Reactive transport in damageable geomaterials · Reactive transport in damageable geomaterials...

Reactive transport in damageablegeomaterials

Thermal-Hydrological-Mechanical-Chemical coupling of

geological processes

This thesis is

presented to the

School of Earth and Environment

for the degree of

Doctor of Philosophy

By

Thomas Poulet

March 2012

c© Copyright 2012

by

Thomas Poulet

iii

Abstract

Numerical modelling is a powerful tool to improve geological understanding, as it is

particularly well suited to simulate and visualise complex scenarios, to validate or

refute hypotheses in conceptual models, and in turn to raise more scientific ques-

tions and to help make predictions. No code exists however which can simulate

nature in its full complexity. Some of the most challenging problems toward that

goal consist in modelling accurately the coupling feedbacks between all processes in-

volved, including mechanical deformation, fluid flow in porous media, heat transfer,

chemical transport and fluid-rock chemical interactions. Thermodynamics focuses

on energy, the common denominator between all processes, and therefore provides

an ideal framework to couple them all in a consistent manner. Within this ap-

proach, all physical processes can be described through the definition of their free

energy and dissipation functions.

This theoretical framework is used to introduce a new mathematical formulation

of continuum damage mechanics (CDM) for geomaterials subjected to thermo-

mechanical loading, implemented using a custom user material subroutine of ABAQUS/-

Standard (2008). The material’s rheology includes isotropic linear elasticity as well

as non-linear visco-plasticity induced by combined creep mechanisms. The formula-

tion is based on the theory of generalized standard materials where the dissipative

processes obey the principle of maximum dissipation.

A second numerical approach, escriptRT, is developed to simulate reactive trans-

port in porous media based on a finite element method (FEM) using the flexible and

v

scalable escript package, combined with three other components: (i) a Gibbs min-

imisation solver for equilibrium modelling of fluid-rock interactions, (ii) an equation

of state for pure water to calculate fluid properties and (iii) a thermodynamically

consistent material database to determine rocks material properties. Thermody-

namic potential functions are used to calculate reversible material properties such

as thermal expansion coefficient, specific heat, elastic shear modulus, bulk modu-

lus and density. The effects of chemical feedbacks are also considered by taking

into account the fluxes exchanged by a Representative Volume Element with its

surroundings.

The two numerical approaches are then linked to create software which simulates re-

active transport in damageable geomaterials, a Thermal-Hydrological-Mechanical-

Chemical (THMC) coupling of geological processes. The resulting code builds

on the modular architecture of escriptRT to solve sequentially these coupled

THMC mechanisms and the various feedback mechanisms considered, including

shear heating and damage. Damage in the host rock is linked to porosity evolu-

tion, which in turn affects permeability. This permeability-damage dependency is

validated against published results. Finally, a simulation of an albitisation scenario

is presented which builds on all components and illustrates the power of this novel

THMC approach for geological modelling, linking structural geology to geochem-

istry and allowing geoscientists to study the competition of the rates of all processes

involved.

vi

Acknowledgements

It is a pleasure to thank those who made this thesis possible. I am heartily thankful

to my supervisor, Prof. Klaus Regenauer-Lieb, for his generosity and availability

in spite of his extremely full schedule, for his continuous encouragement from the

preliminary to the concluding level, and for impersonating the highest level any

researcher can aim at. I would also like to thank my co-supervisors, Dr. Peter

Hornby (1957-2009), A.Prof. Lutz Gross and Dr Ali Karrech, for their continuous

support at different stages of my thesis and above all for their friendship. Dr

Hornby worked relentlessly on some of the most challenging numerical problems

of the reactive transport code, hurdles which would have discouraged any human

person. Dr Gross always responded very quickly to my frequent numerical questions,

and Dr Karrech provided me with much more than regular supervision. I thank him

enormously for the daily discussions which provided the motivation and resources

for the most of the work accomplished.

My thanks also go to Dr. Robert Woodcock for convincing me to embark on this ad-

venture and making the journey feasible within CSIRO. Numerous other researchers

at CSIRO and UWA also contributed to my geological, geophysical and geochemical

learning, including Drs Florian Fusseis, Christoph Schrank, James Cleverley, Peter

Schaubs and Louise Fisher.

My final and undoubtedly most pressing thanks go to my wife Elena for her extra

support with our young family during the time I spent working on this thesis.

vii

Contents

Abstract v

Acknowledgements vii

List of Tables xiv

List of Figures xv

1 Introduction 1

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

1.2 Aims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Published content of the thesis . . . . . . . . . . . . . . . . . . . . . 6

1.5 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Consistent material properties 11

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

ix

2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Reactive transport 29

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Reactive transport formulation . . . . . . . . . . . . . . . . . . . . 32

3.2.1 Pressure equation . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2.2 Temperature transport . . . . . . . . . . . . . . . . . . . . . 35

3.2.3 Transport of chemical elements . . . . . . . . . . . . . . . . 35

3.3 Reactive transport solvers . . . . . . . . . . . . . . . . . . . . . . . 40

3.3.1 The Finley PDE solver . . . . . . . . . . . . . . . . . . . . 40

3.3.2 The GibbsLib solver for chemical equilibrium . . . . . . . . 43

3.3.3 The Equation of State (EOS) solver for fluid properties . . . 44

3.3.4 The PreMDB database for rock properties . . . . . . . . . 45

3.4 Software architecture . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.4.1 python environment . . . . . . . . . . . . . . . . . . . . . . 46

3.4.2 escript framework . . . . . . . . . . . . . . . . . . . . . . . 47

3.4.3 Modelframe approach . . . . . . . . . . . . . . . . . . . . . . 48

3.4.4 EscriptBaseRT layer . . . . . . . . . . . . . . . . . . . . . . 50

3.4.5 PmdPyGC . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.4.6 EscriptFluidHeatChem structure . . . . . . . . . . . . . . . . 51

3.5 escriptRT and hydrothermal gold systems . . . . . . . . . . . . . 52

3.5.1 Physical Conditions . . . . . . . . . . . . . . . . . . . . . . . 52

3.5.2 Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

x

4 Continuum damage mechanics 59

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.2 Thermo-mechanical background . . . . . . . . . . . . . . . . . . . . 61

4.3 From dissipation to flow rules . . . . . . . . . . . . . . . . . . . . . 65

4.3.1 Visco-plasticity . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3.2 Damage potential . . . . . . . . . . . . . . . . . . . . . . . . 67

4.3.3 Flow rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.4 Finite element implementation . . . . . . . . . . . . . . . . . . . . . 68

4.4.1 Prediction-correction algorithm . . . . . . . . . . . . . . . . 69

4.4.2 Consistency factor . . . . . . . . . . . . . . . . . . . . . . . 70

4.4.3 Consistent tangent modulus . . . . . . . . . . . . . . . . . . 71

4.5 Numerical application . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.5.1 Comparative study of damage and heat necking . . . . . . . 73

4.5.2 Energy partitioning . . . . . . . . . . . . . . . . . . . . . . . 78

4.5.3 Effect of loading rate on the structural integrity . . . . . . . 79

4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5 Thermodynamic framework 83

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.3 Multiple scales approach . . . . . . . . . . . . . . . . . . . . . . . . 90

5.4 Representative Volume Element . . . . . . . . . . . . . . . . . . . . 92

5.5 Thermodynamics background . . . . . . . . . . . . . . . . . . . . . 93

5.6 From the Second Law of thermodynamics . . . . . . . . . . . . . . . 97

5.7 Balance equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

xi

5.8 Numerical application . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.9 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . 103

6 Reactive transport with damage mechanics 109

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.2 Constitutive model . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.2.1 Continuum damage mechanics . . . . . . . . . . . . . . . . . 111

6.2.2 Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.2.3 Effective stress . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.2.4 Fluid transport . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.2.5 Heat transport . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.2.6 Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.2.7 Permeability evolution . . . . . . . . . . . . . . . . . . . . . 117

6.3 Numerical approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.3.1 escriptRT, a modular architecture . . . . . . . . . . . . . 119

6.3.2 Mechanical modelling with Abaqus . . . . . . . . . . . . . . 120

6.3.3 Connecting escriptRT and Abaqus . . . . . . . . . . . . . 120

6.4 Application to albitisation . . . . . . . . . . . . . . . . . . . . . . . 121

6.4.1 Problem description . . . . . . . . . . . . . . . . . . . . . . 122

6.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

7 Conclusions and perspectives 133

7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

7.2 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

xii

Bibliography 139

Appendices 163

A Derivations used in chapter 4 163

A.1 Local form of the first principle of thermodynamics . . . . . . . . . 163

A.2 Local second principle . . . . . . . . . . . . . . . . . . . . . . . . . 165

A.3 Derivation of flow rules from a dissipation potential . . . . . . . . . 166

B Derivations used in chapter 5 169

B.1 Dissipation equation . . . . . . . . . . . . . . . . . . . . . . . . . . 169

B.2 State equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

B.3 Heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

B.4 Continuity equations . . . . . . . . . . . . . . . . . . . . . . . . . . 172

xiii

List of Tables

2 Physical properties available in PreMDB . . . . . . . . . . . . . . 14

3 Rocks and minerals available in PreMDB . . . . . . . . . . . . . . 19

4 Nomenclature for chapter 3 . . . . . . . . . . . . . . . . . . . . . . 30

5 Mineralogy of reactive transport example . . . . . . . . . . . . . . . 54

6 Thermo-elastic parameters used in damage mechanics example . . . 74

7 Dissipation constants used in damage mechanics example . . . . . . 74

8 Simulation parameters for CO2 diffusion in damaged rock . . . . . . 102

9 Simulation parameters for THMC simulation . . . . . . . . . . . . 122

xiv

List of Figures

1 Example of PreMDB’s GUI . . . . . . . . . . . . . . . . . . . . . 20

2 Density profile comparisons PreMDB/PREM/ak135 . . . . . . . 25

3 VP profile comparisons PreMDB/PREM/ak135 . . . . . . . . . . 25

4 VS profile comparisons PreMDB/PREM/ak135 . . . . . . . . . . 26

5 Poisson’s ratio profile comparisons PreMDB/PREM/ak135 . . . 26

6 escriptRT’s software architecture . . . . . . . . . . . . . . . . . . 46

7 Simplified modelframe simulation workflow . . . . . . . . . . . . . . 49

8 escriptRT example, hot plume effects . . . . . . . . . . . . . . . . 55

9 escriptRT example, mineralogy evolution . . . . . . . . . . . . . . 56

10 escriptRT example, gold precipitation . . . . . . . . . . . . . . . 57

11 Notched lithospheric layer geometry . . . . . . . . . . . . . . . . . . 75

12 Notched lithosphere, evolution of equivalent inelastic strain . . . . . 76

13 Notched lithosphere, evolution of damage distribution . . . . . . . . 77

14 Notched lithosphere, evolution of temperature . . . . . . . . . . . . 78

15 Mesh independence of shear band width . . . . . . . . . . . . . . . 79

16 Energy responses of damaged and undamaged structures . . . . . . 79

17 Impact of damage on force-strain curves . . . . . . . . . . . . . . . 80

18 Liesegang patterns in sandstone. . . . . . . . . . . . . . . . . . . . . 85

xv

19 Portevin-Le Chatelier bands. . . . . . . . . . . . . . . . . . . . . . . 88

20 Shear heating feedback causing localisation. . . . . . . . . . . . . . 89

21 Chemical feedback causing localisation. . . . . . . . . . . . . . . . . 91

22 Simulated damage zones in reservoir. . . . . . . . . . . . . . . . . . 104

23 Simulated CO2 concentration in reservoir . . . . . . . . . . . . . . . 105

24 Permeability evolution as a function of damage . . . . . . . . . . . 118

25 Flow diagram for coupling escriptRT and Abaqus . . . . . . . . 119

26 Initial geometry of THMC simulation . . . . . . . . . . . . . . . . . 121

27 THMC simulations with and without damage . . . . . . . . . . . . 124

28 Permeability evolution in THMC simulation . . . . . . . . . . . . . 125

29 Fluid flow and temperature in THMC simulation . . . . . . . . . . 126

30 Porosity decomposition in THMC simulation . . . . . . . . . . . . . 127

31 Chemistry distribution in THMC simulation . . . . . . . . . . . . . 128

xvi

Chapter 1

Introduction

1.1 Preamble

Numerical modelling has become a widespread technique to help understand geolog-

ical processes, as it is particularly well adapted to test various geological hypotheses

by studying conceptual models and investigating parameter sensitivities. It is there-

fore broadly used as a forward modelling tool to reproduce geological scenarios, and

in turn as part of inverse modelling workflows to identify some numerical parameters

values regarding the geometry of the system or material properties. This identi-

fication is done by comparing interpretations of numerical simulations with real

data from natural observations, including geological, geophysical and geochemical

measures. Many individual natural processes such as mechanical deformation, fluid

flow in porous media, heat transfer, chemical transport, and fluid-rock chemical

interactions can now be modelled accurately when taken separately, using robust

numerical methods which have been benchmarked thoroughly against analytical

solutions for simple examples. However, while those benchmarks ensure a proper

implementation of the corresponding formulation, they do not necessarily ensure

a perfect validity of the underlying physical or chemical assumptions. Numerical

modelling may not account for some processes responsible for some of the complex

1

2 CHAPTER 1. INTRODUCTION

behaviours observed in nature. Geologists are primarily interested in localisation

patterns (faults, folds, reservoirs, ore bodies...) and not all localisation phenomena

can currently be reproduced numerically using existing tools.

Challenges for numerical simulation tools to reach the next level of realism and

scientific accuracy involve a more accurate modelling of individual processes, but

more importantly a better understanding of all coupling feedbacks between those

processes. The complexity of those various feedbacks is a major reason for scientists

to continue analogue modelling with sand/putty boxes, but no synthetic material

is capable of solving the inherent problem of length and time-scales dependen-

cies associated with scaled-down models supposed to simulate geological scenarios

(Regenauer-Lieb and Yuen, 2003). Extrapolation of experimental results over sev-

eral orders of magnitude is highly questionable, and given the importance of strain

rates in the constitutive behaviour and rheology of materials, it is important to

develop some modelling tools which allow the proper use of geological strain rates

as slow as observed in nature (Kreemer et al., 2003). Numerical modelling pro-

vides such functionality by considering the fundamental principles of conservation

of mass, momentum, and energy.

1.2 Aims

The aim of this thesis is to develop a theoretical and numerical framework to model

the coupling of Thermal-Hydrological-Mechanical-Chemical (THMC) processes for

geological simulations in the application domains of mineral and geothermal explo-

ration. Many dependency feedbacks between various processes have already been

identified that play a critical role to explain natural processes. The chronology of

their discovery has been closely related to the ease of observation of the correspond-

ing phenomena. For example, the importance of density variation was one of the

first mechanisms to be understood, as it represents the main driver for convection,

an important mixing mechanism both for fluids in porous media as well as viscous

1.2. AIMS 3

solids at the larger scale. Convection is driven by density variation, which itself can

be triggered by various mechanisms. Convection has been extensively studied as

a result of temperature dependent density, but also under the influence of salinity

(Nield and Bejan, 2006, and references therein), and the interesting complexity of

couplings is well illustrated when considering both at the same time.

Other couplings have been considered later, as their influence was initially consid-

ered to be of second order only. The thermo-mechanical feedback occurring through

shear heating is, for example, still neglected in many geological simulation codes de-

spite its critical importance on fault localisation. Regenauer-Lieb et al. (2001, 2006)

showed that feedbacks which weaken rocks, such as viscous dissipation and shear

heating, can have considerable influences on the energy thresholds and thereby play

key roles on stress and strain localisation (Hobbs et al., 1990; Sengupta, 2010). Clas-

sical mechanical approaches which ignore such energy balances struggle to explain

the level of forces required to drive tectonics on our planet for example. They re-

quire forces that are at least four times as large as those deemed available from slab

pull or rigid push estimates (Regenauer-Lieb et al., 2008). One possible solution

to this problem is to consider the time-dependent strength reduction caused by the

feedback of deformation, shear-heating and exponential temperature dependence of

flow laws. This feedback is very efficient for materials with high activation energy

such as olivine and it can lead to a substantial reduction in lithospheric strength

(Braeck and Podladchikov, 2007). However, the predicted level of forces is still

an upper limit (Regenauer-Lieb et al., 2010). Additional weakening mechanisms

through damage mechanics, temperature or fluid flow must be taken into account.

The relative importance of the physical mechanisms to consider depends on the

geological application. Initiation of subduction, for example, can be explained by a

double feedback mechanism (thermoelastic and thermal-rheological) promoted by

lubrication due to water (Regenauer-Lieb et al., 2001). Couplings are not limited to

mechanical processes and are widely studied in chemistry as well, which is extremely

relevant to explain processes at a smaller scale. Liesegang rings, for example, can be


seen as a natural example of auto-catalytic chemical reactions in geology (Lebedeva

et al., 2004). Chemistry and mechanics, however, are still often considered as

two separate research fields and bridges between the two are not yet developed

enough to allow geologists to run numerical simulations and understand the complex

couplings between them at all scales. Taron et al. (2009) developed such a bridge

with a THMC framework based on fractured porous media. The main motivation

behind the work presented in this dissertation is to build a novel THMC framework

based on thermodynamics, with a focus on continuum damage mechanics to create

permeable pathways.

1.3 Methodology

Thermodynamics provides an ideal framework to couple various processes and

specifically account for all cross relationships, since energy is the natural common

denominator. Thermodynamic formulations are expressed in terms of a general

internal-variable formalism which describes rocks at the microstructural level to

derive the macroscopic constitutive laws (Rice, 1971). Modern constitutive models

are generally developed within this framework of thermodynamics, which ensures

enough consistency to avoid the artificial creation of energy, numerically, in non

reactive solids subjected to external loading. One of the pioneering contributions,

which emphasise the thermodynamics foundation of inelastic constitutive laws, was

introduced by Rice (1971). It represented a progression of the classic plasticity ap-

proaches, which assumed the existence of a yield function and a flow rule. This ap-

proach extended the normality condition to finite deformation and rate-dependent

cases (Hill, 1950) and broadened as well the stability postulate of Drucker (1959).

