower crust is simultaneously linked to the ability of

crustal blocks to rotate because of lower crustal mobility, leading to MCC formation.

The effect of crustal thickness ratios is more straightforward. For equal values of the integrated

upper crustal strength, a thicker upper crust (largerrh) simply requires more strain until it is

dissected. A thin lower crust (largerrh) relies largely on lateral flow to compensate for unloading,

and thus, at the same integrated viscosity as a thick lower crust, does not accomodate a MCC mode

of extension as easily.

When the different thickness ratiosrh are accounted for, the standard deviation of model be-

haviour is largely reduced. Figure 5.8 shows the more or less direct variation between normalised

fault spacing andrτ/rh for the temperature-dependent simulations withrρ = 0.9 andwf = 0.53.

The greatest fault spacing achieved with the model setup is about 1.6 times the total crustal thick-

ness. The deviation from the trend for the two simulations with highestrτ/rh is most likely due

to boundary conditions, that is, the box is finite and there will always be at least one fault dur-

ing the experiment. The simplistic empirical prediction of fault spacing and extension mode in

Figure 5.8 would be difficult to relate to field observations in a quantitative sense, for example,

through measurements of crustal thickness orβ factor. Other physical formulations for the model

would likely change the absolute values of the defining parameters, but there is a continuum of

behaviour in extension that depends mostly upon the contrast between upper and lower crustal

strength, modulated by crustal thickness ratios. Furthermore, after significant extension (β > 1.8),

there are only two endmember modes. A smallrτ results in a contiguous upper crust even after

extreme stretching, whereas a largerτ initiates more localised strain that evolves into a MCC.

5.4.2 Comparison with Other Modelling and with Nature

As suggested above, the value ofrτ = 1.8 for the distributed faulting mode of Brun et al. (1994)

should lead to an intermediate mode of extension, according to the numerical results above. The

thicker lower crust (rh = 0.33) in Brun et al. (1994) would push the model even further into

the MCC field (c.f. Figure 5.8). A sand upper crust (not viscoplastic), less extension, periodic

extension due to the experimental set-up, and insufficient basal lubrication may all be factors that

hinder meaningful comparison between these numerical and analogue results.

The crustal model in this chapter does not incorporate upper mantle, which precludes direct

comparison with the modes of extension of Buck (1991), for example. However, an equivalent

“narrow rift” mode in the simulations in this chapter is simply a juvenile MCC mode in terms of

65


total extension. Strain is localised in one or a few zones, but the lower crust is not yet exhumed.

Analogue modelling by Benes and Davy (1996), in which they distinguish model behaviour using

a similar measure of layer strength ratios as here, purports to uphold the modes of Buck (1991).

They conclude that very weak lower crust gives rise to a MCC mode of extension, and results

above are broadly consistent with theirs. While Benes and Davy (1996) also classify a narrow

rift mode, they change the upper to lower crust thickness ratio with respect to their core complex

models to achieve this. A narrow rift, which they classify after only 9.3% extension, is the result of

a thicker upper crust and a thinner lower crust, which both contribute to hindering exhumation of

the lower crust. Given greater extension, the narrow rifts of Benes and Davy (1996) are probably

core complexes, whereas the “wide rifts” will remain as such, equivalent to the distributed faulting

models in this chapter. In this sense, narrow rifts are an intermediate stage and not an endmember

behaviour.

The upper mantle will have some influence on the behaviour of the crust during extension.

The thinner or stronger the lower crust, the more likely this influence will manifest itself. The

lower crust is sufficiently thick and ductile in the model above that it effectively decouples the

crust from the upper mantle. If the upper mantle is strong, it will then remain relatively flat, as

the lower crust will flow more quickly than the mantle in response to any unloading. This case is

implicit in the use of a rigid, zero-traction bottom boundary, and corresponds to the observations

of a flat Moho boundary under areas of high extension in Arizona (Hauser et al., 1987). On the

other hand, if the upper mantle is of comparable viscosity to the lower crust, it will participate in

isostatic compensation. In the present model, this would be equivalent to increasing the thickness

of the lower crust.

The geometric origin of low-angle detachment faults remains controversial. According to Scott

and Lister (1992) and Livaccari et al. (1995), for example, structural field relations favour initiation

and activation of some faults at a shallow dip. Others, such as Buck (1988) and Wernicke and

Axen (1988), propose initial high-angle faults that subsequently rotate to shallow dip. In all the

simulations above, initially high-angle normal faults form as planar features that evolve to lower

angles because of lower crustal flow, or isostatic compensation, during unloading of the footwall.