Using the same background, Lubliner (1975) showed that a yield function is mathe-

matically necessary if the dissipation function is positive and homogeneous of degree

one in the internal-variable rates. In a more general formulation, Lubliner (1978)

showed that a viscoplastic potential can be derived in a similar way if the behaviour

1.3. METHODOLOGY 5

is rate dependent. By defining the dissipation as a function of the internal-variable

rates, Ziegler (1963, 1977) proposed the orthogonality principle and showed that

it is equivalent to the principle of maximum dissipation. Based on this princi-

ple, Halphen and Nguyen (1975b) derived the generalised thermodynamic theory

of associated materials by justifying the existence of a plastic potential from which

inelastic strain can be derived. This thermodynamic framework allows deducing

the constitutive relationships, yielding limits and flow rules directly from suitable

free energy and dissipation functions instead of introducing them on ad hoc basis as

commonly presented in the classic theories of soil and rock mechanics. It is also im-

portant to note that such formulations of dissipative processes inherently introduce

some time and length scales associated with the phenomena considered.

The consideration of continuum damage mechanics is also of particular impor-

tance for coupling processes, as damage can be seen as a distribution of cracks and

voids at the micro-scale (Lemaitre, 1985; Chaboche, 1987; Cocks and Ashby, 1982;

Lyakhovsky et al., 1997). This interpretation provides a natural link between the

localisation of thermo-mechanical processes and the evolution of porosity, which in

turn impacts permeability, affects fluid flow and strongly affects fluid-rock chemical

interactions.

Finally, in this thesis, a special attention has been paid to the numerical aspect of

the problem considered, with the aim of developing an efficient tool. This tool can

run in parallel and also provides a high level interface for numerical modellers willing

to focus more on the geological problem at hand than on the programming side.

Software developed as part of this dissertation includes three major contributions:

(i) PreMDB, a material properties database with its graphical user interface, as

presented in chapter 2, (ii) escriptRT, a reactive transport code based on the

open-source escript modelling library and making use of PreMDB, and (iii)

a coupling module between escriptRT and the commercial package Abaqus in

order to add a mechanical deformation module to escriptRT. All code developed is

the property of the Commonwealth Scientific and Industrial Research Organisation


(CSIRO).

This thesis is articulated in five parts. Chapter 2 introduces PreMDB, a tool

for coupling thermochemistry with mechanics. It demonstrates the importance of

material properties variations with temperature and pressure and the need for an

easy-to-use interface for geoscientists. Chapter 3 presents escriptRT, a reactive

transport simulation code for fully saturated porous media which is based on a finite

element method (FEM) combined with three other components: (i) a Gibbs min-

imisation solver for equilibrium modelling of fluid-rock interactions, (ii) an equation

of state for pure water to calculate fluid properties and (iii) PreMDB, a thermo-

dynamically consistent material database to determine rocks material properties.

Chapter 4 establishes a new mathematical formulation of continuum damage me-

chanics (CDM) for geomaterials subjected to thermo-mechanical loading, which is

incorporated in a unified thermodynamical framework in chapter 5 and coupled to

escriptRT in chapter 6 to create a full THMC numerical platform. This last

chapter also presents a simulation of an albitisation scenario which builds on all

the components presented in the previous chapters and illustrates the power of this

novel THMC approach.

1.4 Published content of the thesis

All of the following five chapters have been published, in whole or in part, as journal

articles. The journal articles have been edited into chapters to make the thesis stand

as a coherent body of work. The contributions of all co-authors are summarised

explicitly for each article.

• Chapter 2 was published as Siret et al. (2009). I developed the whole software

architecture of PreMDB and the methodology and scripts to run all models

presented in the various comparisons with PREM and ak135. K. Regenauer-

Lieb contributed the original idea of building such a tool; D. Siret defined the

rocks and mineral compositions and ran all models to populate the database;

1.4. PUBLISHED CONTENT OF THE THESIS 7

J. Connolly participated indirectly by improving Perple X, the independent

software used which permits the extraction of any thermodynamic property

as a generalised formulation of temperature, pressure and composition. The

first three authors (including myself) wrote the manuscript together.

• Chapter 3 was published as Poulet et al. (2012a). I am the main developper

of escriptRT’s architecture and interface, and was helped by D. Georgiev

as he worked on this software package as well to improve the handling of

chemistry (work not presented in this thesis). I also prepared and ran the

numerical application. L. Gross is the main developer of escript, the under-

lying simulation package used to solve partial differential equations, and he is

responsible for the numerical performance of the solver. J. Cleverley defined

conceptually the application presented and drew the conclusions regarding

the chemistry of the selected system.

• Chapter 4 was published as Karrech et al. (2011a). A. Karrech lead this de-

velopment and K. Regenauer-Lieb provided some guidance, especially for the

geological applications. I participated in formulating the various mathemat-

ical problems, conducting parts of the simulations, interpreting results, and

writing the manuscript.

• Chapter 5 was published as Poulet et al. (2010). I derived the framework

presented in this journal article based on useful discussions with K. Regenauer-

Lieb and A. Karrech. A. Karrech is also the author (Karrech et al., 2011a)

of the numerical implementation of damage mechanics that I used to run the

example presented, as presented in chapter 4.

• Chapter 6 was published as Poulet et al. (2012b). I developed the framework

to link damage to porosity and permeability, as well as the software architec-

ture to couple escript with Abaqus. A. Karrech and K. Regenauer-Lieb

contributed through fruitful discussions about the formulation. L. Fisher and

P. Schaubs defined the conceptual application, which I translated into the


numerical model and simulated. They also contributed to the interpretation

of the results and some geological conclusions.

I also participated during the time of my study in four other publications which

are relevant to the subject of this dissertation, but which were not included in this

thesis as I was not the main contributor and they are not essential to the core of

the work. They represent some important extensions however and their reading is

highly recommended.

• Karrech et al. (2011b) develops further the damage formulation presented in

chapter 4 by introducing a pressure-dependent constitutive model.

• Zhang et al. (2012) presents a geological study based on the model presented

in (Karrech et al., 2011b). It models numerically the structural evolution

of the upper 10 km crustal part of a greenstone belt during extension and

shortening, taking into account the influence of a hot pluton emplaced within

a pre-existing granite dome and underlain by a greenstone sequence.

• Karrech et al. (2011c) pushes further the mechanical model for large deforma-

tions and presents a finite strain formulation which overcomes the aberrant

oscillations which appear in shear zones when classical approaches are used.

• Karrech et al. (2012) builds on the framework developed in Karrech et al.

(2011c) and unveils a mathematical formulation and numerical implementa-

tion of a coupled Thermal-Hydrological-Mechanical model for saturated poro-

materials undergoing logarithmic finite deformation.

1.5 Notations

In this dissertation, compact notation is used where X denotes a scalar value, X a

vector, and X a matrix or tensor. Here, the number of underlines shows the order

of the matrix/tensor. For instance, C represents a fourth order tensor, and the

1.5. NOTATIONS 9

notation is expended to the fully indexed tensorial form Cijkl when more clarity is

required. The Einstein convention of implicit summation of repeated indices is also

used unless specified otherwise, leading to the expression of tensorial products as

A.B = AijBjk and double contractions as A : B = AijBji.

A synoptic table of symbols is presented below, and all symbols are defined indi-

vidually in each chapter.

Ω0, Ω Physical domain in reference and current configurations

Γ Physical domain surface in current configuration

D(.)Dt

Material derivative attached to the mixture (material points related

to the solid or fluid phase)

d(.)dt

= ˙(.) Dot notation, material derivative attached to the skeleton

δij Kronecker delta

1 Identity tensor

α Thermal expansion coefficient [K−1]

β Compressibility [Pa−1]

Cv, Cp Isochoric and isobaric heat capacities [J.K−1]

D Damage parameter [-]

ǫ Strain tensor [-]

g Gravity [m.s−2]

G Gibbs free energy [J ]

H Enthalpy [J ]

K Kinetic energy [J ]

k Thermal conductivity [W.m−1.K−1]

K,G Bulk and shear moduli [Pa]

µ Viscosity [Pa.s]

N , b Biot modulus [Pa] and coefficient [-]

P Pore pressure [Pa]

φ Porosity [-]

ψ Helmholtz free energy [J ]

q Darcy flux [m.s−1]

ρ Density [kg.m−3]

σ Cauchy stress tensor [Pa]

S Entropy [J.K−1]

T Absolute temperature [K]


u Fluid velocity [m.s−1]

U Internal energy [J ]

Chapter 2

Consistent material properties

2.1 Introduction

Material properties are essential to realistic simulations of geodynamic and geolog-

ical processes. These properties generally derive from laboratory experiments and

geophysical observations (Bina, 1998a; Bina and Wood, 1987; Cammarano et al.,

2003; Deschamps and Trampert, 2004; Deuss et al., 2006; Irifune and Isshiki, 1998;

Irifune and Ringwood, 1987; Matas et al., 2007; Ringwood, 1991; Trampert et al.,

2001; Weidner, 1985). Additionally, numerical data calculated from thermody-

namic potentials are a complement to empirical data (Ita and Stixrude, 1992; Karki

et al., 2001; Stixrude and Lithgow-Bertelloni, 2005a; Vacher et al., 1998). There has

been considerable progress in geodynamic, thermodynamic and petrological mod-

elling (Connolly, 2005; Connolly and Petrini, 2002; Matas et al., 2007; Stixrude and

Lithgow-Bertelloni, 2005a,b). Here, a tool is presented for coupling thermochem-

istry with mechanics. The main purpose of this tool is to provide geodynamicists

and seismologists easy access to thermochemistry.

The thermodynamic equilibrium problem as solved by Gibbs energy minimisation

determines the basic mechanical properties needed for geophysics and geodynam-

ics. Reversible material properties such as the thermal expansion coefficient, specific

11

12 CHAPTER 2. CONSISTENT MATERIAL PROPERTIES

heat, elastic shear modulus, bulk modulus and density can thus be derived from

thermodynamics and thermodynamic potential functions can be used to model geo-

dynamical processes. Specifically, density differences driving for instance subduc-

tion do not need to be assigned but follow from chemical composition and tem-

perature. Moreover, reversible material property changes at phase transitions can

trigger events that may be of interest to the geodynamicists and seismologists (Bina,

1998b).

Previous works on thermodynamically derived material data have been presented in

the seismic tomographic community with the focus of solving the inverse problem

for chemistry from seismic velocities (Bina and Wood, 1987; Duffy and Ander-

son, 1989; Weidner, 1985). These models gather a collection of physical properties

from independent sources and are not thermodynamically self-consistent as pointed

out by several authors (Connolly, 2005; Connolly and Kerrick, 2002; Stixrude and

Lithgow-Bertelloni, 2005b). Stixrude and Lithgow-Bertelloni (2005b) have extended

the thermodynamic formulation by tensorial presentation of stress and the relation-

ship to entropy and temperature. Therefore, the elastic shear modulus is derived

without ad hoc assumptions. This formulation, among others, is implemented in

the software Perple X (Connolly, 2005) which permits extraction of any ther-

modynamic property as a generalised formulation of temperature, pressure and

composition.

Other codes have also been put forward (ThermoCalc, Domino, FreeGs) with differ-

ent thermodynamic solvers that focus on determining phase equilibria in petrolog-

ical systems regardless of the self consistency of material properties. These codes

use non-linear techniques for Gibbs energy minimisation. The strength of these

non-linear methods is their accuracy. However their weakness is that identification

of the stable mineral assemblage is probable but not certain. This lack of robust-

ness obviates the use of non-linear methods for embedded geodynamic calculations.

In contrast, Perple X utilises a linearised formulation of the minimisation prob-

lem which always converges (Connolly, 1990, 2005; Connolly and Petrini, 2002).

2.2. METHODS 13

For the derivation of self consistent material properties, this linearised algorithm is

convenient because it minimises data gaps. Another important aspect is that for

the purpose of deriving material data the best strategy is to reject mixed formu-

lations, where discrepancies with field observations are adjusted without changing

the underlying basic entropy model. Emergence of thermodynamic solvers for geo-

dynamic processes thus puts a strong constraint on basic consistency of the dataset.

All these reasons lead to use Perple X. A reference database is presented here for

the purpose of standardising material properties to be used particularly in geolog-

ical, geodynamic and geotechnical calculations. The numerical results have been

validated by comparing seismic velocities predicted for a pyrolitic composition to

the seismic models PREM (Dziewonski and Anderson, 1981) and ak135 (Kennett

et al., 1995; Montagner and Kennett, 1996). These models, constructed after travel

time data, give access to detailed information about the average Earth’s structure.

2.2 Methods

The purpose of PreMDB is to provide modellers with a complete and easy access

to fundamental material data for terrestrial rocks and minerals. Another goal is to

standardise material data in order to compare results from various numerical and

experimental techniques. In order to satisfy these requirements, thermodynami-

cally consistent data have been chosen as they are defined over a large range of

temperature and pressure and provide an important complement to experiments

and observations. Non-thermodynamic data such as transport or ad hoc properties

have also been added. Currently, PreMDB lists 20 material properties for each

rock and mineral (table 2). In isothermal-isobaric closed chemical system composed

of Π phases, the phase equilibria are determined minimising the Gibbs free energy of

the system (Gsys). This thermodynamic function is a function of temperature (T ),

pressure (P ) and the chemical composition of the system (Connolly, 2005; Connolly


and Kerrick, 2002). It is defined as:

Gsys(P, T, n) =p∑

i

niGmi (P, T, x1i , x2i , ..., xci) , (1)

where Gmi and ni are respectively the molar Gibbs energy and the number of moles

of phase i; xji is the composition of the ith phase with respect to the jth component

of the system.

Table 2: Physical properties available in PreMDB.

Properties calculated Symbol Units Thermodynamic-consistentin Perple X S&LB model* Other models

Enthalpy h J/kg

Specific enthalpy h× ρ J/m3

Entropy s J/K/kg

Specific entropy s× ρ J/K/m3

Isobaric heat capacity c J/K/kg

Specific heat c× ρ J/K/m3

Density ρ kg/m3

Thermal expansion α 1/K

Compressibility β 1/Pa

Bulk sound velocity Vφ km/s

P-wave velocity VP km/s

S-wave velocity VS km/s

Bulk modulus Ks GPa

Shear modulus G GPa

Elastic modulus E GPa

Poisson’s ratio ν

Gruneisen ratio γ

Additional transport propertiesThermal conductivity k W/K/mThermal diffusivity κ m2/sMelt viscosity η Pa.s

* The equations of state describing the solutions are from (Bina, 1998a; Cammaranoet al., 2003; Connolly and Kerrick, 2002; Duffy and Anderson, 1989). S&LB: Stixrudeand Lithgow-Bertelloni.

In Perple X, all reversible material properties are related to the thermodynamic

potentials G:

2.2. METHODS 15

• Entropy S

S = −(

∂G∂T

)

P

(2)

• Isochemical enthalpy H

H = G + TS (3)

• Internal Energy U

U = H − PV (4)

V being the volume.

• Heat capacity CP

CP = −T(

∂2G∂T 2

)

P

(5)

• Density ρ

ρ =N

V= N

∂G∂P

(6)

where N is the molar formula weight.

• Thermal expansion α

α = − 1

V

(

∂S

∂P

)

P

=1

V

(

∂2G∂P∂T

)

(7)

• Compressibility β

β = − 1

V

(

∂2G∂P 2

)

T

(8)

• Adiabatic bulk modulus Ks

Ks = − ∂G∂P

[

∂2G∂P 2

+

(

∂2G∂P∂T

)2

/∂2G∂T 2

]−1

(9)

• Gruneisen parameter γ

γ = V

(

∂P

∂U

)

V

(10)


Connolly and Kerrick (Connolly and Kerrick, 2002) use an ad hoc empirical

model (see equation 5 in (Connolly and Kerrick, 2002)) to calculate the shear

modulus. Stixrude and Lithgow-Bertelloni (2005b)’s formulation of Gibbs

energy for isotropic material permits to relate the shear and elastic moduli to

thermodynamic potentials:

• Elastic compliance tensor sijkl

sijkl = − 1

V

∂2G∂σij∂σkl

(11)

where σijkl is the stress tensor and the subscript on the derivative defining the

compliance means stress components except those involved in the derivative

are held constant. The elastic compliance tensor describes the general elastic

material behaviour in compression and shear. For an isotropic body there

are two independent quantities, e.g. bulk and shear moduli. For anisotropic

assemblages of phases, the shear modulus is essentially interpreted as that of

an isotropic polycrystalline aggregate.

• Shear modulus G, from one of its simplest forms

1

G=

1

V

n∑

ψ

(

xψVψ1

Gψ

)

(12)

For any equations of state other than those of Stixrude and Lithgow-Bertelloni

(2005a,b), the shear modulus calculated in Perple X is not computed self-

consistently from thermodynamic potentials (Connolly and Kerrick, 2002).

For isotropic elasticity the elastic modulus, as well as seismic properties are obtained

from the following equations:

• Elastic modulus E

E =9KsG

3Ks +G(13)

2.2. METHODS 17

• Poisson’s ratio ν

ν =3Ks − 2G

6Ks + 2G(14)

• Bulk sound velocity Vφ (Ita and Stixrude, 1992)

Vφ =

√

Ks

ρ(15)

• S-wave velocity VS (Ita and Stixrude, 1992)

VS =

√

G

ρ(16)

• P-wave velocity VP (Ita and Stixrude, 1992)

VP =

√

Ks + 4G/3

ρ(17)

Seismic wave velocities in a single crystal are thermodynamically self-consistent.

On the contrary, they are not in an aggregate of crystals and must be obtained by

an ad hoc averaging scheme such as the Voight-Reuss-Hill theory (Connolly and

Kerrick, 2002; Watt et al., 1976).

Transport properties are estimated from empirical models that are derived from

laboratory experiments and expressed as functions of the computed thermodynamic

properties:

• Thermal conductivity k Two equations have been implemented in PreMDB

to define k:

– As a function of the temperature T

k = A+B

350 + T(18)


where constants A and B are defined for different rock types. k is ex-

pressed in W/m/K and T in Celsius (Clauser and Huenges, 1995; Zoth

and Hanel, 1988).

– As a function of the P-wave velocity VP

k = 0.0681e0.0006VP (19)

with k in W/m/K, VP in m/s (Ozkahraman et al., 2004).

• Thermal diffusivity κ

κ =k

ρCP(20)

where k is defined according to eq. (19), ρ and CP derived from Perple X.

• Melt viscosity η

η = A+B

T+ exp

(

C +D

T

)

(21)

where A, B, C and D are linear functions of mole fractions of oxide compo-

nents, except for H2O (Hui and Zhang, 2007).

Depending on the complexity of the system and the precision required, the linearised

Gibbs energy minimisation problem can be time consuming. For this reason, com-

puted phase diagram sections and material properties are stored in the PreMDB

database. At present compositions of 48 major rock forming dry and wet miner-

als and 9 terrestrial rocks have been incorporated, representing a standard for the

sedimentary part of the crust (Plank and Langmuir, 1998), the upper and lower

continental crust (Rudnick and Fountain, 1995; Taylor and McLennan, 1985), the

oceanic crust (Staudigel et al., 1996) and the mantle (pyrolite and peridotite) (Hart

and Zindler, 1986; Ringwood, 1979). The rocks and minerals currently described in

PreMDB are listed in Table 3.

A graphical user interface has been developed to browse the database and plot all

2.2. METHODS 19

Table 3: Terrestrial rocks and minerals available in PreMDB

Minerals Rocks

Dry (25) Wet (22) Composition(10)

Watercontent

References

Albite Amesite Sediment Wet (Bina, 1998a)Almandine Annite Granite Dry/wet (Cammarano et al.,

2003)Anorthite Anthophyllite Granitoid Dry/wet (Behn and Kelemen,

2003)Cordierite Antigorite Basalt Wet (Bina and Wood,

1987)Diopside A-phase Gabbro Wet (Behn and Kelemen,

2003)Enstatite Brucite Peridotite Dry/wet (Bina, 1991)Fayalite Celadonite Pyrolite Dry (Bina, 1998b)Fe-Cordierite

Clinochlore

Ferrosilite DaphniteForsterite EastoniteGrossular Fe-

AnthophylliteHedenber-gite

Fe-Celadonite

Hematite FerroactinoliteHercynite GlaucophaneIlmenite Hydrous

CordieriteJadeite MargariteKalsilite MuscoviteMicrocline ParagoniteNepheline PargasitePyrope PhlogopiteQuartz TalcSanidine TremoliteSap-phirine442Sap-phirine793Spinel


properties as 2D graphs. The visual representation of all properties is an impor-

tant role of PreMDB as it is generally cumbersome to interpret or validate data

in a tabulated format directly from text files as Perple X produces them. Re-

versible properties computed from Perple X can be visualised as a scale coloured

2D map function of temperature and pressure. Transport properties are either vi-

sualised the same way (thermal conductivity derived from eq. (19) and diffusivity

from eq. (20) or represented as functions of the temperature (thermal conductiv-

ity from eq. (18) and melt viscosity from eq. (21)). Source (Perple X/equation)

and references of each property are presented in the graphical interface. Python

(http://www.python.org) is used as a scripting language to post-process data from

Perple X and convert them to a specific internal data structure. The graphical

user interface is implemented with wxPython (http://www.wxpython.org) and the

plots are rendered with matplotlib (http://matplotlib.sourceforge.net/).

Figure 1: Example of PreMDB’s GUI showing a wet peridotite composition satu-rated in H2O. The major density jumps reflect dewatering reactions predicted fromthermodynamic modelling.

2.3. RESULTS 21

PreMDB’s main window lists all materials of the database regrouped by cate-

gories under different tabs. There are currently two categories: earth materials and

major rock forming minerals. Each category displays a table listing sequentially dif-

ferent rocks/minerals along with their name, composition, comments, Perple X

input files (when applicable) and resulting pseudo-sections. Double clicking on a

rock/mineral line opens a new window composed of two distinct parts. The first

one lists all material properties available and the second one displays extra infor-

mation and references on any property selected. Select a property to visualise it.

In the case of transport properties, an option is offered for each available equation.

The visualisation pops up in another window as the graphical representation of

the underlying equation or tabulated data (Fig. 1). Basic functionalities are then

available, for instance to magnify the picture, obtain data values under the mouse

in the status bar or save the pictures in different file formats. It is possible to open

up as many windows as needed to easily compare properties of one or several ma-

terials. The main window also displays a toolbar with a unit converter and search

functionality.

2.3 Results

Comparison of petrological data between Perple X and other thermodynamical

software has been presented elsewhere (Hoschek, 2004). The emphasis is here on

the validation of the simulated material properties comparing thermodynamic re-

sults to independent sources. The focus of this first presentation of PreMDB is

on bulk material properties not on individual mineral data, which will be done

in a future contribution. Standard rock compositions are assemblages of different

minerals and their properties here are derived from averaging of these constituents.

Moreover, simulations calculate ideal rock phase diagrams at thermodynamic equi-

librium. Therefore, because of the large variability of rock composition and nat-

ural inhomogeneities, it is impossible to perform an exact one to one comparison


with a real rock. Another limitation to the validation procedure is that material

property data are conventionally derived at ambient laboratory temperature and

pressure conditions while thermodynamic simulations cover an extended TP range.

Therefore, this study focuses here on a comparison with PREM (Dziewonski and

Anderson, 1981) and ak135 (Kennett et al., 1995; McKenzie and Bickle, 1988)

models for the Earth’s mantle, down to the core-mantle boundary. To assess the

influence of chemistry on seismic properties, three pyrolitic compositions have been

considered presenting variable Al2O3, CaO and FeO contents (Ringwood, 1979;

Saxena, 1996; Stixrude and Lithgow-Bertelloni, 2005a). These compositions are re-

spectively named “Sax”, “Rng” and “Stx”. Calculations have been performed using

the thermodynamic datafile developed by Stixrude and Lithgow-Bertelloni (2005a)

and augmented for the lower mantle as described by Khan et al. (Khan et al., 2006).

Both seismic models PREM and ak135 have been constructed from travel time

data. These 1D models provide a good description of the elastic moduli of the

mantle, but inferring average temperature and composition from them requires a

model of the equation of state (EoS) and accurate knowledge of the thermoelastic

properties of minerals. For the purpose of the comparison to seismic datasets,

the temperature in the mantle is calculated self-consistently following an approach

proposed by Ita and Stixrude (Ita and Stixrude, 1992). An adiabatic interior is

considered to be overlain by a lithosphere defined by the half-space conductive

cooling solution. A potential temperature of 1,600K for the 100 Ma geotherm is

considered, as this temperature lies between two estimates of potential temperature

required to produce oceanic crust of average thickness (Klein and Langmuir, 1987;

McKenzie and Bickle, 1988). The isentrope is computed self-consistently by finding

the PT path along which the total entropy of the assemblage is constant. While

the isentropic assumption follows the principle of self-consistency, the isentropic

character of the mantle is not argued.

The density, P- and S-wave velocity and Poisson’s ratio adopted in the seismic

models PREM and ak135 are compared to the ones calculated for pyrolite mantle

2.3. RESULTS 23

composition with varying Al2O3, CaO and FeO contents figs. 2 to 5. A global

agreement is obtained between seismological and thermochemical models (autocor-

relation functions ACF respectively varying between 0.996-0.998, 0.996-0.998,0.988-

0.995 and 0.748-0.779 for PreMDB/PREM and 0.992-0.996, 0.994-0.997, 0.987-

0.994 and 0.763-0.782 for PreMDB/ak135), except for the Poisson’s ratio. The

discrepancy for this parameter results from the square root relation between the P-

and S-wave velocity ratio (VP/VS) and the Poisson’s ratio (ν):

VPVS

=

√

2(1− ν)

1− 2ν(22)

Therefore, the deviation for this latter parameter is squared, making the Poisson’s

ratio more sensitive to chemistry.

The mixture of 60% of pyrolitic and 40% of chondritic composition proposed by

Matas et al. (2007) has also been tested but did not fit any better the seismic

models.

A robust result of the comparison presented is that the low-velocity layer underneath

the lithosphere emerges out of phase changes and does not necessitate partial melt

as pointed out by Stixrude and Lithgow-Bertelloni (2005a). The results presented

show the abrupt discontinuity around 410-420 km depth, with a better agreement

with ak135 than PREM (best fit obtained with the “Stx” composition). It cor-

responds to the formation of wadsleyite (“Stx” and “Rng” compositions) or the

disappearance of olivine (“Sax” composition). This result is in good agreement

with seismological studies (Bina, 1998a; Deuss et al., 2006; Irifune and Isshiki,

1998). The calculation also predicts two other seismic discontinuities around 536

and 565 km depth only in the case of a composition depleted in Al2O3 and CaO

but enriched in FeO (“Sax”). They correspond to the formation of majorite that

then transforms in akimotoite. This result fits seismic observations (Bina, 1991;

Irifune and Isshiki, 1998) and suggests that heterogeneities in the upper mantle

composition lead to the presence or absence of this seismic discontinuity. Another


significant result is that in none of the chemical models considered, the 660 km

discontinuity comes out as a sharp jump in physical properties. There is rather a

series of phase changes in the 650-890 km depth range depending on the compo-

sition considered. This result differs from PREM and ak135 seismological data

and may result from an incomplete description of the solid solutions considered

in the thermodynamic model. The chemical composition and temperature profile

considered, in particular the underlying equilibrium assumption, may also be re-

sponsible for this discrepancy. However, compositions depleted in Al2O3 and CaO

but enriched in FeO (Saxena, 1996) fit the jump more closely than the other com-

positions considered. The calculations predict the formation of perovskite at the

detriment of akimotoite at 649 km depth, followed by the appearance of periclase at

697 km depth and finally the transformation of ringwoodite in magnesiowustite at

758 km depth. This result supports the hypothesis that the 660 km discontinuity

is a transition zone as inferred by receiver function analyses (Deuss et al., 2006;

Vacher et al., 1998) and is thus in agreement with more recent observations than

ak135 and PREM.

A third deviation of the model is observed in the lower mantle increasing towards

the core-mantle boundary and particularly around 2,830-2,840 km depth that cor-

responds to the transformation of perovskite into post-perovskite. The deviation

to the seismic data suggests that the pyrolite composition model may not accu-

rately reflect the chemistry of the lowermost mantle. This observation supports

the hypothesis that the lowermost mantle has a different chemical composition re-

sulting possibly from an accumulation of subducted slabs (Grand, 2002). Other

parameters such as sub-adiabatic conditions in the lower mantle, anisotropy or iron

spin transition could also be responsible for the shift observed in the lower mantle

results.

2.3. RESULTS 25

Figure 2: Density-depth profile of PreMDB compared to the seismic models PREM

and ak135.

Figure 3: VP -depth profile of PreMDB compared to the seismic models PREM

and ak135.


Figure 4: VS-depth profile of PreMDB compared to the seismic models PREM

and ak135.

Figure 5: Poisson’s ratio profile of PreMDB compared to the seismic modelsPREM and ak135.

2.4. DISCUSSION 27

An interesting observation is the discrepancy between simulated and seismologi-

cal Poisson’s ratio (autocorrelation functions respectively of 0.748-0.779 and 0.763-

0.783 for PREM and ak135, depending on the pyrolitic composition considered-

fig. 5), especially around the 660 km discontinuity. This result shows that this

parameter is very sensitive to chemistry variation. Therefore it can be a useful

marker of the variation of mantle composition with depth and could be used for

future fine tuning of mantle chemistry.

2.4 Discussion

Recent studies (Stixrude and Lithgow-Bertelloni, 2005a,b) have shown that phys-

ical properties of terrestrial rocks and minerals derived from thermodynamic po-

tentials complement geophysical observations and experimental measurements. In

the benchmark study presented here, the model fits the PREM and ak135 seismic

models and may even record more precise discontinuities due to thermodynami-

cally predicted phase transitions. Discrepancies in the thermodynamic simulations

may result from an incomplete description of the solid solutions considered in the

thermodynamic model, as well as from the chemical composition and temperature

profile considered, in particular the underlying equilibrium assumption. Another

component of the uncertainties is that the PREM and ak135 seismic reference

data derive from experimental measurements.

The properties of terrestrial rocks and minerals previously determined from labora-

tory analyses can now be derived more accurately from thermodynamic modelling

for the entire temperature and pressure range of the Earth’s mantle. An advantage

of this approach is that it permits self-consistent extrapolation beyond the range of

the laboratory. The computational tool used for this purpose is Perple X, a Gibbs

energy minimisation algorithm that computes phase equilibria, maps phase relations

and extracts mineral physical properties of geodynamical interest. Perple X is ro-

bust and computes stable mineral assemblages. It computes the thermochemical,


thermal, seismic and other elastic properties from fundamental thermodynamic re-

lations. In particular, Perple X predicts phase transitions that are important for

seismic tomography and geodynamic modelling. In addition to input from Per-

ple X, complementary transport properties derived from laboratory experiments

were added to enhance the information for the user. The resulting reference mate-

rial database is a foundation for performing realistic simulations of rock behaviour

during geological and geodynamic processes.

2.5 Conclusion

PreMDB provides a database for material properties derived from thermodynam-

ics at a range of temperatures and pressures that are not always available from

laboratories, as well as for complementary transport properties. The influence of

the variation of those petrological properties with temperature and pressure was

demonstrated and validated against two accepted seismic models, focusing mainly

on the Earth mantle. These material properties variations however are also criti-

cal at shallower depths, even though the thermodynamic equilibrium assumption

is reaching its limit towards the surface. The graphical interface built around the

database also represents a valuable step to help scientists quickly visualise and

realise the effect of those changes. In the following Chapter 3 those important feed-

backs are incorporated in a numerical simulation package, escriptRT, which not

only accounts for material properties variations but also incorporate many more

chemical feedbacks by modelling directly the fluid-rock chemical interactions.

Chapter 3

Reactive transport

Symbols definition

This chapter uses the nomenclature defined in Table 4.

3.1 Introduction

This chapter presents escriptRT, a new reactive transport simulation code for

fully saturated porous media which is based on a finite element method (FEM)

combined with three other components: (i) a Gibbs minimisation solver for equi-

librium modelling of fluidrock interactions, (ii) an equation of state for pure water

to calculate fluid properties and (iii) the thermodynamically consistent material

database presented in chapter 2 to determine rocks’ material properties. Using de-

coupling of most of the standard governing equations, this code solves sequentially

for temperature, pressure, mass transport and chemical equilibrium. In contrast,

pressure and Darcy flow velocities are solved as a coupled system (Gross et al.,

2009).

Transport of heat and chemical species in porous media is a critical component

in understanding many geological processes such as convection, precipitation and

29

30 CHAPTER 3. REACTIVE TRANSPORT

Table 4: Nomenclature definition

Symbol Definition

ρ density [ kgm3 ]

φ porosity [-]Ω representative volume element [m3]q Darcy flux (in co-moving frame) [m

s]

κ intrinsic permeability [m2]λ thermal conductivity [ W

m K]

µ viscosity [Pa s]γf fluid acceleration [m s−2]g gravity [m

s2]

e unit vector [-]u fluid velocity [m

s]

p fluid pressure [Pa]T temperature [K]Cp specific heat capacity [ J

kg K]

Q radiogenic heat source [ Wm3 ]

M masses [mol]S reactive masses [mol]L reactive mass ratios[-]V discrete volume of the mesh [m3]U molal concentrations [mol

kg]

P molal reaction rates [ molkg s

]

C molar concentrations [molm3 ]

R molar reaction rates [ molm3 s

]

D total mass dispersion [m2

s]

D diffusion coefficient of heat [m2

s]

Superscriptp component in porous spacel component in liquid phases component in solid phaseSubscriptf fluid componenti index of chemical elementz z-coordinate directionVectors and tensorsX vector XX tensor X

3.1. INTRODUCTION 31

dissolution of minerals, which can be applied to geothermal systems or the for-

mation of ore deposits. There already exists some good descriptions and robust

codes which address this class of problems such as those presented by Pruess et al.

(1999), Bartels et al. (2003), Diersch (2005), Geiger et al. (2006) or Ingebritsen

et al. (2006) for example. It is usually no trivial task, however, for researchers to

modify or extend these codes as those developments require a full set of different

skills, from a deep geo-scientific knowledge to a high proficiency in software de-

velopment and numerical analysis. The task can be even more daunting as the

code gets more sophisticated and the changes required need to be compliant with

all underlying software and hardware architecture choices. Object Oriented Pro-

gramming is fortunately increasingly used and facilitates the researcher’s work but

none of the mentioned codes satisfy for example the full separation of concerns

(SoC)1 paradigm (Hursch and Lopes, 1995) in order to let the expert user focus on

geosciences only.

The escript module (Gross et al., 2008) provides a generic environment for mod-

ellers to develop simulations solving partial differential equations (PDE) in python.

Its numerical solver can run in parallel and also offers the ability to work on un-

structured 2D and 3D meshes, which enables one to model complex and geologically

realistic structures for example. This modelling library is separated by design from

the underlying solver and provides with its modelframe approach an ideal modu-

larity which is key to escriptRT’s architecture. The development of escriptRT

was indeed motivated by the need of a powerful but also user-friendly, flexible, and

extensible platform. The purpose of this code is to allow geologists with different

levels of numerical skills to focus on the definition of their problem, with the pos-

sibility to interact with the code at different levels but no obligation to become

experts in applied mathematics and numerical programming.

This chapter is articulated in four parts. Section 3.2 presents the classical system of

1Separation of concerns in computer science is the process of building a clear architecture withseparate building blocks to minimise as much as possible the overlap of common functionalitiesbetween those blocks.


equations describing all geological processes considered. Section 3.3 introduces all

solvers within the code dealing with those equations. The modular software archi-

tecture uses a conceptual-mathematical-numerical (CMN) pattern and is discussed

in Section 3.4. Modellers benefit from this abstraction process to describe their

problem at a conceptual level, independently from the corresponding mathematical

representation and in turn from the underlying numerical implementation. Finally,

Section 3.5 demonstrates the use of escriptRT by simulating the precipitation

of gold within a complex granite-greenstone sequence conceptualised from Archean

gold deposits.

3.2 Reactive transport formulation

The escriptRT code models the transport of heat and solutes in porous media,

as well as the chemical reactions between the fluid and the host rock. The physical

formulation of the problem is based on the assumption of a porous medium fully

saturated with a single phase fluid, for which Darcy’s law applies (see Bear, 1979).