The faults are not listric but initially dip at approximately 45 due to the nature of the implemented

rheology. Listric faults should enhance block rotation, so that the trend from distributed faulting to

MCC mode will be moved towards lower values ofrτ in systems of listric faults. The rotation of

high-angle faults first manifests itself at the base of the upper crust, where the changes in the local

stress field are greatest. Continued unroofing and lower crustal flow can eventually produce very

gently-dipping faults such as those of Figure 5.3c. These results do not contradict the hypothesis

that detachment faults may form at low angles. There is simply no condition present for an initial

stress field rotated from the vertical, and new faults form at a specific angle toσ1. However, the

results do illustrate and support the hypothesis that initial high-angle faults can rotate to low angle

and develop into flat-lying detachment faults that continue to slip due to significant weakening.

It is worth noting a Coulomb failure analysis by Wills and Buck (1997) that determines the

likelihood of fault slip at shallow dips due to specific boundary or loading conditions. They con-

66


clude that slip will not occur on low-angle surfaces unless there is an unlikely combination of

localised, near-lithostatic pore pressure, with unsustainably high tensile stresses in the upper 5 km

of the crust. On the other hand, if normal faults form at shallow dips because of pre-existing weak-

nesses, for example, earlier, gently-dipping thrust faults (e.g., Horvath, 1993), such structures can

alter the stress field and also override the control of vertical rheological strength contrasts on fault

spacing.

Through numerical experiments on a single elasto-plastic layer, Lavier et al. (2000) find that

both the layer thickness and the amount of fault weakening can affect the degree of offset per

fault. However, because the Lavier et al. (2000) experiments are over an inviscid substrate, they

nucleate a single fault zone only (this includes a single graben or horst with secondary faults in

the same zone), that is, the endmember of MCC mode (Figure 5.6a). They acheive fault rotation

by allowing influx of filler material from the base. By including the important role of the lower

crust in distributing stresses, results in this chapter determine fault weakening to be less important

than the rheological contrast between upper and lower layers of the crust.

A more sophisticated model by Lavier and Buck (2002) includes a visco-elastic lower crust.

While their purpose is to investigate the influence of cooling rates on faulting style, their results fit

within the description of modes given above. Estimated values ofrτ for the simulations presented

by Lavier and Buck (2002) are all very high, such that the experiments are well within a MCC

mode and consequently nucleate single fault zones, albeit sometimes composed of more than one

individual fault.

Translation ofrτ values into natural viscosity for the lower crust allows a comparison with

independent estimates. It is difficult to express the integrated viscosity as a single value, since the

profile varies considerably with depth. Observational estimates based on, for example, post-glacial

rebound or post-seismic relaxation of crustal velocities are probably reflective of the most mobile

part of the lower crust. The minimum viscosity of the lower crust (belowzc) is therefore used

for crude comparison. The distributed faulting mode withrτ = 0.53 has a lower crust viscosity

of 1022 Pa·s. The MCC mode in simulation C returns a viscosity of 4.5×1020 Pa·s. An estimate

of lower crustal viscosity is lacking for western Turkey, but, in the western U.S.A., Pollitz et al.

(2001) calculate an upper limit of 5×1020 Pa·s based on geodetic measurements of post-seismic

velocity fields after the 1999 Hector Mine earthquake in California. This value is very close to

the model value for the MCC mode, but the distributed mode viscosity appears very high. For

a better comparison with the western U.S.A., the initial lower crust thickness is scaled to 20 km

(rh = 1), leaving a total crustal thickness of between 20 and 25 km after extension, consistent

with the regional crustal thickness model of Chulick and Mooney (2002) for the Basin and Range.

Now, the distributed faulting mode occurs for a viscosity of 6×1021 Pa·s, and the MCC mode for

a viscosity of 2.5×1020 Pa·s or lower.

Byerlee’s law (Byerlee, 1968), the result of numerous laboratory experiments, postulates a

profile of brittle yield strength versus depth that is relatively independant of rock type. Pore

pressure also exerts a strong influence on rock strength, but departures from hydrostatic level are

usually not sustained over the length and time scales of the modelling above. The thickness of the

67


upper crust, or, equivalently, the depth to the brittle to ductile transition, will therefore determine

the integrated strength of the brittle layer. The temperature gradient will be the first-order control

on this thickness, and it will also govern the viscosity of the lower crust, so that the mode of

extension can reflect the thermal regime at the time of deformation. The mineral composition

(including grain size) and water content of the lower crust are similarly responsible for its viscosity

(e.g., Wilks and Carter, 1990; Ross and Wilks, 1995; Rybacki and Dresen, 2004).

5.5 Chapter Summary

Numerical modelling shows that the mechanical stratification of the crust provides the funda-

mental control on fault spacing, and, ultimately, the mode of extension. Pre-existing structures

and weaknesses are often thought to govern the behaviour of continental crust under extension,

and many prior studies have focussed on the effect of heterogeneities in triggering faulting and ex-

humation of lower crustal material. The role played by such features is, in fact, subordinate to that

exerted by the rheological contrast from upper to lower crust, quantified by the ratio of integrated

upper to lower crustal strength. This ratio dictates fault spacing in the upper crust through the

stress transfer from ductile lower crust to brittle upper crust. The fault spacing, naturally linked to

the ability of the lower crust to flow, controls the subsequent evolution of the normal fault systems.

A small ratio of upper to lower crustal strength leads to a distributed mode of faulting, where many

faults take up limited strain, and the upper crust is never completely pulled apart. A large strength

ratio results in few active fault zones, each accomodating a large amount of strain. This leads

to block rotation and complete dissection of the upper crust, with the consequent exhumation of

lower crustal rocks. Examples of this mode of extension may be the metamorphic core complexes

of the Western U.S.A. and the Aegean. The actual critical strength ratio for the transition between

modes will depend upon such factors as the relative thickness of the lower crust with respect to

the upper crust, and the degree of fault weakening. These are secondary factors that do not alter

the primary importance of vertical rheological contrasts in determining the mode of extension.

Changes in the relative strengths of crustal layers can most easily be related to heat flow, and the

mineral composition and water content of the lower crust.

68

Chapter 6

CRUSTAL-SCALE FLUID FLOW: CARLIN SYSTEM

6.1 Introduction

From the generic modelling of Chapter 5, the present chapter pushes the use of numerical sim-

ulations towards practical integration into a geologically-based industry. The Carlin gold trend, in

northern Nevada, U.S.A., serves as the example of crustal-scale tectonic and fluid flow modelling

applied to a mineral exploration problem.

The results in Chapter 5 demonstrate that the vertical stratification of strength controls fault

spacing and the large-scale tectonic behaviour of the crust. However, pre-existing structures are

often important in actually localising the strain patterns that describe this large-scale behaviour.

During a compressional orogeny, a crustal weakness may serve as the locus of a thrust fault. Rhe-

ological contrasts between basement blocks can also influence the degree and location of faulting

and relative uplift. In northern Nevada, basement architecture in the form of early rifted con-

tinental margins, formed during pre-Cambrian extension, may dictate the subsequent structural

geometry of overlying sedimentary sequences during large-scale compression. Within the region

of the Carlin gold trend, specific anticlinal fold and thrust geometries in the sedimentary rocks,

involved in various orogenies up until the Laramide, may focus fluid movement and provide effec-

tive traps to the system, resulting in the unique gold endowment of the area. Most mineralisation is

situated less than 100 m below the Roberts Mountain thrust, which defines the lower boundary of

the sequence of deep-water sedimentary rocks that has ridden over both the basement and younger

sedimentary layers.

Muntean et al. (2003) argue that the Carlin and Battle Mountain–Eureka (BME) gold trends

correspond to reactivated normal faults that likely had their origins in Proterozoic rifting. Numer-

ical modelling offers a way to test the basic hypothesis by which old normal faults, or “steps”

that are relics of continental rifting, control the subsequent location of upper crustal faults and

anticlinal structures during compression.

6.2 Geological Setting

Following continental rifting in the pre-Cambrian through to Devonian, present-day north-

ern Nevada has been subject to a number of compressional episodes of varying duration (Fig-

ure 6.1). These range from the Antler orogeny, approximately 340 Ma ago, through to the

Laramide orogeny, which ended with the onset of Basin and Range extension about 50 Ma ago

(Miller et al., 1992). The Roberts Mountain thrust, which defines a regional cap to the mineralisa-

69

CHAPTER 6. CRUSTAL-SCALE FLUID FLOW: CARLIN SYSTEM Wijns Ph.D. Thesis

Figure 6.1: Geological history of northern Nevada. Arrows indicate eastward (right) or westward (left)

orientation of driving forces. Timing and direction of events from Link et al. (1993), Bird (2002), Wyld

et al. (2003), and J. Muntean (pers. comm).

tion and probably acted as a permeability seal, occurred during the Antler orogeny. The deep-water

sedimentary units that were thrust overtop of inland units are commonly referred to as the “upper

plate”. The “lower plate” refers to the inland units that actually have their origin at higher strati-

graphic levels. Orogenic events subsequent to the Laramide emplaced more sedimetary sequences

over the Roberts Mountain allochthon. Although the direction of maximum stress changed for dif-

ferent orogenic events, the overall evolution of northern Nevada was dominated by east-directed

compression.

The linear arrangements of gold deposits along the BME and Carlin trends have prompted

many researchers to look for evidence of large-scale structural controls, especially in geophysical

data (e.g., Rodriguez, 1998; Grauch et al., 2003). The demarcation between ancient continental

crust and younger oceanic crust is well established through Pb and Sr isotope ratios (Wooden

et al., 1998; Grauch et al., 2003), but this boundary, although close, is not coincident with the

major mineral trends. Processing of gravity and magnetic data by Grauch et al. (2003) has revealed

features that align with mineral occurences; these are more persuasive for the BME than the Carlin

trend (Figure 6.2). A 2D inversion of magnetotelluric data also shows narrow, vertically extensive,

electrically conductive zones under the two trends, which Rodriguez (1998) interprets as crustal

faults.