The fluid and hosting rock are assumed to be always in thermodynamic and chemical

equilibrium, and capillary pressure effects are neglected (Bear, 1979). Following the

finite element methodology a representative volume element Ω of rock is considered

and decomposed as a sum of its solid and porous components

Ω = Ωs + Ωp . (23)

In this section the current status of escriptRT is presented to solve sequentially

a set of decoupled governing equations. The following subsections introduce con-

secutively all process models involved, which employ direct couplings and indirect

feedbacks (see chapter 5 or Poulet et al., 2010). A direct coupling is defined to

be the proper handling of the dynamical updates of all variables involved in the

solved system of equations. An indirect feedback refers to the sequential update of

selected variables in some of the equations, while their time evolution is neglected

3.2. REACTIVE TRANSPORT FORMULATION 33

in others.

3.2.1 Pressure equation

All processes simulated by escriptRT are based on a description of fluid flow

in saturated porous media and it is therefore natural to present firstly the pore

pressure equation. This equation follows from the fluid mass conservation which

can be written asd(φρf )

dt+∇ · (ρfq) = 0 , (24)

The flux q is defined using Darcy’s law under the assumption of quasi static fluid

flow with zero acceleration (γf ≈ 0) in absence of tortuosity

q = − k

µf(∇p− ρfg) . (25)

escriptRT proposes several constitutive models, supporting both compressible

and incompressible fluids. In the case of an incompressible fluid with no porosity

evolution, equations (24) and (25) provide directly the pressure equation

∇ · (ρfk

µf∇p) = ∇ · (ρ2f

k

µfg) (26)

When compressible fluid is considered, the fluid density (ρf ) is dependent on tem-

perature (T ) and pressure (p). A simplified mechanism is also introduced to modify

ρf as a function of salinity X (wrt CNaCl). The following fluid property definitions

are used:

• the thermal expansivity αf , defined as αf = − 1ρf

∂ρf∂T

;

• the compressibility βf , defined as βf =1ρf

∂ρf∂p

;

• the chemical expansivity γf , defined as γf =1ρf

∂ρf∂X

.


Using the chain rule, the variation of fluid density becomes

dρfdt

= ρf

(

−αfdT

dt+ βf

dp

dt+ γf

dX

dt

)

. (27)

The porosity φ is also considered to be only dependent on pressure p (Geiger et al.,

2006)dφ

dt=∂φ

∂p

dp

dt. (28)

Equation 23 and the definition of porosity φ ≡ Ωp

Ωlead to

dΩt

Ωt

= (1− φ)dΩs

t

Ωst

+ φdΩp

t

Ωpt

. (29)

The compressibility of the rock βr ≡ − 1Ω∂Ω∂p

is then given (Bear, 1972) by

βr = −(1− φ)1

Ωst

dΩst

dp− φ

1

Ωpt

dΩpt

dp= (1− φ)βs + φβf . (30)

From the definition of the porosity and (23), a simple derivation leads to

∂φ

∂p= −φ(1− φ)(βf − βs) . (31)

Combining (27), (28) and (31) provides

d(φρf )

dt= φρf

[

βrdp

dt+ γf

dX

dt− αf

dT

dt

]

. (32)

Using expressions (24), (25) and (32), the pressure equation can finally be written

as

βrφρfdp

dt−∇ · (ρf

k

µf∇p) = −∇ · (ρ2f

k

µfb)

+αfφρfdT

dt− γfφρf

dX

dt. (33)


This equation can then be solved directly using escript as described in Sec-

tion 3.4.2.

3.2.2 Temperature transport

The second process model considered is heat transport, where thermal equilibrium

is assumed between the solid and liquid components of the medium. The effects

of mechanical deformation and the related thermal expansion of the solid part are

neglected. In respect to the liquid component, any heating due to viscous dissipation

is also neglected. The temperature is then transported via the standard advection

diffusion equation (Nield and Bejan, 2006)

ρ Cp∂T

∂t+ (ρ Cp)fv.∇T −∇.(D ∇T ) = Q , (34)

where Q is a radiogenic heat source, ρ Cp = φ(ρ Cp)f + (1 − φ)(ρ Cp)s and D =

φDf + (1− φ)Ds is the diffusion coefficient.

3.2.3 Transport of chemical elements

In the mass transport model, the fluid is regarded as being composed of a set of

chemical species in aqueous phase. Those species are transported with the fluid and

considered to be in chemical equilibrium with the solid minerals of the surrounding

rock matrix at the end of every discrete time step. These assumptions allow the

transport problem to be solved for a limited set of reaction invariants instead of the

full set of chemical species. This approach is similar to the concepts of “tenads” in-

troduced by Rubin (1983), as well as the tableaux defined by Morel (1983). We use

as dependent variables the masses of chemical elements constituting the chemical

species, as well as the total electrical charge of the species. The achieved reductions

of the size of the system and the computations in total are considerable. For exam-

ple, a typical simulation would involve tens of chemical elements but hundreds of


chemical species, and therefore the codes transporting chemical species would solve

much larger systems of PDEs.

Another advantage of using the masses of chemical element as dependent variables is

the improved numerical stability of the system of transport equations. Indeed,when

the full list of chemical species is treated as dependent variables, intensive precip-

itation/dissolution of minerals can cause appearances and disappearances of some

species according to local conditions, resulting in pulses of zero/non-zero masses.

This can cause numerical difficulties to the solver and requires special attention,

see Guimaraes et al. (2007). This issue is generally not present when transporting

chemical elements.

The molal concentrations of these elements are defined at each mesh node of volume

V as

U l ≡ M l

Mf

, (35)

where M l represent the masses of chemical elements within all aqueous species at

the mesh node of interest and Mf is the corresponding mass of the fluid there

Mf = ρfVf . (36)

Similar ratios are defined for the solid phase of chemical elements

U s ≡ M s

Mf

, (37)

where M s are their masses within all mineral species.

Physically the solute is assumed to be pure water, which properties are computed

from available empirical data for the thermal equilibrium of its equation of state

(see Section 3.3.3). As a chemical species, the mas of water is updated according

to the current equilibrium state of the chemical system. In such a way the mass

conservation of the i-th chemical element is represented by the conservation of its


mole number, written in terms of molality as

∂(φρfUli )

∂t+ ∇ ·

[

φρfuUli − φD∇(ρfU

li )]

+ φρfPi = 0 , (38)

d(φρfUsi )

dt− φρfPi = 0 , (39)

where Pi is the molal production/consumption rate of this element in liquid phase.

The dispersion tensor D accounts for molecular diffusion, longitudinal and trans-

verse dispersions. The effects of tortuosity can also be considered in this term as

they are neglected in their full treatment. All chemical elements in liquid phase are

also assumed to disperse in the same way.

The conservation of fluid mass (24) allows (38) to be written as

φρf∂U l

i

∂t+ φρfu · ∇U l

i −∇ ·[

φD∇(ρU li )]

+ φρfPi = 0 . (40)

Since by Fick’s law the diffusive flux is driven by volumetric concentrations, mo-

lalities U are converted to molarities C = ρfU , and (39) and (40) are rewritten

as

φρf∂

∂t

(

C li

ρf

)

+ φρfu · ∇(

C li

ρf

)

− ∇ · (φD∇C li) + φRi = 0 , (41)

d(φCsi )

dt− φRi = 0 . (42)

In (41)–(42) C l and Cs are computed using the fluid volume Vf , similarly to

U l and U s in (35) and (37) respectively. The variables R represent the molar

production/consumption rates of chemical elements in liquid phase over the cor-

responding time step; they depend on the extent of the heterogeneous chemical

reactions.


During transport, the temporal and spatial variations of the fluid density are as-

sumed to be small, keeping therefore ρf approximately constant. Equation (41) can

then be simplified to

φ∂C l

i

∂t+ φu · ∇C l

i −∇ · (φD∇C li) + φRi = 0 . (43)

The effects of chemical reactions are presented in (42)–(43) via the production/con-

sumption rates of chemical elements R , which are computed separately from the

transport by a chemical solver presented in Section 3.3.2. In such a way the

advection-reaction-diffusion (ARD) equation (43) is uncoupled and replaced with

an advection-diffusion equation and a system of mass updates for the chemical

elements in solid and liquid phase. Equations (42)–(43) then become

φ∂C l

i

∂t+ φu · ∇C l

i −∇ · (φD∇C li) = 0 , (44)

and

dC li

dt+ Ri = 0 , (45)

dCsi

dt− Ri = 0 . (46)

The total mass of each chemical element is preserved because the advection-diffusion

equation (44) is conservative and the production/consumption rates in (45)–(46)

have the same amplitude but opposite signs. The influence of the chemical reactions

is indirectly presented by the production/consumption mass rates in equations 45

and 46.

The basic assumptions behind the transition from the ARD equation (43) to the

system (44)–(45) are that all chemical species are always at equilibrium and the

chemical reactions between them are infinitely faster than the advective and diffu-

sive transport processes for the geological scenarios considered. These assumptions

allow the influence of the production/consumption term to be taken into account at


the beginning and at the end of every time step when solving (44). This translates

in this workflow by computing the initial mass of the chemical elements in liquid

phase with the chemical solver, and then updating these masses at every time step

by solving firstly (44) and then (45)–(46). Those equations use production/con-

sumption rates computed from the current reactive mass Sl in liquid phase

S =M leq −M l

tr , (47)

where

M ltr are the masses of chemical elements in liquid phase after solving (44);

M leq are the masses of chemical elements in liquid phase after chemical equilibrium

is obtained by the solver.

The aqueous masses in equilibrium Maeq vary within the range

0 ≤Maeq ≤M , (48)

whereM =Ma+M s are the total masses of chemical elements and the comparison

is component wise. In accordance with (47) the range of reactive masses is

−Matr ≤ S ≤M −Ma

tr . (49)

Then the production/consumption rates are

R =S

Vf∆t, (50)

where ∆t denotes the discrete time step.

To estimate the extent of the influence of reactive mass on the decoupling of (43)


another parameter is introduced

L =S

Matr

=Ma

eq −Matr

Matr

=Caeq − Ca

tr

Catr

, (51)

where L is the reactive mass ratio and the division is component wise. It evaluates

the aqueous mass generated or consumed relative to the aqueous mass present at

the end of the transport. In accordance with (49) the range of reactive mass ratio

is

−1 ≤ L ≤ M

Matr

− 1 , (52)

and again the division is component wise. Since the current approach is using

transport of aqueous chemical elements, the relations (48)–(49) and (52) define

precise lower and upper limits of equilibrated and reactive masses. In contrast,

other approaches using transport of chemical reagents can only define the lower

limits of these parameters.

3.3 Reactive transport solvers

This section presents various solvers used by escriptRT to solve the constitutive

equations presented in Section 3.2. All PDE equations are solved directly using the

escript solver; the fluid-rock chemical reactions, as well as the equations of state

for the fluid and solid material properties are computed using separate external

packages.

3.3.1 The Finley PDE solver

escript can potentially use different FEM solvers and is linked in its current

implementation to the solver library Finley (Davies et al., 2004), which solves a

PDE given in weak form. For the case of a scalar unknown solution u the weak

3.3. REACTIVE TRANSPORT SOLVERS 41

form is

∫

Ω

(

Mv∂u

∂t+∇v · A · ∇u+∇v ·Bu+ vC · ∇u

+vDu) dΩ =

∫

Ω

(∇ ·X + vY ) dΩ , (53)

which needs to be fulfilled for all so-called test functions v. To simplify the presen-

tation, boundary conditions are ignored. It is assumed that this equation is solved

for a sufficiently small time interval such that it can be assumed that the PDE coef-

ficientsM , A, B, C, D, X and Y are constant in time and are independent from the

unknown solution u. Details on how the weak formulation of the PDE is applied

will be presented in Section 3.4.2. This equation is discretised using standard con-

form FEM in 2D and 3D . Finley supports first and second order approximations

of triangle, tetrahedron, quadrilateral, and hexahedron elements (see Zienkiewicz

et al., 2005). In order to support the usage of compute clusters and multi-core

architectures, Finley is parallelised for both shared and distributed memory using

element and node colouring on shared memory via OpenMP2 and domain decom-

position on distributed memory via the Message Passing Interface (MPI) library.

Finley offers therefore flexibility but also performance for our geological scenarios

as it includes optimised solvers for such transport problems (Gross et al., 2009),

including a coupled solver for the pore pressure and Darcy flow velocities.

In the case of a static problem (M = 0) Finley uses iterative techniques such

as preconditioned conjugate gradient method (PCG) (Weiss, 1996) to solve the

discretised version of the weak formulation (53) of the PDE. In the cases discussed

in this paper the coefficients may have large jumps across element boundaries,

typically at the conjunction of different geological structures. This situations leads

to slow convergence or even divergence when solving the discrete problem. To

accelerate the convergence, algebraic multi-grid is used as a preconditioner.

2http://www.openmp.org


When solving the time dependent transport of chemical species (44) or tempera-

ture (34) the term Mv ∂u∂t

is present in equation (53). In this case Finley applies

the backward Euler or Crank-Nicolson scheme to discretise in time. In cases of an

advection dominated problem, i.e. large values for B, C in comparison to the diffu-

sion term A, unphysical oscillations in the solution can be observed, in particular in

the presence of a steep front in the solution as it is typically seen in reactive trans-

port models. To stabilise the solution artificial diffusion is introduced and limited

via a flux control scheme to restrict the application of artificial diffusion to locations

of large gradients in the solution. Using the framework introduced by Kuzmin et al.

(2004) the scheme is directly applied to the spatially discretised problem without

referring to the continuous problem or the underlying conform FEM mesh.

The incompressible fluid equation (26) can directly be solved within the Finley

framework (53); however, when using FEM the direct reconstruction of the flux

in (25) via gradient calculation can be unreliable and produce numerical artifacts.

This is caused by the fact that the flux constructed in this straight forward way

defines the defect of the FEM approximation of the pressure. In order to rectify the

problem one needs to solve for pressure and flux simultaneously. A least squares

approach is used, combining the incompressibility condition for the flux and the

definition of the flux and solve the optimisation

minq,p

∫

Ω

(

|µfkq −∇p+ ρfb|2 + λ · |∇ · ρfq|2

)

dΩ (54)

This optimisation problem can be reformulated as a weak PDE of the form (53)

for the solution u = (p, q). As in practice values of p and q are different by several

orders of magnitude, iterative schemes have problems to converge. Therefore, the

implementation successively solves for q and p starting from the pressure solution

of the incompressible fluid equation (26). In each step two PDE s need to be

solved, one for pressure and one for flux, using the Finley library. The iteration

is accelerated using the PCG method on the pressure corrections.

3.3. REACTIVE TRANSPORT SOLVERS 43

3.3.2 The GibbsLib solver for chemical equilibrium

Chemical equilibrium of the mass composition is solved using a Gibbs free energy

minimisation technique at prescribed temperature and pressure. It includes the

masses of the chemical elements and water (so called bulk composition), which were

computed during the transport step (see Section 3.2.3 and equation 44). The chem-

ical species existing at equilibrium are found using the GibbsLib solver (Shvarov,

1978) (via GibbsLib.dll) , which is the solver used within the HCh geochemical

modelling package (Shvarov and Bastrakov, 1999). The GibbsLib solver requires

access to an associated thermodynamic database, Unitherm, which is tuned for

hydrothermal ore systems and also forms part of the HCh software bundle. This

database specifies the thermodynamic properties of aqueous species, gases and min-

erals and is valid within the limits 0−5 kbar, 25−1000 C of the modified Helgeson-

Kirkham-Flowers (HKF) model (Helgeson et al., 1981; Shock et al., 1992; Tanger

and Helgeson, 1988). The user specifies a subset of the thermodynamic database

for use in the simulation (as a plane text and binary file) and this is done using

the CSIRO option in the UT2K software3. This option allows users to select

the underlying thermodynamic database of their choice according to the available

empirical data for the involved chemical species.

Several codes have already been developed to simulate reactive transport problems

and can be divided in two categories: Law of Mass Action (LMA) and Gibbs En-

ergy Minimisation (GEM). A presentation of their respective strengths and weak-

nesses can be found in (Shao et al., 2009), including a list of popular codes such

as TOUGHREACT (Xu and Pruess, 2001), PHT3D (Prommer, 2002) or SHE-

MAT(Clauser, 2003) for example. TheGEM solvers require more thermodynamical

data and more computation resources. They also have advantages however, includ-

ing the ability to work from fixed bulk compositions as a starting condition. In

this sense the starting mineral assemblages at equilibrium in the model vary across

3UT2K is available from Evgeniy Bastrakov at Geoscience Australia, www.ga.gov.au. Seealso (Cleverley and Bastrakov, 2005).


the defined Pressure-Temperature gradient (at fixed total composition), when the

LMA approach requires close-to-equilibrium starting assemblage conditions. This

difference can be critical for the large-scale hydrothermal systems we are targeting.

A comparison of various geochemical modelling approaches and packages which in-

cludes the specific use of HCh in hydrothermal modelling was also demonstrated

in (Cleverley and Oliver, 2005).

There are a number of optional methods for passing chemical state information

to GibbsLib and in escriptRT the chemical composition are passed as moles

of elements at the specified pressure-temperature conditions. Users can choose

to include solute water in the total basis set of elements or pass this separately to

GibbsLib. The resultant chemical equilibrium state is returned as moles of aqueous

species and minerals. Although possible, gas phases or mineral solid solutions are

currently not dealt with explicitly.

3.3.3 The Equation of State (EOS) solver for fluid proper-

ties

escriptRT currently handles a single phase flow, and as a starting point of its fluid

EOS three different models for pure water were selected: the IAPWS-95 (Wagner

and Pruß, 2002), IAPWS-97 (Wagner et al., 2000) and revised HKF model (Helge-

son et al., 1981; Shock et al., 1992; Tanger and Helgeson, 1988). The HKF model

EOS provides thermodynamic consistency with GibbsLib, the chemical solver de-

scribed in Section 3.3.2. The fluid EOS module of escriptRT is critical to the

modelling accuracy obtained. For example fluid density is the main driver for con-

vection (Nield and Bejan, 2006). The water density EOS covers a wide range from

0 to 1000 C and 0 to 5,000 bar. Other water properties are defined on narrower

ranges, but nevertheless are sufficient to define the general conditions of many ge-

ological scenarios with a single liquid phase.

Note that the fluid density ρf (and corresponding chemical expansivity) can also

3.4. SOFTWARE ARCHITECTURE 45

be computed using other options:

• from a formula provided by the user to define ρf from chosen concentrations

ci through the definition of constants Ai, Bi and Ci as

ρf = ρ0

(

∑

i

Aici + exp

(

∑

i

Bici

))

+∑

i

Cici (55)

• as an interpolation table with temperature and pressure dependent values.