If the geophysical data are highlighting major crustal faults that control the locations of the

mineral trends, these may be expressions of the reactivation of early normal rift faults at even

deeper levels (Figure 6.3). The relative offsets between reactivated normal faults and their prop-

agated thrusts in overlying sedimentary rocks are likely to be complicated by multiple orogenies,

gravitational slumping, and widespread extension in the Eocene and later.

70

Wijns Ph.D. Thesis CHAPTER 6. CRUSTAL-SCALE FLUID FLOW: CARLIN SYSTEM

a b

Figure 6.2: Gold deposits (circles and squares) on top of (a) 5 km upward-continued Bouguer gravity

anomaly with basin effects removed, and (b) magnetic potential with Pb and Sr isotope ratio boundaries.

Images from Grauch et al. (2003).

Figure 6.3: Schematic conceptual model, slightly modified from Muntean (2000), for the structural setting

of the Carlin trend above a basement normal fault that is reactivated during compression. The steps in the

basement represent the tops of pre-Cambrian extensional faults that are the product of a rifted continen-

tal margin. The actual cross-sectional distance between the Shoshone Mountains and the Carlin trend is

approximately 50 km.

71


a

bWest East

Figure 6.4: Initial numerical model, consisting of three basement blocks (Archaean in the east, Proterozoic

in the center, oceanic in the west) separated by faults and overlain by flat-lying sedimentary sequences.

Stripes in the sedimentary layers are marker units for visualising deformation. All bounding surfaces are

free-slip, and compression is from the west. (a) Entire crust, initially 20km thick. (b) Upper crust, 10 km

thick, where the top of the oceanic basement block is represented by the bottom boundary of the model.

Table 6.1: Natural values for Nevada model parameters.

Parameter Value

Whole crust model

depth of crust 20 km

compression velocity 10 mm/yr

Upper crust model

depth of crust 10 km

compression velocity 6 mm/yr

All models

densityρ of sedimentary rocks 2500 kg/m3

densityρ of basement rocks 3000 kg/m3

permeabilityk of sedimentary rocks, initial 10−14 m2

permeabilityk of basement rocks, initial 10−16 m2

cohesionco 16 MPa

friction coefficientcp of sedimentary rocks 0.6

friction coefficientcp of basement rocks 0.8

strain weakeninga 0.2

“saturation” strainεo 0.5

strain rateε, initial 1×10−15/s

thermal diffusivityκ 10−6 m2/s

gravityg 10 m/s2

72


6.3 Numerical Modelling

Section models in 2D follow an approximate ENE – WSW transect running from Archaean

cratonic crust in present-day western Nevada, U.S.A, to the oceanic crust along the protomargin

of western North America. The numerical model represents a number of sedimentary layers de-

posited over three basement crustal blocks – Archaean, transitional or Proterozoic, and oceanic –

that are the product of continental rifting during the Proterozoic through Devonian (Figure 6.4).

These blocks are separated by normal faults that provide zones of weakness that will reactivate.

Conceptually, the western rift fault represents the setting under the BME trend, whereas the east-

ern fault lies under the present-day Carlin trend. The model distance of 50 km between rift faults

is approximately the true distance today.

Basement blocks are much stronger than sedimentary layers. Relevant parameter values appear

in Table 6.1, along with the boundary conditions. All units are fluid saturated, but the permeability

of the sedimentary layers is initially two orders of magnitude greater than that of the basement

blocks. With continuing compression, this permeability evolves, but never reaches the low values

of the basement. Under accumulated shear, a dilation factor (c.f. equation 2.13 and Figure 2.2)

causes the permeability to increase up to a maximum of 50 times its original value, enhancing flow

along fault zones. Temperature is not modelled, therefore fluid flow is a result of mechanical and

gravitational forces only.

6.3.1 Whole crust tectonics

In the first instance, the entire crust, of initial thickness 20 km, undergoes compression over a

length of 200 km (Figure 6.4a). In the case of successively weaker basement blocks from east to

west (Figure 6.5a), most slip occurs along the first fault, and the eastern fault is barely reactivated.

The oceanic crust is thrust up against the transitional crust, providing the elevation for contin-

ued transport of western sedimentary rocks over eastern ones, mostly by gravitational slumping.

Eastern sedimentary layers are relatively isolated from deformation.

With the objective of determining the importance of basement rheological contrasts, the oceanic

and transitional blocks in Figure 6.5b are made the same strength. The lack of rheological contrast

does not affect the outcome in terms of strain partioning between basement faults. However,

the more competent oceanic crust undergoes faulting rather than homogeneous thickening, which

provides locally greater elevation for the sedimentary rocks in the vicinity of the basement thrust.

The eastern fault is only significantly reactivated when the western fault is considerably stronger

(Figure 6.5c). More compressive stress is transferred through the western fault, deforming the sed-

iments over the transitional crust. There is noa priori reason for a difference in fault strength, and

the greater elevation of the inland sedimentary rocks hinders the gravitational transport of western

units.

73


Figure 6.5: Simulations after 15% compression of initial model in Figure 6.4a. (a) Successively less com-

petent basement blocks from east to west, (b) equally competent oceanic and transitional blocks, and (c)

stronger western fault. Dark bands indicate accumulated plastic shear strain (faults).

6.3.2 Upper crust tectonics

Simulations on the upper crust alone assume that the top of the oceanic basement block sits

under the western sedimentary layers at the lower boundary of the model in Figure 6.4b. This

mimics the situation where oceanic crust subducts below the transitional and Archaean blocks. An

erosion surface near the top of the model box keeps the maximum elevation of material, relative

to the undeformed eastern level, at 5 km. Thus, gravitational slumping is not as pronounced as in

the whole-crust simulations, where erosion is neglected.

Figure 6.6 contains three snapshots of the model at different stages of shortening. Once again,

the eastern fault is markedly inactive, all strain being accomodated by the western fault. An early

backthrust forms off the asperity between oceanic and transitional crust (Figure 6.6a), and is car-

ried up and over the eastern sedimentary rocks. As this backthrust steepens due to continued

compression, a new one forms in the same original location (Figure 6.6b). This behaviour contin-

ues, with successive backthrusts becoming upright and inactive towards the east (Figure 6.6c), in

similar fashion to the results of Erickson et al. (2001). Only minor gravitational spreading occurs

in the thrust foreland.

When the basement step from oceanic to transitional crust is smaller, greater direct compressive

stress can be transmitted across the western fault. This leads to slightly more deformation of

eastern sedimentary units (Figure 6.7). However, even with 50% shortening (Figure 6.7c), the

deep-water sedimentary rocks (upper plate) are not thrust as far as the eastern fault, as is the case

postulated for the Carlin trend. This is probably dependent on the steep thrust angle, which is

discussed below.

74


Figure 6.6: Simulation based on initial model in Figure 6.4b, after (a) 15% shortening, (b) 25% shortening,

and (c) 34% shortening. An erosion surface limits the maximum elevation.

Figure 6.7: Simulation based on initial model in Figure 6.4b, but with a lower basement step from oceanic

to transitional crust. (a) 17% shortening, (b) 30% shortening, and (c) 50% shortening. An erosion surface

limits the maximum elevation.

6.3.3 Fluid flow

Fluid flow patterns are first dictated by the evolution of more permeable shear zones, and

initial compression moves fluid upwards along the reactivated western fault (Figure 6.8a). As the

thrust elevation builds, compressively driven upward flow competes with topographically driven

downward flow (Figure 6.8b). The isolation of the eastern sedimentary rocks to both deformation

and high fluid flux remains constant throughout the simulation, with eastward fluid flow, from both

compression and topography, directed along the main thrust shear.

75


Figure 6.8: Average representative fluid flow vectors, in white, for upper crust simulation from Figure 6.6.

(a) 5% shortening, before topographic build-up, (b) 15% shortening, with topographically driven flow, and

(c) 34% shortening.

6.3.4 Local scale

Figure 6.9 shows the effect on folding, at a more local scale, of compression against the ramp

between two basement blocks. Folding depends to a great degree on anisotropic strength (vis-

cosity) contrasts within layers, rather than strength contrasts between isotropic layers (Muhlhaus

et al., 2002a,b). The version ofEllipsis used for these simulations cannot simultaneously model

material anisotropy and failure, so yielding is omitted in this test. Results concentrate on local

folding that is beyond the resolution of the crustal-scale simulations above, and show that a ramp,

such as exists between basement blocks, is effective in localising an anticline in the sedimentary

rocks above it. This behaviour would be expected in the vicinity of both reactivated basement

faults.

6.4 Discussion

The structural evolution of the sedimentary sequences that host Carlin-type deposits depends

less on rheological differences between basement blocks (Archaean, transitional, oceanic) than

the ability to reactivate deep faults, which then propagate into the overlying sedimentary rocks.

Where erosion is neglected in the whole-crust model, high topography promotes upper plate mo-

tion across the lower plate by gravitational slumping rather than thrusting due to the driving com-

pressive stress. In the upper crust model that limits topography through an erosion surface, the

movement of the upper plate is more a result of the compressive stress. In neither case does the

upper plate reach the eastern fault that represents the structural setting of the Carlin trend. This

shortfall in the thrust distance is probably due mostly to the steep angle of failure (initial 45) that

is an outcome of the constitutive model in the numerical code. Successive thrust faults should

form at low angles, and this would facilitate eastward transport of the upper plate.

76


20 km

a b

Figure 6.9: (a) Initial model for testing folding at a local scale, seeded with small random anisotropy per-

turbations in the dark layer. (b) Final state after 33% compression. Compression velocity as in Figure 6.4b.