Users can then apply any pre-computed equation of state of their choice;

• using an equation of state for saline water proposed by Driesner (2007).

Note that the effect of salinity on fluid properties is an important research subject

and the thermodynamic treatment of salt solutions has not been considered in this

thesis beyond the use of the equations of state mentioned above.

3.3.4 The PreMDB database for rock properties

Solid material properties are calculated in a similar way to fluid properties and

are considered as dependent on temperature and pressure. PreMDB (Siret et al.,

2009) is used for that purpose, which is a thermodynamically consistent material

database based on thermodynamic potential functions to calculate all reversible

material properties, as presented in chapter 2. The code allows the solid density ρs,

specific heat Cp,s and thermal conductivity ks to be interpolated from tabulated

values for temperature and pressure. PreMDB also employs various published

empirical relationships.

3.4 Software architecture

This section presents the modular software architecture of escriptRT (see Fig-

ure 6) and the derived advantages of using such an approach. The underlying idea


Figure 6: software blocks

is to apply the CMN pattern, which is achieved through the usage of five successive

layers presented in the following subsections.

3.4.1 python environment

python is a powerful and easy-to-use interpreted programming language (van

Rossum and Drake, 2009), ideally suited for rapid application development and

connecting existing components together. It benefits from a large development com-

munity and provides a wide range of additional packages, including very mature and

actively developed numerical extensions like numpy (Oliphant, 2006). The usage of

a high level language like python provides the full flexibility of an Object-Oriented

Programming language. It also enhances the clarity of the code and certainly adds

some user-friendliness when compared to lower level programming languages. It

is important to note that these advantages are not obtained at the detriment of

efficiency. python is used as a binding language to simplify the handling of other

optimised libraries implemented using lower level languages (Sanner, 1999). Within

escriptRT this is illustrated by the four following examples (see Figure 6):

• The escript python interface is used to expose the efficient underlying C++

and C implementation.


• Some equations of state of water are implemented in fortran and compiled

into a python library using f2py.

• The initial implementation of the geochemistry module was implemented in

C and linked to python using the boost4 library.

• The current implementation of the geochemistry module uses python na-

tively and makes direct calls to the GibbsLib Windows dll.

3.4.2 escript framework

A cornerstone of escriptRT is the escript5 module (Gross et al., 2007), a high

level simulation environment written in python and providing access to underlying

linear PDE solver Finley as discussed in Section 3.3.1. This solver is available in

escript through the LinearPDE class object which defines a general second-order

linear PDE of the form

−∇ · (A · ∇u+ Bu) + C · ∇u+Du = −∇ ·X + Y (56)

where u represents an unknown scalar function and A, B, C, D, X and Y are

functions of their location in the domain. The boundary conditions are set in a

similar way (see Gross et al., 2007).

escript was designed to solve general, coupled, time-dependent, non-linear systems

of PDEs. Users can achieve that goal by appropriately setting the coefficients of

the LinearPDE. For example the compressible fluid equation (33) can be rewritten

as

−∇ · (ρfk

µf∇p) + βrφρf

p

∆t= −∇ · (ρ2f

k

µfb)

−βrφρfpold∆t

+ αfφρf∆T

∆t− γfφρf

∆X

∆t(57)

4http://www.boost.org5see https://launchpad.net/escript-finley


where all time derivatives were discretised numerically ∂p∂t

= ∆p∆t

= p−pold∆t

. ∆t

represents the current time step used and pold the value of p at the previous step.

Identifying all terms with (56) allows to define all parameters as

• A = ρfkµf1

• X = ρ2fkµfg

• D = βfφρf1∆t

• Y = βfφρfpold∆t

+ αfφρf∆T∆t

− γfφρf∆X∆t

• y = ρfqBC where qBC

denotes the boundary flux

This approach represents a major step towards the SoC paradigm as it encapsulates

and hides all the numerical implementation aspect at the C level, letting developers

focus on the scientific aspect of their problem. In such a way, by simple manipulation

of python objects, users can produce portable code that runs (without any change)

on multiple platforms and supports parallelisation through both MPI for distributed

memory and OpenMP for shared memory. This is the easiest process for developers

to benefit over time from escript efficiency improvements, new functionalities

or wider platform support as they don’t have to modify their code (unless the

Application Programming Interface gets modified).

3.4.3 Modelframe approach

The escriptRT code relies heavily on the modelframe module, a higher-level com-

ponent of escript, to clearly design the code architecture. This module provides a

framework to implement mathematical models as python objects using the Model

class. This class handles a generic time stepping workflow (see Figure 7) com-

ing from solving a time dependent PDE. It implements a set of generic methods

including doInitialization, getSafeTimeStepSize, doStepPreprocessing,


Figure 7: Simplified modelframe simulation workflow

doStep, terminateIteration, doStepPostprocessing, finalize and do-

Finalization (see Gross et al., 2008). This breakdown of such a generic workflow

provides a very convenient framework to implement all the couplings and feed-

backs most developers are interested in. Different Model derived classes can access

each other’s updated parameters through the use of Link objects (see Gross et al.,

2008) which provides the simplest mechanism to implement indirect feedbacks as

all variables from different classes get updated sequentially. Both the compressible

and incompressible pore pressure models were implemented in two different Model

classes. The power of the modelframe module is illustrated by the fact that in

the simulation script those two implementations can be seamlessly interchanged

without modifications to other parts of the code.

The modelframe module is a major component to ensure escriptRT’s extensibility

and re-usability (see Gross et al., 2008). Each Model derived class can be easily

extended with new features at the appropriate point in the workflow thanks to all

available methods. The code produced is also easily reusable as each Model derived

class can be used as is in a different simulation. Involving different process models


then only becomes a matter of plugging together such classes to build simulations.

Coupling the existing code with other process models also becomes quite easy as

the only step required is to implement (possibly wrap) the new process models in

new Model derived classes. The modelframe module serves as the basis of the next

two higher-level modules escriptBaseRT and escriptFluidHeatChem.

3.4.4 EscriptBaseRT layer

escriptBaseRT module is a collection of utilities and basic classes which deals with

the numerical aspects of the solvers involved. It contains base Model classes to

map the top level classes to the corresponding numerical schemes. The advection-

diffusion equations dealing with the transport of heat and solutes can for example

use either the traditional backward Euler or Crank-Nicholson schemes. The calcula-

tion of material and fluid properties can be accelerated via the interpolateTable

functionality of escript. In short, escriptBaseRT encapsulates all external pack-

ages like GibbsLib to provide the required level of abstraction for escriptRT by

allowing the top level module to focus as much as possible on the geophysical and

geochemical concepts.

3.4.5 PmdPyGC

PmdPyGC is a generic package supporting a wide range of functionalities for geo-

chemical simulation. It plays four essential roles: (i) parsing chemical system def-

inition files with their associated thermodynamic database (see Section 3.3.2), (ii)

creating and updating chemical systems, (iii) updating their current state and (iv)

applying a prescribed external chemical solver to set the system into equilibrium.

The package is written in C++ and contains hierarchically ordered classes pertain-

ing to the building components of the chemical system, such as elements, charges

and reactants, and to quantitative aspects of the chemical state, such as masses,


volumes, concentrations, phases and reactions, corresponding to the current temper-

ature and pressure. Various interfaces are supported to provide access to wide range

of applicable fluid EOS modules and chemical solvers. The current chemical solver

of usage is GibbsLib (see Section 3.3.2); via dynamical interaction of PmdPyGC

with its Windows dll, updates are provided for the chemical state at equilibrium,

including produced aqueous and mineral species and related phase transitions. The

access of escriptRT to PmdPyGC is done using the boost library.

The fluid EOS modules presented in Section 3.3.3 are implemented in escriptRT

in three different manners, as illustrated on Figure 6. For example they involve

fortran to C translation using f2c (Feldman et al., 1993), so they can be used

directly by escript. This is much more efficient for example than another pos-

sibility which would consist of translating fortran packages directly to python

using f2py (Peterson, 2009). As mentioned previously, python is mainly used as

a binding language to connect components while the numerics are more efficiently

implemented using lower-level languages (like C in escript). The final option

for access to the EOS of water is to save tabulated values in an ASCII file for a

given regular grid of temperatures and pressures. These file can then be loaded in

escript and used to interpolate the corresponding fluid properties for each mesh

point very efficiently.

3.4.6 EscriptFluidHeatChem structure

The escriptFluidHeatChem module is the top level interface to the whole pack-

age and represents the conceptual level of the CMN pattern. Its purpose is to

provide easy access to numerical modelling to geologists without much numerical

experience but also to expert users willing to develop new constitutive models.

escriptFluidHeatChem provides a clear interface for geologists to describe their geo-

physical/geochemical problem at a conceptual level and run numerical simulations

with or without any numerical knowledge of the underlying implementations.


A simulation can be defined through this interface as a sequence of time slices, where

each time slice defines a period of time for a given set of parameters, including each

process model, constitutive model, stopping conditions, boundary conditions or

numerical parameters. Each time slice is defined using a large python dictionary

(mapping object) on multiple levels and allows users to set each variable using

values or predefined keywords to chose between all available methods. An extensive

check is run before launching simulations to ensure that all initial parameters have

been set up properly for all time slices. This avoids therefore any bad surprise

after several days of simulation. This top level module has also allowed an easy

integration of escriptRT within a graphical user interface, which allows geologists

and geochemists to use it without any specific programming knowledge.

3.5 escriptRT and hydrothermal gold systems

The escriptRT code was written to allow the simulation of complex coupled heat-

mass transport and chemical reaction problems associated with understanding pro-

cesses important for the formation of ore deposits. This section presents an example

of the application of the escriptRT code with model of transport and precipita-

tion of gold associated with oxidised granites in a granite-greenstone terrane. This

is a conceptual model developed from observations in some Archean hydrothermal

gold deposits in the Eastern Goldfields of the Yilgarn craton, Western Australia.

3.5.1 Physical Conditions

The model is constructed to represent one half of a symmetrical 2D vertical cross-

section of a mafic basin (top) underlain by gently folded mafic-ultramafic sequence

with granitic basement (bottom). Hot oxidised, gold-bearing granite is added into

the granitic basement (see Figure 8). A fault passes through all layers with a

permeability increase of one order of magnitude. Porosity in this fault is calculated

3.5. ESCRIPTRT AND HYDROTHERMAL GOLD SYSTEMS 53

accordingly with the classical Blake-Kozeny equation presented in McCune et al.

(1979). The model is 10 km deep and 12.5 km wide with the top buried to 3 km.

Temperature and pressure are fixed at the top to 100 C and 70 MPa while at

the bottom, pressure is free to vary dynamically throughout the model and fluid is

allowed to pass across this boundary. A fixed heat flux is applied to the bottom

boundary (47.95 mW/m2). This is derived from crustal thermal modelling for

a 35 km thickness Archean crust using material properties from PreMDB. The

model starts with a basal temperature of ∼ 350–370 C because of the variation

in thermal properties of the rocks related to temperature-pressure feedbacks. The

granite is initialised as a 650 C body superimposed on the steady state thermal

gradient, and is then allowed to cool due to advective-conductive heat transport to

the surrounding rocks.

3.5.2 Chemistry

The initial chemistry of the units in the model is defined as a bulk composition by

using volume fractions of the mineral phases and the chemistry of the fluid phase (if

different from pure water). As a starting condition HCh (Shvarov and Bastrakov,

1999) is used to help define the initial chemistry using the chemical composition of

standard Archean rocks. The initial fluid phase is defined as water — 0.5 molar

NaCl in most units except the granite which used H2O — 2MNaCl — 0.5MKCl

— 1MSO2 — 1.14e−4MAu. The Au concentration is set below saturation. The

major mineralogy of the various units are described in Table 5. The first step of

the simulation solves for chemical equilibrium between the model fluid and rock at

each node before any mass transport takes place. The simulation was run until all

transient effects on gold mineralisation were observed and the results are presented

for a total simulation time of 200 ky.


Table 5: Initial major mineralogy of the various units in the model, from top tobottom. Faults are 1 order of magnitude more permeable than the surrounding rocks.

Domain Mineralogy Porosity Log permeability(%) (m2)

Mafic basin Tremolite-amphibole-albite-quartz-magnetite-chlorite

10 −16

Upper Basalt Epidote-hematite-quartz-chlorite-albite

10 −13.7

Ultramafic Antigorite-magnetite-chlorite-diopside-pyrrhotite

10 −16

Lower Basalt Epidote-hematite-quartz-chlorite-albite

10 −15

Granitic Basement Quartz-albite-muscovite-k-feldspar-epidote

2 −17

Granite Albite-K-feldspar-quartz-anhydrite-biotite

5 −15

3.5.3 Results

After 200 ky the model has developed a series of convective cells within the upper

basalt zone (Figures 8a and 8b) as well as a broad thermal anomaly above the

granite (which has cooled to ∼ 450 C ). If we look in detail at the mineralogical

development in the dome above the granite cupola we see the development of a C

and S anomaly shown by calcite, pyrite and pyrrhotite distribution (Figures 9a and

9b). The pyrrhotite, which was stable in the ultramafic unit at the start, forms as

a halo to the pyrite probably related to the fluids becoming more reduced as they

move away from the core of the upflow into the reducing rocks. Phlogopite (Mg-rich

biotite) is also developed in this zone related to the K-rich fluids coming from the

granite (Figure 9b).

Gold is developed within two distinct zones (Figure 10): the upper basalt top con-

tact in the upflow zone, and as a lithology perpendicular reaction front within the

ultramafics but away from the core upflow zone. The upper gold enrichment ap-

pears to be driven by a chemical contrast and the strong lateral flow within the

3.6. CONCLUSION 55

upper basalt unit as a function of enhanced permeability. The reaction front gold

zone is located at a thermal and redox gradient within the ultramafic, although

other mineralogical changes in this zone are subtle at best. As with all reactive

transport modelling, these results can be non intuitive as there is little gold pre-

served within the strong alteration above the granite. This model does not only

present interesting results about the nature of the deposition but it also shows that

for dynamic thermal-flows it is difficult to preserve concentrations of gold within

the core of the system. In addition, there are many subtle but important feedbacks

which could affect the gold trapping process, including porosity evolution.

(a) (b)

Figure 8: Results after 200 ky showing the strong control from the passage of thehot plume above the granite and the strong convection cell development in the upperbasalt unit: (a) Magnetite (total moles); and (b) Temperature distribution. Domainsare from top to bottom: 1) mafic basin, 2) upper basalt, 3) ultramafic, 4) lower basalt,5) granitic basement and 6) granite and 7) fault with enhanced permeability crossingall layers. Arrows represent the Darcy flux with velocities of about 1m/y in the leftpart of the upper basalt layer.

3.6 Conclusion

The motivation behind the development of escriptRT was to provide a powerful

but also user-friendly, flexible, and extensible platform. This allows numerical mod-

elling geologists to focus on the definition of their geophysical/geochemical problem


(a) (b)

Figure 9: Distribution of minerals within the upflow zone inside the ultramafic after200 ky for a subset of the whole region: (a) Pyrrhotite and pyrite (contour, max 2.5m); and (b) Calcite and phlogopite (contour, max 0.6m).

at hand at the constitutive level without necessarily having to become experts at the

same time in applied mathematics and programming. The software architecture of

escriptRT presents some advantages in terms of flexibility (capacity to be adapted

or modified), extensibility (simplicity to add new features) and re-usability (possi-

bility to easily use some components or to couple with other codes), all derived from

the usage of carefully chosen components. A modular object-oriented architecture

was implemented using escript and its modelframe module. python was mainly

used as a high-level glue language to connect components while the numerics were

efficiently implemented using lower-level languages. This chapter also presented an

example that illustrates the complex coupling between thermal response and mass

transfer, as well as the complex mineral assemblages that develop and localise with

the stratigraphy. This simulation emphasises how critical process coupling is when

attempting to simulate large-scale hydrothermal systems.

3.6. CONCLUSION 57

Figure 10: Distribution of gold and fluid flow at the end of the model in a subsetof the whole region. There are two key areas of gold mineralisation : 1) reactionfront gold in the ultramafic and 2) boundary related gold at the upper mafic contact.Temperature contours (C) are added as dashed lines.

Chapter 4

Continuum damage mechanics

4.1 Introduction

The strength of the lithosphere can be affected by various weakening mechanisms

that turn rigid tectonic plates into deforming plates whenever a critical energy

threshold is overcome for the onset of lithospheric failure. In previous contribu-

tions (Regenauer-Lieb et al., 2001, 2006) it was shown that weakening feedbacks

such as viscous dissipation and shear heating can have considerable influences on

the energy thresholds and thereby play key roles on stress and strain localisation

(Hobbs et al., 1990; Sengupta, 2010). Classical mechanical approaches which ignore

such energy balances are struggling to explain the level of forces required to drive

tectonics on our planet. They require forces that are at least four times as large

as those deemed available from slab pull or rigid push estimates (Regenauer-Lieb

et al., 2008). One possible solution to this problem is to consider the time-dependent

strength reduction caused by the feedback of deformation, shear-heating and ex-

ponential temperature dependence of flow laws. This feedback is very efficient for

materials with high activation energy such as olivine and it can lead to a substantial

reduction in lithospheric strength (Braeck and Podladchikov, 2007). However, the

59

60 CHAPTER 4. CONTINUUM DAMAGE MECHANICS

predicted level of forces is still an upper limit (Regenauer-Lieb et al., 2010). Addi-

tional weakening mechanisms through damage mechanics, temperature or fluid flow

must be taken into account.

Damage mechanics is commonly described by (i) intensive variables which ac-

count for the sensitivity of macroscopic elastic properties to distributed local weak-

nesses (ii) damage evolution under external loading. There is a general agreement

(Lemaitre, 1985; Chaboche, 1987; Cocks and Ashby, 1982; Lyakhovsky et al., 1997)

on the first point meaning that damage can be seen as distribution of cracks and

voids at the micro-scale. Consequently, the intensive variables of damage are in-

terpreted geometrically as the ratio of damaged over intact sections (Cocks and

Ashby, 1980) and used to describe the effective stresses in damageable materials

(Lemaitre and Dufailly, 1987). However, damage evolution is still a challenging

subject especially in case of geological materials. The available models in this con-

text dealt with several aspects which involve the thermodynamics of fluid materials

(Bercovici, 1998; Bercovici and Ricard, 2003; Landuyt and Bercovici, 2009; Ricard

et al., 2001), the phenomenological description of weakening (Regenauer-Lieb, 1998;

Hieronymous, 2004), the brittle damage of crustal materials (Lyakhovsky et al.,

1997; Hamiel et al., 2004a,b; Lyakhovsky et al., 2005; Nanjo et al., 2005). These

models cover a wide range of applications and can be used to study seismic events

or wave propagation in geological structures under small perturbations, the brittle

regime of crust deformation, the materials weakening due to thermal and chemi-

cal feedbacks etc. However, these models do not allow elasto-visco-plastic analyses

where reversible and permanent deformations both play crucial roles. Observations

of shear zones deformation at relatively high temperature show that visco-plastic

processes have considerable effects on void nucleation and are dominating the long

term response of geological materials (Fusseis et al., 2009).