Gravitational collapse causes the formation of extensional normal faults while overall compres-

sion is still active (most apparent in Figure 6.6a and b). Uplift is important for promoting slumping

over great distances, so the erosional setting plays a large part in the effectiveness of this mecha-

nism as a contributor to transport of upper plate units. It is likely that such a gravitational driver

is responsible for the widespread extension apparent in the Basin and Range when compression

ceased after the Laramide orogeny (Axen et al., 1993).

Apart from the propagation of the reactivated rift fault, a ubiquitous feature of both crustal

models is the early backthrust formed off the asperity between basement blocks. The implication

is that an early backthrust should exist in such a field situation, but, depending on the amount

of shortening, it, and subsequent backthrusts, will be transported eastward along the main thrust

while rotating towards the vertical. A backthrust, always present at the location of the oceanic

to transitional contact of the model, can explain anomalous west-vergent deformation associated

with a geanticline that is locally, but not everywhere, subparallel to the Sr 0.706 line – the inferred

edge of the original continent (Madden-McGuire and Marsh, 1991; Saucier, 1997).

Field evidence shows that anticlines are important for hosting Carlin-type gold deposits, and

these would act as natural fluid ponding sites if seals are present. A ramp anticline often devel-

ops when there is compression across existing faults (e.g., Cooke and Pollard, 1997), or, as in

the model, against a ramp between two basement blocks. The fluid ponding potential of such

structures is enhanced if they remain unbroken by thrusting.

The fluid flow paths in Figure 6.8 illustrate the potential for mixing between fluids from deeper

sources that travel up the main thrust, and meteoric fluids that are driven by topography. The

maximum depth of meteoric fluid penetration is approximately 5 km, equal to the maximum rela-

tive topographic elevation. Thus, erosion will also be critical to determining the depth of mixing

between these fluids from different sources. The simplicity of Darcy flow and the homogeneous

nature of the sedimentary layers establish this determination of mixing level as a first-order con-

clusion. Probable anisotropic permeability from sedimentary layering during deposition should

move the level towards the surface.

Fluid flow through fault zones is naturally more complicated than what is modelled. There is no

provision in these simulations for the development of anisotropic permeability during shear. Such

a phenomenon might act as a barrier to flow perpendicular to fault zones, and this would serve

to further reduce fluid flux between upper and lower plate rocks. Darcy’s Law, which assumes

a laminar flow field, may not be appropriate for high fluxes along faults. Shear zones in the

brittle regime will evolve with extremely variable slip rates, periods of complete quiescence, and,

77


possibly, fault healing. All of these factors contribute to the great difficulty in modelling flow

along shear zones at long time scales.

Elevated pore pressure is one of the most effective ways to change the stress regime and pro-

mote failure and fracturing where it would not otherwise occur. A low-permeability cap in the

sedimentary layers, perhaps in the form of strong, shear-parallel, anisotropic permeability devel-

opment in the Roberts Mountain thrust, or layers of low-permeability shale units, would lead to

elevated pore pressures in the rock column below as more material is thrust above (e.g., Hubbert

and Rubey, 1959). This has the immediate effect of extending the depth of faulting towards the

basement, possibly reaching more deeply sourced fluids and enhancing the control that basement

features exert on fault locations. If a permeability seal can be kept contiguous, deeper faulting will

be allied to an effective fluid trap until a significant change in the stress regime allows venting and

mineral precipitation.

6.5 Chapter Summary

The reactivation of basement normal faults during subsequent compression leads to propagated

thrust faults in overlying, initially intact rock units. In northern Nevada, pre-Cambrian rift faults

dating from the stages of continental break-up may serve as the loci for thrust faults formed during

a series of compressional orogenies from 340 to 50 Ma. These thrusts carried deep-water sedimen-

tary rocks (upper plate) from the west over younger units to the east (lower plate). The failure of

the model simulations to carry the upper plate as far as the proposed location of the Carlin trend is

most likely due to a high angle of thrust initiation (45). This angle is determined by the material

constitutive law in the numerical code, and does not adequately reflect nature. In all simulations,

the western fault accomodates the vast majority of strain, and the eastern (Carlin) fault is protected

from reactivation unless it is initially much weaker than the western fault. The importance of (ex-

tensional) gravitational slumping in transporting the upper plate eastwards will be a function of

maximum topography, hence of erosion. Fluid flow is equally driven by the compressional stress

and by topography. Mixing of fluids from deep versus surface sources will occur at a level that

is a function of topographic relief, and which is probably not deeper than the maximum relative

surface elevation.