This chapter proposes a new theory of continuum damage suitable for plate tec-

tonics which is characterised by large time and length scales. Classifying olivine

as a generalised standard material (Halphen and Nguyen, 1975a), a mathematical

4.2. THERMO-MECHANICAL BACKGROUND 61

model is formulated which describes constitutive behaviour of a damageable litho-

sphere. Although damage is interpreted as an intensive variable which accounts

for the sensitivity of the elastic properties to local weaknesses in accordance with

Lemaitre (1985) and Lyakhovsky et al. (1997), its evolution is described through a

new dissipation potential which is active in the inelastic regime.

The developed constitutive model is integrated using the user material subroutine

UMAT of ABAQUS/Standard (2008). The numerical approach developed in this

chapter is based on relatively new techniques of thermal and rate dependency sug-

gested by Ponthot (1995); Voyiadjis and Abed (2006) and Karrech et al. (2010).

The technique of return mapping developed by Simo and Taylor (1985); Karrech

et al. (2012) is used to guarantee a robust integration of the constitutive model.

The effectiveness of this prediction-correction method is also confirmed by Paulino

and Liu (2001); Kang (2004); Kumar and Nukala (2006). Based on the normal-

ity condition, a consistent tangent modulus is derived by taking into account the

CDM description with a combination of creep mechanisms, water content, rate, and

temperature dependency.

4.2 Thermo-mechanical background

Consider a representative volume element of material which is statistically homo-

geneous with its surrounding such that its averaged properties do not change if its

boundaries are expanded. Employing the notion of sequential equilibrium states,

an additive decomposition of strain increments is used to develop the constitutive

model:

dǫ = dǫe + dǫin (58)

The superscripts e and in denote respectively the elastic and inelastic strains. This

decomposition is valid in case of small deformation of standard materials (Karrech

and Seibi, 2010). The Helmholtz free energy ψ can be used to derive constitutive

relationships. It can be defined as a functional of observable variables (such as


strain, ǫ, and Temperature, T) and the internal variables (such as damage D,

elastic strain, inelastic strain and other dissipation quantities, which are ignored in

this study). Hence, the Helmholtz energy can be expressed in terms of its variables

as follows: ψ(ǫe, T,D). It is worthwhile noting that other forms such as Gibb’s free

energy, enthalpy or internal energy can be deduced in terms of ψ using Legendre’s

transform. In particular, the Legendre transform of ψ with respect to temperature

results in the internal energy:

u(ǫe, s,D) = ψ(ǫe, T,D) + sT (59)

where s denotes the specific entropy. It is the dual of temperature in the sense of

the Legendre transform. Using the time derivative of equation (59) and equation

(176) in appendix A.1, one deduces that:

ρψ + ρT s+ ρsT = σ : ǫ+ r − div(q) (60)

where ρ is the material density, q is the heat flux vector, and σ is Cauchy’s stress

tensor. In accordance with Fourier’s law, the heat flux can be expressed as q =

−kgrad(T ), where k is the thermal conductivity. Equation (60) summarises the

first principle of thermodynamics in its local form. It shows that the local internal

energy is equal to the internal work augmented with the local heat production and

transfer. Combining equation (60) with equation (179) in appendix A.2 results in

the following inequality:

σ : ǫ− ρψ − ρsT −q

T.grad(T ) ≥ 0 (61)

The above equation contains the fundamental second principle of thermodynamics

stating that the rate of entropy production is higher or equal to the heat supply

over temperature. Equation (61) represents the local rate of irreversible entropy

production as discussed in details by Regenauer-Lieb et al. (2010). Using the addi-

tive decomposition of strain (58) as well as the derivative of Helmholtz free energy

4.2. THERMO-MECHANICAL BACKGROUND 63

with respect to time1: ψ = ∂ǫeψ : ǫ+ ∂TψT + ∂DψD, equation (61) reduces to:

σ : ǫin +

(

σ − ρ∂ψ

∂ǫe

)

: ǫe − ρ

(

s+∂ψ

∂T

)

T − ρ∂ψe

∂DD −

q

T.grad(T ) ≥ 0 (62)

This expression, which invokes the positivity of the irreversible entropy production

rate, is often called Clausius-Duhem inequality. In the above derivations, the stress

is considered to be the same across the elastic and viscoplastic regimes. Applying

the postulate of Coleman and Noll (1963) by considering that equation (62) must

hold for every admissible process, leads to the following relationships:

σ = ρ ∂ψ∂ǫe

(a) and s = − ∂ψ∂T

(b) (63)

Combining equations (62) and (63) results in the following local inequality, which

states that the rate of energy dissipation D is always positive:

D = σ : ˙ǫin − ρ∂ψe

∂DD −

q

T.grad(T ) ≥ 0 (64)

The thermodynamic force of damage, also known as triaxiality, can be defined as:

Y = −ρ∂Dψ, in analogy with the definitions of stress and entropy. The dissi-

pation D is a scalar product of duals involving the thermodynamic forces, σ, Y ,

T = −grad(T )/T , with their respective fluxes, ˙ǫin, D, q. With these notations,

Clausius-Duhem inequality (64) can be rewritten as:

D = σ : ˙ǫin + Y D + q.T ≥ 0 (65)

This expression includes intrinsic and thermal terms: D = Di + Dt. Since the two

dissipations can take place independently, the postulate of Coleman and Noll (1963)

requires that Di and DT are both positive. If the elastic behaviour is linear and

1∂xψ is the partial derivative of ψ with respect to the variable x


isotropic then equation (63-a) reduces to:

σ = C : ǫe (66)

where C denotes the fourth order elasticity tensor of the damaged material. The

Cauchy stress σ is an homogenised quantity representing the average force on a

given surface. Therefore, the “effective” stress which applies on the actual un-

damaged skeleton can be tracked through the following definition suggested by

Kachanov (1986) and taken into account in the formulations of Lemaitre (1985)

and Lyakhovsky et al. (1997, 2005):

σ =σ

(1−D)(67)

The variable D can be seen as the ratio of damaged over undamaged sections.

Consequently, it can also be understood as an energy portioning variable. When

damage is isotropic, equations (66) and (67) can be used to deduce a relationship

between Cauchy stress, the damage variable D and the elastic properties of the

intact material:

σ = C : ǫe = (1−D)C0 : ǫe (68)

where Cijkl =(

K − 23G)

δijδkl+G (δikδjl + δilδjk) is the fourth order elasticity tensor

of the damaged material, C0ijkl is its equivalent in the undamaged configuration2, K

is the bulk modulus, G is a shear modulus, δij denote the Kronecker symbol, and

the indices (i, j, l, k) represent the directions of a Cartesian space. The selection

of the above form of elastic energy also implies that the thermodynamic force, Y ,

associated to the damage parameter, D, is given by:

Y = −ρ ∂ψ∂D

=1

2ǫe : C0 : ǫe (69)

2The term “undamaged” refers to the material at its original state when no degradation tookplace.

4.3. FROM DISSIPATION TO FLOW RULES 65

From the above definition (69) and following the derivation of Lemaıtre and Chaboche

(2001), it can be shown that Y is given by:

Y =σ2eq

2(1−D)

[

1

3G+

1

K

(

σHσeq

)2]

(70)

where σH = 13tr(σ) is the hydrostatic pressure, and σeq =

√

32s : s is the equivalent

stress. The deviatoric stress is expressed by s = σ − σH1 and 1 is the third order

unity matrix.

4.3 From dissipation to flow rules

In order to derive the flow rules, the existence of a regular potential of intrinsic

dissipation φ(σ, Y ) is assumed. As it is defined within a constant, the dissipation

potential could contain other variables which do not play any role in deriving the

flow rules. This section explains at first the different processes that contribute to

this potential and then uses them to deduce the flow rules.

4.3.1 Visco-plasticity

Plastic behaviour requires the definition of a yield function f(σ) (elasticity enve-

lope) and plastic potential g(σ). In case of associated materials, such as olivine,

these two functions coincide. This study uses the following expression:

f(σ) = g(σ) = σeq − (1−D)σ0 = σeq − σ0 (71)

where σ0 is the yield limit of the intact material which can be taken as a constant

if there is no plastic hardening or dependent on plastic deformation otherwise. σeq

in this case depends only on the second invariant for simplification, however, the

first and third invariants can be included depending on experimental evidence.


As the lithospheric materials generally deform at high temperatures, the viscous

effects are considered in the inelastic regime. In its basic form, viscoplasticity

is commonly introduced through an overstress which was introduced by Perzyna

(1966). Unlike rate-independent plasticity, the equivalent stress, σeq, is no longer

constrained to remain less than the limit σ0. The exceeding quantity is known as

overstress, it is defined as follows:

O =< σeq − σ0 > (72)

where 2 < x >= x + ‖x‖. Therefore the magnitude of the permanent deformation

is proportional to the overstress can be written as:

ǫv =Oη

(73)

where ǫv =√

˙ǫin : ˙ǫin is an equivalent viscoplastic deformation and η is a fluidity

parameter. Notice that the original viscoplasticity introduced by Perzyna (1966) is

linear and athermal. It was later modified by Perzyna1966 himself to include non

linear effects of the form: ǫv = φ(O) =⟨

O

η

⟩m

. By inverting the latter relation-

ship, Ponthot (1995) noticed that φ[−1](ǫv) = ηǫv1/m

= O = σeq − σ0. Hence, he

introduced the new “continuous” condition which can be written as:

f(σ, ǫv) = σeq − σ0 − ηǫv1/m

= f(σ)− φ[−1](ǫv) = 0 (74)

This study considers combined dislocation and diffusion creep mechanisms. There

are well know temperature and water content dependent measurements describ-

ing them for olivine (Goetze, 1978; Mei and Kohlstedt, 2000a,b). In accordance

with Regenauer-Lieb et al. (2001) and Regenauer-Lieb (2006), their combination is

4.3. FROM DISSIPATION TO FLOW RULES 67

expressed as follows:

ǫv = φ(O) = AdO

g3d

exp(

−Qd+σH(Vd−rk∆VOH)

RT

)

+ ApOnexp(

−Qp+σH(Vp−rk∆VOH)

RT

)

(75)

where Aq (q=d,p) are the pre-factors of the diffusion and dislocation mechanisms

respectively, Qq are the activation energies, gd denotes the grain size, σH is the

hydrostatic pressure, ∆VOH is weakening term for increasing pressure (change in

molar volume associated with the incorporation of hydroxyl ions into forsterite), r

and k are fitting parameters of order 1, Vq is an activation volume, and the subscript

q refers to the corresponding creep mechanism. From equation (75), it can be seen

that the different creep mechanisms act in series in accordance with the model of

Regenauer-Lieb et al. (2001). By proceeding the same way as Ponthot (1995), one

obtains

f(σ, ǫv) = f(σ)− φ[−1](ǫv) = 0 (76)

where φ[−1] is the invert of equation (75). For this study the function φ defined

by (75) is more complex than the power law used by Ponthot (1995), therefore the

inversion is performed numerically using the dichotomy method. From equations

(71) and (76) one can obtain a potential of inelastic deformation:

φi(σ, ǫv) = σeq − σ0 − φ[−1](ǫv) (77)

4.3.2 Damage potential

In light of the formulations of Cocks and Ashby (1980, 1982); Chang et al. (1987),

the damage evolution is described by the following potential:

φDe = Y(

1(1−D)n+1 − 1

)

(78)

where n denotes the dislocation law exponent. Unlike the expressions postulated

by Lemaitre (1985) (cf. equation 2.18), Bonora (1997) (cf. equation 20), Chaboche


(1987) (cf. equation 28), the damage evolution in this case is zero in the absence of

initial damage or voids nucleation.

For damage to take place, it is necessary to introduce a second potential which

describes nucleation of voids. This study proceeds similarly to Dhar et al. (1996)

and describes damage nucleation by

φDn = H

κ+1(YH)κ+1 (79)

where H and κ are material constants. An additive form of equations (78) and (79)

is used to describe the dissipative potential of damage:

φD = Y(

1(1−D)n+1 − 1

)

+ H

κ+1(YH)κ+1 (80)

4.3.3 Flow rules

In accordance with Bonora (1997), the dissipation potential includes an additive de-

composition of inelastic deformation (77) and damage (80) potentials. Applying the

principle of maximum dissipation, it was shown (see equation (184) in appendix A.3)

that the flow rules read:

˙ǫin = λ ∂g∂σ

= λ32

s

σeq

D = λ(

1(1−D)n+1 − 1 + (Y

H)κ) (81)

where s is the deviatoric stress. From the above equation, the equivalent inelastic

deformation is expressed as follows: ǫv =√

˙ǫin : ˙ǫin = λ.

4.4 Finite element implementation

In this section, the mathematical model derived in the last section is implemented

numerically. The algorithm used to integrate the non-linear part of the developed

4.4. FINITE ELEMENT IMPLEMENTATION 69

material behaviour is based on an implicit scheme where the main steps can be

summarised as follows: (i) the flow rules (of inelastic deformation and damage) are

calculated based on a trial stress prediction (ii); the optimisation problem is solved

using Newton’s method, where in each increment a Lagrange multiplier is calculated

and used to correct the trial stress, the damage parameter, as well as the hardening

parameters until convergence is achieved (iii); the Lagrange multiplier is then used

to calculate the inelastic deformation, evaluate the damage parameter and correct

the stress field. (iv) Once all parameters that can be used in the next step are

known, the consistent tangent modulus is calculated. These steps are performed by

taking into account the effect of degradation growth, temperature, rate dependency

and water content. The incremental form used in the numerical method is detailed

in the following paragraph.

4.4.1 Prediction-correction algorithm

The integration method consists of starting from a converged configuration to pre-

dict a new one (Simo and Taylor, 1985). This assumes that the material is still

linear elastic, hence the solution can be estimated by using Hooke’s law (68), and

the prescribed deformation increment, ∆ǫ, to calculate a trial stress tensor as fol-

lows:

σTij = (σij)n + Cijkl∆ǫkl (82)

A first correction step is required if the trial equivalent stress is outside the elasticity

envelope. It is worthwhile noting that one of the main features of a non-linear

numerical constitutive model is the correction step, which can be written as follows:

(σij)n+1 = (σij)n + Cijkl(

∆ǫkl −∆ǫinkl)

(83)

where σTij is the trial stress and (σij)n+1 is the corrected stress, Cijkl is the elasticity

tensor, ǫinkl denotes the inelastic strain tensor. If a damage increment takes place,


an eventual second correction can be expressed as follows:

(σij)n+1 = (σij)n + Cijkl(


−∆DC0ijklǫ

nkl (84)

where ǫn denotes the strain at the previous converged configuration. Using Hooke’s

law (68), it can be seen that:

(σij)n+1 = (1− ǫD)(σij)n + Cijkl(


(85)

where ǫD = ∆D/(1−D). By neglecting ǫD, the above equation can also be approx-

imated by:

(σij)n+1 = σTij − Cijkl∆ǫinkl (86)

The correction can be performed only if the viscoplastic strain increment is known.

The necessary steps leading to this quantity are detailed in the following paragraph.

4.4.2 Consistency factor

The algorithm described herein uses Newton’s method to calculate the consis-

tency factor. The procedure is based on the enforcement of the persistent yield-

ing condition. It also requires a time integration of the flow rule (81) as follows:

∆ǫinij = ∆λ∂σijg. f is firstly expanded in Taylor series to the first order as follows.

f(

σTij − Cijkl∆ǫinkl

)

= f(

σTij)

−∆λ∂f

∂σijCijkl

∂g

∂σkl(87)

The arguments of f are obtained from equation (86). As it is an explicit function

of ∆λ, the obtained expression is denoted in a symbolic form Γ(∆λ) = f(

σTij)

−∆λ∂σij fCijkl∂σijg. Expanding Γ in a Taylor series to the first order results in the

following expression:

Γ(∆λ+ δ(∆λ)) = Γ(∆λ) + δ(∆λ)∂Γ(∆λ)

∂(∆λ)(88)

4.4. FINITE ELEMENT IMPLEMENTATION 71

The solution method requires that Γ(∆λ + δ(∆λ)) = 0. Hence substitution for Γ

from equation (87) into equation (88) leads to:

δ(∆λ) =f(σn+1

ij )− φ[−1](ǫineq)−∆λ ∂f∂σkl

Cijkl∂g∂σkl

∂f∂σkl

Cijkl∂g∂σkl

(89)

Using Newton method, an iterative correction of the consistency factor can be

performed. At each iteration a new factor is evaluated as follows ∆λnew = ∆λold +

δ(∆λ) until f(

σTij)

tends to zero.

4.4.3 Consistent tangent modulus

The tangent modulus is a fourth order tensor which relates the stress increment to

the total strain increment. When the stress state is within the limit of elasticity,

this modulus coincides with the elastic tensor Cijkl defined in the second section. In

the opposite case, additional terms have to be included to account for the relative

softening due to the non-linear behaviour. Proceeding in the same way as to obtain

the consistency factor, equation (86) can be symbolically written as:

σij(∆ǫinkl + δ∆ǫinkl) = (σij)n + Cijkl

(

∆ǫkl −∆λ∂g

∂σij

)

−∆λCijkl∂2g

∂σij∂σkl∆σij (90)

Rearranging the terms containing ∆σij = σij−(σij)n, equation (90) can be rewritten

as:(

C−1ijkl +∆λ

∂2g

∂σij∂σkl

)

∆σij =

(

∆ǫkl −∆λ∂g

∂σij

)

(91)

In order to obtain an incremental relationship between stress and strain, the fourth

order effective modulus Dijkl is firstly calculated as Dijkl =(

C−1ijkl +∆λ ∂2g

∂σij∂σkl

)−1

and in turn the multiplier ∆λ. The second derivative of g with respect to the stress

tensor can be written as follows:

∂2g

∂σij∂σkl=

3

2σeq

(

δikδjl + δilδjk2

− δijδkl3

− 3

2

sij sklσ2eq

)

(92)


Therefore, the effective modulus is given by:

Dijkl =

(

C−1ijkl +

3

2

∆λ

σeq

(


− δijδkl3

− 3

2

sij sklσ2eq

))−1

(93)

Substituting for the compliance C−1ijkl expressed in terms of bulk and shear modules

in the above equation and rearranging the different terms results in:

Dijkl =

(

E−1ijkl −

∆λ

σeqvijvkl

)−1

(94)

where vij =32

sijσeq

and

E−1ijkl = −

(

1

2G+

3

2

∆λ

σeq− 1

3K

)

δijδkl3

+

(

1

2G+

3

2

∆λ

σeq

)


(95)

Using the notation Gγ = G/(1 + 3∆λGσeq

), the above expression can be simplified

further:

E−1ijkl = −

(

1

2Gγ

− 1

3K

)

δijδkl3

+1

2Gγ


(96)

Inversion of this tensor results in the following expression:

Eijkl =

(

K − 2

3Gγ

)

δijδkl + 2Gγδikδjl + δilδjk

2(97)

Applying Sherman-Morisson formula, equation (94) can be rewritten as:

Dijkl = Eijkl −vijEijklEijklvklvijEijklvkl − σeq

∆λ

(98)

Substituting for vij and Eijkl in equation (98) leads to the following expression:

Dijkl =

(

K − 2

3Gγ

)

δijδkl + 2Gγδikδjl + δilδjk

2+ β

sijσeq

sklσeq

(99)

4.5. NUMERICAL APPLICATION 73

where β = − (3Gγ)2

3Gγ−σeq∆λ

. The consistency factor ∆λ in equation (91) can be obtained

by applying the consistency condition:

df =∂f

∂σijdσij +

∂f

∂σ0dσ0 +

∂f

∂λdλ = 0 (100)

Hence, the incremental relationship between stress and total strain can be expressed

as:

d∆(σij)n+1 =

(

Dijkl −BijB

∗kl

H∗

)

d∆ǫkl (101)

where Bkl = Dijkl∂f∂σij

, B∗kl = Dijkl

∂g∂σij

, and H∗ = ∂f∂σij

Dijkl∂g∂σij

+ 1∆t

∂φ[−1]

∂ǫineq. Using

equation (99), the terms of equation (101) representing the consistent operator can

be deduced:

Bij = Dijkl∂f

∂σkl= (3Gγ + β)

sijσeq

(102)

B∗kl = Dijkl

∂g

∂σij= (3Gγ + β)

sklσeq

(103)

∂f

∂σijDijkl

∂g

∂σkl= (3Gγ + β) (104)

In the following section, this incremental technique will be applied on a particular

case study in order to verify its applicability.

4.5 Numerical application

4.5.1 Comparative study of damage and heat necking

The model is applied to study the weakening and localisation of an idealised litho-

sphere layer subjected to different loading rates. This application aims to point

out the effects of intrinsic, thermal and damage dissipation on localisation. As a

first step, the viscoplastic model is compared to the results of Regenauer-Lieb and

Yuen (1998) on numerical simulation of a notched lithospheric layer under regular

extension (see figure 11). The model is defined by a 100 km deep and 800 km long


plane strain cross-section, with a 4 km square groove in the middle of the upper

edge in order to simulate a perturbation or a material defect. A biased mesh which

is refined in the neighbourhood of the notch was used to avoid any numerical dis-

crepancy in the shear localisation zone. The boundary conditions consist of a free

slip on the top and right edges, a free displacement on the bottom side, and an

extension velocity of 20 mm/yr at the right hand side of the lithospheric layer. In

terms of thermal conditions, the cross section is subject to an initial temperature of

978o K throughout the whole domain. As the problem includes thermo-mechanical

coupling, the elements used are isoparametric elements of type CPE4T and size of

about 250m. The material behaviour, summarised in tables 6 and 7, was considered

elastic-visco-plastic where thermo-mechanical coupling, inelastic energy feedback,

effect of temperature and water content were taken into account as described in the

above sections.

Mass Density ρ(Kgm−3) 3300Elasticity Young modulus, E(GPa) 10

Poisson’s ratio, ν 0.3Thermo-mechanics Thermal conductivity, k(Wm−1K−1) 3.4

Specific heat, cp(JKg−1K−1) 1240

Expansion coefficient, α(K−1) 1.2× 10−5

Table 6: Simulation Parameters: Thermo-elasticity

Thermal feedback Inelastic heat fraction, ξ 0.9Creep Mecanisms Universal gas constant, R(Jmol−1K−1) 8.3144

Weakening term, rk∆VOH(m3mol−1) 10.6× 10−6

Activation Volume Power (m3mol−1) 2.0× 10−5

Mechanism’s Exponent, n∗q np = 3 nd = 1

Prefactor Aq(µm3∗MPa−ns−1K−1) 1.5× 103 4.8× 104

Activation energy Qq(kJmol−1) 470.0 295.0

Activation energy Aq(Jmol−1K−1) 8.3144

Grain size gd(µm) 15Damage paramaters Critical damage, Dcr 0.85

Normalising term, H(MPa) 6Triaxiality exponent, κ 2Critical inekastic deformation, ǫ0 0.01

Table 7: Simulation Parameters: Dissipation constants (*the subscripts q=p,d referto power law and diffusion respectively. The same order is valid for the following tworows. ** µm3 is included only if q=d)


Figure 11: Finite element modelling of a lithospheric layer containing a notch (seezoom)

Effect of temperature

In case of undamaged viscoplastic behaviour, the numerical results are similar to

those obtained by Regenauer-Lieb and Yuen (1998). Two shear bands take place

at the notch and propagate progressively in a 45o inclination with respect to the

horizontal, through the lithospheric cross section. Figure 12-a shows the equivalent

inelastic strain that takes place before t = 1 Myrs. It can be seen that regular peak

lines of high deformation with a maximum magnitude of ǫineq = 0.25 are concentrated

in the neighbourhood of the notch. The magnitude decreases sharply by about 75%,

4 km away from the centre of the notch and becomes insignificant in magnitude

outside this region. Figure 12-a also shows the initiation of propagating inelastic

strain along the shear bands. At t = 3Myrs, the propagation continues and a

high cumulative inelastic deformation reaches most of the shear bands as can be

shown in Figure 12-c. However, the peak of maximum equivalent deformation,

ǫineq = 0.5 is still in the neighbourhood of the notch centre. A sharp decrease of

about 75% in magnitude can still be noticed around 4 km away from the centre of

the notch. On the top of the propagating inelastic strain, Figure 12-d shows the

necking behaviour on both sides of the lithospheric layer. It can be seen that in

general the flow is towards the region of maximum deformation except at the notch


itself where a high thermal expansion takes place and allows the material diffusion

to be accelerated. The model also shows a temperature increase at the shear bands

with contours of the same patterns as those obtained in case of inelastic strain.

Figure 14-a shows that at t = 1 Myrs the temperature increases up to 980o K with

a peak in the neighbourhood of the notch. This temperature variation represents

the conversion of dissipated energy into heat and the acceleration of dissipation due

to the increase of temperature. Figure 14-c shows the temperature distribution at

t = 3 Myrs. Note, as expected, that the temperature reaches a much higher peak

magnitude but with more uniform gradient along the shear bands.

(a) (b)

(c) (d)

Figure 12: Equivalent inelastic deformation of the rate-dependent undamaged (aand c) and damaged (b and d) lithospheric layer after 1 Myrs (a and b) and 3 Myrs(c and d)

Damage weakening

This section shows that continuum damage accelerates material weakening and

affects the magnitude of inelastic deformations in shear zones. Figures 13 show

the contours of damage distribution with respect to time. They also show that

damage initiates in the neighbourhood of the notch and diffuses along the shear

zones. It can be seen that damage varies from 0 to 0.85 as maximum critical value.


Theoretically, damage must always be smaller that unity (total degradation); in this

case, a critical value of the order 0.85 is considered as shear zones in geology can still

have a certain strength after material degradation. Comparison between Figures

12-a and 12-b shows that damage increases the equivalent inelastic deformation

by around 20% at the beginning of the loading process. At later stage and as

damage increases, this effect accelerates. Figures 12-c and 12-d shows that inelastic

deformation is three times higher when the geological structure is damaged. This

increase of inelastic strain is explained by the effect of damage on the strength of

materials which results in increase of the total deformation. In addition, the increase

of the damage parameter D results in a shrinkage of the yield surface, which in turn

increases the inelastic deformation. Figures 14-b and 14-d show the distributions

of temperature at t = 1 Myrs and t = 3 Myrs, respectively. The contours show

that the pattern is similar to the one obtained in the case of undamaged structure.

However, the overall differential magnitude in this case is much lower, as expected.

It varies from about 7 degrees down to 4 degrees when the structure is damageable.

This decrease of temperature is due to the decrease of heat generation related to

the higher thermal feedback of the inelastic dissipation.

(a) (b)

Figure 13: Damage distribution after (a) 1 Myrs (b) 3 Myrs

The former figures also show that the necking process is accelerated by damage since

the lithospheric shape undergoes higher material flow when it is subjected to severe

degradation than when it is continuously yielding. This result can be interpreted

qualitatively by comparing the undamaged and damaged deformed shapes at the

bottom of the plate in all the above mentioned figures. It is worthwhile noticing that


(a) (b)

(c) (d)

Figure 14: Shear heating effect due to inelastic deformation of the undamaged (aand c) and damaged (b and d) lithospheric layer after 1 Myrs (a and b) and 3 Myrs(c and d)

all the deformed shapes are presented at a scale factor of 1 and are not enhanced.

It is also important to mention that the obtained shear bands width is independent

of the element size as shown in figure 15.

4.5.2 Energy partitioning

Figure 16 shows that the variation of elastic energy with respect to deformation is

quadratic before yielding, decreases continuously if damage takes place and main-

tains a plateau-like shape if no damage occurs. The decrease of the elastic energy is

attributed to the increase of the damage parameter which results in partial conver-

sion of the elastic energy into damage, whereas the plateau-like behaviour is due to

the non-hardening effect of the material. The figure also shows that the viscoplastic

energy increases linearly if the structure is undamaged. Again, this response can

be explained by the non-hardening of the material and the constancy of the applied

velocity. Figure 16 also shows that the viscoplastic energy is higher if the structure

is undamaged; the result is in accordance with the heat generation shown in figure


Figure 15: Independence of the shear band width on the size of elements

14. Therefore, it can be concluded that if damage is not taken into account the

necessary forces to deform lithospheric structures are overestimated.

(a) (b)

Figure 16: Elastic and viscoplastic energies for (a) undamaged and (b) damagedstructures.

4.5.3 Effect of loading rate on the structural integrity

Depending on the geological events lithospheric layers can undergo different levels

of loading rates. In this paragraph the above described model was slightly modified

as follows: (i) since the problem is symmetric, only the right hand side of it is

considered for simulation. (ii) The rectangular notch is replaced by a circular notch


of radius 25 km to obtain more pronounced weakening. The rest of boundary and

initial conditions are all similar. Figure 17 shows the response of the new structure

under different loading velocities which are multiples of 0.63 mm/yr , in both cases

when damage is and is not taken into account. In the case of non-damaged material,

it can be seen that the responses are almost indiscernible for small loading rates;

however, the viscous effects are amplified gradually with larger loading rates. This

results in the main features of a rate-dependent behaviour which is characterised by

(1) the expansion of the yield envelope and (2) the increase of the viscous hardening,

as can be shown in Figure 17. These phenomena can be encountered in different

materials and represent the main feedback of viscoplastic behaviour. The first

phenomenon can be explained by the expansion of the yield function (increase of

radius due to the increase of φ[−1](ǫineq)) and the second is related to the increase of

hardening (derivative of φ[−1](ǫineq)).

Figure 17: Materials behaviour depending on viscoplasticity and damage: (a) re-sponse of the un-damaged structure (b) response of the damaged structure, underdifferent loading velocities (mm/yr).

A comparison between fig. 17-a and fig. 17-b also highlights the importance of

damage mechanics in the modelisation of brittle and ductile behaviours. The first

subfigure is characteristic of ductile behaviours for all rates of loading, whereas the

second one displays brittle responses for small loading rates and ductile responses

for higher loading rates. Loading rate provides a switching mechanism between

brittle and ductile mechanisms, allowing a smooth transition from one to the other.

4.6. CONCLUSION 81

4.6 Conclusion

A numerical model was suggested to study the geodynamics of a notched cross-

sectional lithospheric layer using finite element. It takes into account thermo-

mechanical coupling, continuum damage mechanics and viscoplastic rheology with

multiple creep mechanisms. The formulation was implemented using relatively new

integration techniques. The approach took into account persistent viscoplastic

yielding equality introduced by Ponthot (1995) and Carosio et al. (2000) as well

as the thermal and athermal effects on yielding as described by Voyiadjis and Abed

(2006) who noted that temperature and strain rates have small effects on harden-

ing curves, but mainly contribute to the change of yielding points. The numerical

approach also involved the predictor-corrector algorithm of return mapping and

Newton’s method which are characterised by high rates of convergence. The re-

sults in case of rate-dependent non damaging materials showed a good agreement

with those obtained by Regenauer-Lieb and Yuen (1998); they confirmed that shear

heating produces weakening. A comparative study showed that continuum dam-

age contributes significantly to the material softening and reduces the previously

estimated reaction forces in the lithosphere. Damage also plays the role of a loading-

rate-controlled switching mechanism between brittle and ductile responses. It was

also shown that continuum damage can reduce considerably the strength of the

lithosphere, especially at low loading rates. Plate tectonic deformation may indeed

be lubricated by damage mechanics.

Chapter 5

Thermodynamic framework

5.1 Introduction

Thermodynamics provides a unified framework to couple mechanics and chemistry,

with practical numerical applications for realistic geodynamics problems. The work

presented in this chapter is developed within the framework of classical thermody-

namics, where the effect of chemical feedbacks are included by taking into account

the fluxes exchanged by a Representative Volume Element with its surrounding.

Earlier suggestions are followed using full explicit coupling, only to find that they

are numerically intractable for realistic geodynamics problems with a number of de-

grees of freedom potentially exceeding one hundred. As a method of reducing this

number a multi-scale approach is employed where the given scale of interest allows a

separation of direct and indirect feedbacks. Indirect feedbacks are not solved in the

system of equations but are incorporated through pre-calculated thermodynamic

databases. Direct feedbacks are calculated in the framework of thermodynamic

equations and solved explicitly to define the dissipative structures emerging out of

those feedbacks. Thus a framework is proposed that can be used to extend in a

computationally manageable manner the linear far-from-equilibrium theory from

Prigogine and co-workers into the non-linear regime for thermo-chemo-mechanical

83

84 CHAPTER 5. THERMODYNAMIC FRAMEWORK

problems.

The emergence of localisation phenomena is a key to understand problems rang-

ing from small scale mechanical behaviour to the large behaviour of lithospheric

plates. Those problems can be understood in a thermodynamical sense as dissi-

pative structures (Prigogine and Lefever, 1968). This term describes the patterns

which self-organise in far-from-equilibrium dissipative systems such as the chem-

ical oscillations shown in fig. 18. Application of this theory to mechanics is not

novel but since its introduction by Prigogine it has been frequently regarded as in-

herently too difficult in the geomechanical community because of two fundamental

problems. First, explicit calculation of thermodynamical fluxes requires very large

computational engines that were not commonly available until recently. Second,

and more fundamentally it was perceived that thermodynamics can not uniquely

define a mechanical problem. The reason for this concern is that knowing all

macroscopic variables one can only determine the material parameters in the con-

stitutive laws if there are fewer parameters than observable constitutive relations.

However it was pointed out by Kocks et al. (1975) that in crystal plasticity the

number of relevant material parameters may far exceed the number of macroscopi-

cally observable relations; the use of energy, volume, etc., as “state variables” then

becomes meaningless. The focus of this text is on the first problem by develop-

ing a thermodynamical framework that allows computational solutions for coupled

thermo-chemo-mechanical problems within reasonable computing time. Moreover

this framework is self-consistent as the heat and chemical feedbacks are tracked ex-

plicitly. The approach presented also addresses the second point of concern, where

the dimensionality of the mechanical problem is reduced into a numerically tractable

unique solution through a method of averaging variables (Rice, 1971). This allows

to extend far-from-equilibrium thermodynamic theories such as Prigogine’s theory

into the non-linear regime, with simplifications that make it possible for solutions

to be practical from the geological outcrop to the large geodynamics scale.

5.2. LITERATURE REVIEW 85

Figure 18: Liesegang patterns in a sandstone showing diffusion and chemical oscil-lations. Those patterns can be explained by auto-catalytic chemical reactions. Photocourtesy of Ron Vernon.

5.2 Literature review

Classical solid mechanics was initially formalised on mathematical concepts for de-

scribing the different relationships between actions and motion. The introduction of

thermodynamics offered a rigorous framework in which mechanical models can be

developed at least under the assumption of thermodynamic equilibrium. However,

this assumption is particularly deficient in describing dynamical processes that are

inherently time dependent as the heat is generated, conducted and dissipated in

the neighbouring domains (Baker, 2005). The notion of quasi-static deformation is

incompatible with the inherent time dependency introduced by the energy equation

and is an extreme end-member valid for a quasi-steady state. This incompatibil-

ity is mainly due to the difference between the primary length scales involved in

the equations of motion, heat transfer, mass transport and the secondary length

scales related to the different rheology mechanisms. Deviating slightly a system out


of thermodynamic equilibrium can result in coupled reactions which may involve

motion, heat transfer, matter transfer, etc. Within this framework of small ther-

modynamic perturbations, thermo-chemico-mechanical processes can be described

in a coupled manner where equations of balance and dissipation can be identified

and used in a closed loop. In the following a brief overview is given of some existing

thermodynamical approaches which tie the disciplines of chemistry and mechanics.

Prigogine and Glansdorff were the first ones to look into the stability of non-

equilibrium stationary thermodynamic states (Glansdorff et al., 1973). They used

a linear stability analysis to relate the rate equations of a chemical system to the

emergence of dissipative structures. The limitation of this approach was that far-

from-equilibrium systems were considered. However they were linear systems char-

acterised by linear partial differential equations. This allowed these authors to

obtain solution from a linear stability analysis and describe the system from the

stability of non-equilibrium states through the assessment of Lyapunov functions.