78

Chapter 7

CONCLUSION

Numerical modelling is a basic tool for tectonic studies. In the obvious absence of experi-

ments at a field scale, analogue models and computer simulations are two means by which to

extrapolate laboratory rock mechanics to a scale of interest for geology. With both techniques,

the greatest challenge is the accurate representation of the behaviour of geologic materials: in the

analogue case, this requires confidence in the validity of using proxy materials, and in the numer-

ical case, confidence in the validity of the physical equations. As limitations on computational

loads disappear with technical advances in the computer hardware industry, the richness in detail

of three-dimensional numerical modelling will equal that of analogue modelling. This will leave

the numerical approach as the preferred technique for geological simulations, due to the control

over measurements and material properties, the ease of visualisation of results, and the rapidity

with which parameters can be varied in order to test new scenarios. Analogue models will remain

valuable for testing numerical codes on the physics of known materials.

In line with the vision of expanding the scope of numerical modelling, the inclusion of porous

fluid flow into a high-deformation code such asEllipsis opens the door to new areas of modelling

not possible with the analogue technique. In Appendix A, a test with coupled pore fluid pressure

verifies the Byerlee approximation used in the non-porous modelling of Chapters 3 through 5.

Pathways for possible mineral-bearing fluids are calculated in Chapter 6 during the evolution of a

compressional orogen. The application of ever-increasing computational sophistication, however,

must be tempered by constant referral to field observations. Chapters 5 and 6 compare results to

cooling curves, seismic sections, viscosity estimates, and field mapping, and make suggestions

for guiding new observations. This feedback loop is crucial for establishing numerical modelling

as a real tool in applied geology, from generic studies to specific cases. It should influence both

the collection of observations and subsequent refinements to the initial conceptual and numerical

models.

The application of computer modelling to any problem, but especially to specific field situa-

tions, comes with a warning against over-interpretation. Although this warning is based on the

self-evident truth that a result is only as good as its input, the pitfalls may not always be obvi-

ous, especially to those less aware of the numerics behind whatever software is being used. As

an example, Appendix B illustrates different results for the exact same geological problem, due

simply to changes in computational grid density and solution accuracy in theEllipsis algorithms.

More obvious candidates for influencing the results are the material parameter values of the model.

Changes in some parameters will influence a result more than changes in other parameters – this

is referred to as model sensitivity. It is essential to explore the model sensitivity with respect to all

79

CHAPTER 7. CONCLUSION Wijns Ph.D. Thesis

parameters being used. This allows conclusions about model behaviour to be formulated with due

consideration for uncertainties.

A method for handling the often daunting task of exploring model sensitivity is available

through the interactive evolutionary computation (IEC) and visualisation system of Chapters 3

and 4. Not only is the navigation through parameter space towards one or more final solutions

made simple, but associated techniques for visualising all solutions become automatic sensitivity

maps. The effectiveness of the IEC system declines with an increase in the number of model pa-

rameters, eventually reaching a stage where the number of required outputs for ranking cannot be

matched by either the available computer time or the ability of a human to evaluate all solutions,

and the reduction of dimensionality for visualisation is equally strained. This breakdown of the

method, however, is a reminder of the point at which the usefulness of a model is probably suffer-

ing from the inclusion of too many parameters, to the detriment of a firm grasp of the fundamentals

of the problem.

The development of porous flow capabilities within theEllipsis code, the use and refinement

of the IEC system, and the adoption of various visualisation techniques for parameter space all

culminate in the ability to perform meaningful numerical modelling for new problems in tecton-

ics and fluid flow, with an understanding of the expanded solution space around the conceptual

model. In the larger geological modelling community, the transition to three dimensions, minus

porous flow, has already been made in a successor toEllipsisand in independent high-deformation

codes; the coupling of chemical reactions to stress and fluid flow is a reality with other software;

and research is being undertaken on the physics of translating laboratory-scale rock mechanics

phenomena to ever larger scales. The growing computational complexity made possible by this

research will demand continued attention to the basic concepts of responsible modelling that have

been outlined throughout this thesis, and the adoption of systems, such as IEC plus visualisation,

that will render tractable the task of analysing model results and formulating viable conclusions.

80

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89

APPENDIX A

PORE PRESSURE APPROXIMATION

As mentioned in Chapter 3, from laboratory experiments by Byerlee (1968), there appears to be

a universal friction coefficientcp of 0.6 – 0.85 for dry rock. In a saturated rock column, assuming

an average hydrostatic pore pressure, the solid pressure is reduced, so that the zone of failure is

deeper. For dry rock of densityρs, the pressurepsdry = ρsgz (without deformation). In rock

saturated with water of densityρp, the solid pressurepssat is related to the effective pressurepe by

pssat = pe − pp ≈ ps

dry − ρpgz

usingpe ≈ psdry for small porosity. As a fraction of the dry pressure,

pssat

psdry

=(ρs − ρp) gz

ρsgz=

ρs − ρp

ρs.

For the trial in Figure A.1, the pore pressure is kept hydrostatic through negligible pore space

compression and a permeability that does not evolve. Extension velocity, dimensions, and all

other material parameters are equal to those for the IEC experiment of Chapter 3. Usingρs = 2700

kg/m3 andρp = 1000 kg/m3, a Byerlee coefficient of 0.6 for dry rock becomes 0.38 for saturated

rock. Although this approach seems valid in terms of the initial depth of yielding, it is certainly

too simplistic to deal with the effect of pore pressure during deformation of the rock.

Figure A.1: Extension experiment (33% extension) for a single crustal layer. (a) with a Byerlee coefficient

of 0.6 for dry rock, (b) with fully coupled pore pressure, and (c) with a Byerlee coefficient of 0.38, to

account for an average hydrostatic pore pressure. The brittle zone is the same depth as in (b).

91

APPENDIX B

MESH DEPENDENCE OF MODEL RESULTS

A numerical modelling code that has no explicit internal length scale for deformation, such as

the version ofEllipsis used for this dissertation, is callednon-regularised. The results of such a

code exhibit a dependence on the resolution of the computational grid. In effect, the grid spac-

ing defines an implicit length scale for deformation. It also controls the width of shear bands.

This phenomenon can sometimes be used to advantage in choosing a grid resolution, but, at the

same time, requires that conclusions with respect to the results, particularly in a spatial sense, be

drawn with caution. Figure B.1a contains an extension experiment, against which are compared

the results of two other simulations with identical input files, save the finite element mesh density

(Figure B.1b), and computational accuracy of the finite element loops (Figure B.1c). The differ-

ences are due to the nucleation of shear zones at sites of small perturbations in the stress field, and

minute changes in calculated velocities are sufficient to displace these perturbations.

Figure B.1: (a) Extension experiment similar to simulation C from Chapter 5 (c.f. Figure 5.3). (b) Same

experiment at a finite element grid resolution of 83% of original. (c) Same experiment with a computational

accuracy twice as large as original.

The above cases underline the importance of running the same simulation at different grid spac-

ings or accuracy. This will highlight the results that can be adopted with confidence. For example,

the location of fault zones changes dramatically, and should therefore not form part of any mod-

elling conclusions. The relatively large spacing between fault zones is quite consistent, showing

that a weak crust does indeed promote widely spaced faulting. The most robust conclusion is that,

in all cases, the upper crust is completely pulled apart, exhuming the lower crust.

All the modelling in this dissertation heeds the caution just given. Any conclusions are based

on multiple runs at different resolutions, and sometimes different accuracies.

93

APPENDIX C

CRUSTAL VISCOSITY PROFILE

C.1 Viscosity at Brittle to Ductile Transition

In order to create a continuous maximum shear stress profile with depth, the viscosityη at the

brittle to ductile transitionzu must satisfy both the brittle yield equation (2.8) and the temperature-

dependent viscosity equation (5.1).

η(zu) = ηu =τyield(zu)

ε= ηoe

−cT (zu) .

Having fixed the strain rate and bothco andcp, ηu is derived from equation (2.8) for the yield

stress before any strain softening, noting that the total overburden pressurep is the lithostatic

stress reduced by the extensional stress.

co + cp (ρugzu − ηu ε) = ηuε

ηu =co + ρugzucp

(1 + cp) ε.

The constantsηo and c control the viscosity profile through the ductile region, and are related

through the equations above.

e−cT (zu) =ηu

ηo

−c∂T

∂zzu = ln

(co + ρugzucp

(1 + cp) ηoε

)

c = −(

∂T

∂zzu

)−1

ln(

co + ρugzucp

(1 + cp) ηoε

).

C.2 Integrated Crustal Strength

Referring to Figure 5.1b, the integrated strengthτint of the crust is simply the area between 0

and the maximum shear stress. This maximum stress is defined by the yield envelope in the brittle

zone (equation 2.8), and by the viscous stressη ε in the ductile zone. The crustal response to

extension is defined by the ratiorτ of the integrated strength of the upper crust (initially all brittle)

to that of the lower crust (initially all ductile):

rτ =τuint

τ lint

.

95

APPENDIX C. CRUSTAL VISCOSITY PROFILE Wijns Ph.D. Thesis

For the upper crust at the onset of extension,

τuint =

∫ zu

0τyield dz

= cozu +12zu (τyield(zu)− co)

= cozu +12zu

(co + ρgzucp

1 + cp− co

)

=zu

2

(co +

co + ρgzucp

1 + cp

).

Integrating the viscous stress profile for the lower crust at an initial constant strain rate and linear

temperature profile,

τl(int) =∫ zl

zu

η(T )ε dz

= ε

∫ zl

zu

ηoe−cT dz

= ε

∫ zc

zu

ηoe−c ∂T

∂zz dz + ε

∫ zl

zc

ηoe−cT (zc) dz

= − ηoε

c∂T∂z

[e−c ∂T

∂zz]zc

zu

+ ηoεe−cT (zc) (zl − zc)

= − ηoε

c∂T∂z

[e−cT (zc) − e−cT (zu)

]+ ηoεe

−cT (zc) (zl − zc) .

96

Appendix D (on CD-ROM) is available with the print copy of the thesis, which is held in the University Library.

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