The predictive power of this method was limited for the particular case where the

ordinary differential equation is linear and temperature and pressure variations were

neglected. Since this assumption is a good first approach to chemical systems the

theory gained wide support for the description of non-equilibrium states that were

called dissipative structures.

The basic underlying principles for the pattern formation seen in fig. 18 go back

to the so-called Brusselator concept introduced by Prigogine and Lefever (1968).

The Brusselator shows how a non-equilibrium system can develop into oscillations

through a self-accelerating feedback process called an auto-catalytic reaction and

how non-linearity can lead to the spontaneous generation of ordered patterns. The

Liesegang rings shown in fig. 18 can then be seen as a natural example of auto-

catalytic chemical reactions in geology (Lebedeva et al., 2004). Many more such

oscillating chemical systems have been identified since then also in other fields such

as bio-chemistry, e.g. the life cycle of the cellular slime mould (Kondepudi and


Prigogine, 1998). The theory can readily be extended into the non linear far-from-

equilibrium regime through explicitly solving the non-linear differential equation

that is encountered in more realistic systems. Kuhl and Schmid (2007) for example

solved a fourth order equation and obtained results very similar to the Brusselator.

This is encouraging to pursue a general development for non-linear systems that

can be solved by numerical methods.

The above described processes did not deal explicitly with deformed solids under

large strain. In fact, deviatoric stresses were neglected in all approaches (Kocks

et al., 1975) . Most classical approaches also make the very strong assumption

of isothermal deformation which ignores the basic underlying physical theory of

thermodynamics and limits these approaches to the classical plasticity theory. The

consideration of thermodynamics in solid bodies that sustain shear and not only

volumetric deformation was pioneered by Ziegler (1977) who coined the term ther-

momechanics. His theory applies to the case of quasi-static deformation and pro-

vides such a thermodynamic consistency check in the strong form of the second

law of thermodynamics. The principal merit of the thermodynamic approach is

indeed to ensure that the mathematical framework suggested by plasticity theory

does not violate the second law of thermodynamics (Halphen and Nguyen, 1975b).

For frictional materials, an extension was suggested by (Collins and Houlsby, 1997)

who define self-consistently the yield criterion and the flow law from the postulate

of a dissipation function (see also (Houlsby and Puzrin, 2007)).

The incorporation of temperature dependency of the material properties into the

criteria for localisation has a long history but has not specifically been tied to

thermodynamics. Explicit calculation of a thermal feedback mechanism was found

by Gruntfest (1963) who analysed the behaviour of temperature sensitive fluids.

Shear heating feedback is based on the mechanics of conversion of deformational

work into heat for cases where a higher temperature lowers the viscosity; a small

fluctuation in temperature then accelerates the strain rate and in turn produces

even more shear heating, leading to a self propagating “thermal runaway”.


In a thermomechanical interpretation of infrared thermography deformation exper-

iments relaxation of the isothermal assumption was proposed by Chrysochoos and

Dupre (1991). A Portevin-Le Chatelier localisation shown in fig. 19 is an illustra-

tion of this non-isothermal localisation feedback. This feedback has been applied

to geoscience applications to explain deep earthquakes (Braeck and Podladchikov,

2007; Hobbs et al., 1986; Ogawa, 1987; Orowan, 1960) and localisation phenomena

in plate tectonics (Kaus et al., 2005; Regenauer-Lieb and Yuen, 1998). In geody-

namics a thermodynamics framework was put forward by Regenauer-Lieb and Yuen

(2003). An illustration of such localisation is shown in fig. 20.

Figure 19: Heat sources (in W.m−3) from infrared camera showing the evolution ofPortevin-Le Chatelier bands in a time series of AlMg alloy in extension. Picture fromLouche et al. (2005).

An extension of the approach to the chemical system with large shear deformation

was recently proposed (Regenauer-Lieb et al., 2009). It differs from Prigogine’s


Figure 20: A stiff and straight layer in a soft matrix is compressed (Hobbs etal., 2009). The lateral dimension after 54% shortening is 9.9 km. The localisationphenomenon from shear heating feedback causes the formation of shear bands andfolding of the layer. Localisation operates on the scale of hundreds of metres to km.The colour legend indicates the shear stress in Pa.


approach by considering shear deformation and non-linear thermodynamics. How-

ever it is still taking the extreme end-member view of isothermal deformation. In

this formulation the chemical diffusion process is formally equivalent to the thermal

diffusion process and the chemical reaction process to the thermal shear heating.

Consequently the shear heating theory can be mapped one to one over into a chem-

ical localisation theory. However an important difference between the two diffusion

processes is that the characteristic length scales are very different for the same

time scale. The thermal diffusion operates at kilometre scales while the granular

chemical diffusion takes place on the sub-centimetre scale (see fig. 21). Another

important difference to note is that geological velocities (with strain rates of the

order of 10−16 to 10−14 s− 1) are on the same order as the chemical diffusion rates,

leading to potential kinematic trapping of chemically reacting species.

A full thermo-chemo-mechanical framework was suggested recently (Rambert et al.,

2007). The approach suggested there is based on a gradient plasticity approach

which inherently considers all possible length scales in a single framework. This is

an elegant solution but is computationally expensive as those gradients are added

as extra degrees of freedom in the numerical solution. This approach has not been

applied to geodynamics to date.

A different approach is presented here that also relies on a non-linear far-from-

equilibrium thermodynamic context but using a multi-scale formulation. This ap-

proach is closer to the one presented in (Coussy, 2004) but presents a novel for-

mulation focusing on the thermodynamical fluxes exchanged by the system at its

surface.

5.3 Multiple scales approach

To describe a thermodynamic system one needs to define three different scales of

interest. The micro-scale can be thought of as the scale of atoms and molecules,

the meso-scale is the next scale up, at least at the level of grains or bigger, and the

5.3. MULTIPLE SCALES APPROACH 91

Figure 21: A stiff and straight layer in a soft matrix is compressed (Regenauer-Liebet al., 2009). The lateral dimension after 54% shortening is 3.7 cm. The localisationphenomenon resulting from chemical diffusion causes folding of the stiff layer. Thelocalisation mechanism operates on the mmcm scale. Contours are showing the shearstress in Pa.


macro-scale is at the level of geodynamic, geo-mechanical problems. A continuum

framework is considered where every material point can at least conceptually be

imagined as a continuum itself. This allows to simulate the essential features of the

true interactions between real physical entities at any given scale within the same

framework despite the extreme difference in nature of the processes considered at

different scales.

5.4 Representative Volume Element

A representative volume element (RVE) Ω is considered of a material specimen un-

dergoing non-uniform inelastic deformation from an arbitrary fixed reference con-

figuration. Its evolution is studied with an Eulerian approach, that is involving

only its current configuration. This RVE is defined as an open domain exchanging

mass, work, momentum, and heat with its surroundings through its surface δΩ. The

choice of a suitable RVE size for a given observation time is the crucial step for the

assumption of continuum RVE in thermodynamic equilibrium, RVE embedded in

a larger system that is not in equilibrium.

Consider two thermodynamics processes happening at two very different times and

length scales. Define t1 as the time to reach local equilibrium in a reference volume

much smaller than the size of the system under study and t2 as the time required to

reach the equilibrium in the entire system, t1 ≪ t2. Following Onsager’s regression

hypothesis (Onsager, 1931) the time evolution of the fluctuation of a given physical

value in an equilibrium system obeys the same laws on average as the change of

the corresponding macroscopic variable in a non-equilibrium system. t1 is chosen

as the smallest time step that will be used in the numerical simulation of the whole

model and derive a corresponding size that defines the arbitrary RVE in local

thermodynamic equilibrium. Note that the time and length scales of this RVE

are large enough for the ergodic hypothesis of thermodynamics to hold, as the

discrete nature of the underlying processes are not considered. The axiom of local

5.5. THERMODYNAMICS BACKGROUND 93

state is then adopted as in the classical theory of non-equilibrium thermodynamics

(de Groot and Mazur, 1962).

5.5 Thermodynamics background

This section defines all classical thermodynamic properties and equations involved

in the framework. T denotes the local absolute temperature and s the specific

entropy of the system (considered per unit mass). The specific internal energy e

can be decomposed as the sum of the internal heat energy Ts and the specific

Helmholtz free energy ψ

e = ψ + T s . (105)

ψ is a thermodynamic potential and must then be a function of a set of independent

state variables. It is a difficult problem to know which variables to use as state

variables and Collins and Houlsby (1997) showed that elastic and plastic strains

can only be taken as state variables when the elastic behaviour at the micro-level

is linear, which is a good approximation in geodynamics for example but not so

good for micro-polar materials (Collins, 2005). At this stage, the smallest scale is

selected in order to avoid micro-polar complexity. For geodynamic modelling the

stress tensor is therefore symmetric.

The local state of a material point is assumed to be characterised by the set

(T, ǫe, D, αk) of independent variables, where ǫe = (ǫeij)1≤i,j≤3 is the elastic part

of the local total strain tensor ǫ relative to a fixed reference configuration, D is a

damage parameter and αk1≤k≤n represent a set of n local internal state variables

considered in the reference configuration. The damage formulation chosen follows

the original thermodynamic framework from Lemaitre (1985) and Chaboche (1987).

An extension to geodynamic applications where multiple mechanism of void growth

and nucleation was suggested by Karrech et al. (2011c). At the arbitrary but macro-

scopic scale of the RVE, those state variables are averaging variables (Rice, 1971)


derived from the underlying micro-level for all processes considered. The identifica-

tion of those variables depends on the processes targeted and this chapter mainly

focuses on the diffusivity of chemical components and therefore denotes by αk the

number of moles of the kth chemical constituent. The notation for αk can however

be read in a more generic way and can also include other parameters such as those

related to damage mechanics, gravitational or electrical potentials, surface tension,

variation in fluid content, dislocation density, crystallographic preferred orientation,

etc. That choice of variables leads to

ψ = ψ(T, ǫe, D, αk) . (106)

To simplify the notation, the same symbol is used for a function and its value and

the set of independent variables is omitted whenever it is evident or has been defined

previously.

For this RVE the First Law of thermodynamics spells out the energy conservation.

The time rate of change of kinetic energy K and total internal energy E is equal to

the sum of the external power and heat supplied.

dE

dt+dK

dt= Pext +Q (107)

Using the principle of virtual power to relate the internal and external power with

the acceleration power

Pacc = Pext + Pint (108)

and the fact that dKdt

= Pacc the First Law can be rewritten as

dE

dt= −Pint +Q (109)

The heat term can be expressed through a volumetric source term r and an outgoing

5.5. THERMODYNAMICS BACKGROUND 95

heat flow vector q as

Q =

∫

Ω

r dV −∫

δΩ

q.n da =

∫

Ω

r − dev(q) dV (110)

where da is a material surface oriented by the outward unit normal n. The internal

power can be written as the sum of mechanical and chemical terms as defined in

(Nguyen et al., 2007):

Pint = −∫

Ω

σ : ǫ dV −∫

Ω

µkαk dV (111)

where µk represents the chemical potential of the kth chemical constituent. There-

fore the last term in eq. (111) is a sum over all constituents. The term “chemical

potential” must be taken here in a generalised sense as it is not necessarily restricted

to chemistry. Each variable µk is the dual variable of the associated state variable

αk and can also denote a gravitational, electrical potential or surface potential for

example as explained earlier in the definition of the set (αk) in its generalised sense.

In eq. (111) σ = (σij)1≤i,j≤3 represents the elastic stress and is work-conjugate to

the elastic strain ǫ. The usual dot notation is used for the material derivative ǫ =dǫ

dt.

The energy equation can then be written as

dE

dt=

∫

Ω

σ : ǫ dV +

∫

Ω

µkαk dV +

∫

Ω

[

r − div(q)]

dV (112)

It is important to note that the presence of the heat term r in equation eq. (112)

is a consequence of the choice of RVE which is the lowest scale considered for all

equations derived. The mechanisms (e.g. radioactive decay) that account for the

source term on the right hand side of eq. (112) are therefore hidden in the total

internal energy term E without decomposing E in different internal energies (e.g.

for all isotopes involved).

Some useful calculus formulas are mentioned now (see for example (Coussy, 2004))


which are valid for any arbitrary field G, using the velocity of a given particle:

d

dt

∫

Ω

GdV =

∫

Ω

∂G

∂t+ div(Gu)dV (113)

div(Gu) = u.grad(G) +G.div(u) (114)

G =G

dt=∂G

∂t+ u.grad(G) (115)

From the definitions of specific internal energy e, specific entropy s and density ρ,

the total internal energy E, entropy S and mass M get written as

E =

∫

Ω

ρe dV (116)

S =

∫

Ω

ρs dV (117)

M =

∫

Ω

ρ dV (118)

Using eqs. (113) and (115) to (118) and considering the thermodynamical fluxes

crossing the RVE, the balance equations of energy, entropy and mass are written

differently as

dE

dt=

∫

Ω

[ρ+ ρdiv(u)] e dV +

∫

Ω

ρedV +

∫

δΩ

ρke vk.nda (119)

dS

dt=

∫

Ω

[ρ+ ρdiv(u)] s dV +

∫

Ω

ρsdV +

∫

δΩ

ρks vk.nda (120)

dM

dt=

∫

Ω

[ρ+ ρdiv(u)] dV +

∫

δΩ

ρke vk.nda (121)

where ρk and vk represent the density and the entering velocity of the kth chemical

component in Ω through the surface da following an underlying process considered

which does not need to be detailed in this framework. This is a generic way to

consider any process and can conveniently model for example fluid flow in a porous

medium without having to consider a full framework as developed by (Coussy, 1995).

5.6. FROM THE SECOND LAW OF THERMODYNAMICS 97

Equations eqs. (112) and (119) provide the local form of the energy equation as

(

(ρ) + ρdiv(u))

e+ ρe+ div(ρkevk) = σ : ǫ+ µkαk + r − div(q) (122)

Having defined the thermodynamic equilibrium scale, the non-equilibrium processes

may be considered at the large scale.

5.6 From the Second Law of thermodynamics

The Second Law of thermodynamics relates the evolution of entropy with the heat

supplied to the open systemdS

dt≥ Q

T(123)

Deriving eq. (105) with respect to time one obtains the relationship

e = ψ + sT + sT (124)

The second term on the right hand side of eq. (124) describes the rate of entropy

production and the third term the coupled variation of temperature with time.

Feedbacks between the three terms on the right hand side of eq. (124) are at the

source of localisation phenomena as discussed by Regenauer-Lieb and Yuen (2004);

Regenauer-Lieb et al. (2006). For instance the temperature-creep feedback operates

on reducing the yield potential exponentially upon a local temperature perturba-

tion. Those feedbacks express a competition of rates of processes, which happen at

vastly different time and length scales.

From eq. (106) the chain rule yields

ψ =∂ψ

∂ǫeǫe +

∂ψ

∂TT +

∂ψ

∂DD +

∂ψ

∂αkαk (125)

From eq. (123) the dissipation equation can be derived as (see appendix B.1 for


details)

− (ρ+ ρdiv(u))ψ − ρ(ψ + (T )s)− div(ρkψvk)− ρksvk.grad(T )

≥q.grad(T )

T− σ : ǫ− µkαk (126)

By decomposing the total strain in its elastic and plastic components

ǫ = ǫe + ǫp (127)

and by considering specific thermodynamic mechanisms, as described in (Coleman

and Gurtin, 1967) for example, the state equations can be classically identified as

(see appendix B.2 for details)

σ = ρ∂ψ

∂ǫe(128)

s = −∂ψ∂T

(129)

µk = ρ∂ψ

∂αkfor 1 ≤ k ≤ n (130)

The notation Y is defined as

Y = ρ∂ψ

∂D(131)

With those definitions the dissipation equation can be rewritten as

− ρkvk.(grad(ψ) + s grad(T ))− ρY (D)−q.grad(T )

T+ σ : ǫp ≥ 0 (132)

By definition of the specific Helmholtz free energy (1) and by neglecting all second

order terms, one obtains

grad(ψ) =µkρkgrad(αk)− s grad(T ) (133)

As diffusive chemical phenomena only are considered in this chapter, Fick’s law for

the mass transport of each constituent (with no sum intended) can be introduced

5.7. BALANCE EQUATIONS 99

with a diffusive coefficient ωk defined such that:

vk = −ωkµk.grad(αk) (134)

5.7 Balance equations

The main three balance equations which fully define the system are now ready to be

spelt out. The Clausius-Duhem inequality can be derived from eqs. (132) to (134)

and reads

ωkgrad(αk).grad(αk)− ρY D −q.grad(T )

T+ σ : ǫp ≥ 0 (135)

Using Fick’s law eq. (134) allows to rewrite the first balance equation, the equation

of mass conservation eq. (121) in the form of

Mkαk +Mkαkdiv(u)− div

(

Mkωkαkµk

grad(αk)

)

= 0 (136)

WThe simplest form of Fourier’s law is used which relates the heat efflux vector

linearly to the gradient of temperature:

q = −K grad(t) (137)

A simple derivation (see appendix B.3 for details) leads to the second balance

equation: the heat equation

ρCT − div(Kgrad(T )) + Y D − ∂µk∂T

αkT

− ωkgrad(αk).grad(αk)−ρkωkµk

grad(αk)grad(s)T = σ : ǫp + r (138)

where C denotes the specific heat. The last equation required to close the system is

the continuity equation. To derive it, the external power if first defined as coming


from an external volumetric force f and contact force T

Pext =

∫

Ω

f.udV +

∫

δΩ

T .n da (139)

Considering a quasi-static context and neglecting the inertia force leads to the

definition of the continuity equation (see appendix B.4 for details) and its boundary

conditions as

div(σ)− αkgrad(µk) + f = 0 (140)

T = σ.n− µkαkn (141)

The mass conservation eq. (136), energy equation eq. (137) and continuity equation

eq. (140), along with their boundary conditions, fully determine the problem. In

order to apply this framework to specific geophysical processes, the key idea is to

select the appropriate underlying physical processes to describe these fluxes and

analyse the inherent length and time scales. Because the choice of the physical

processes to be incorporated relies on the individual working hypothesis it becomes

immediately apparent that the proposed thermodynamic framework must be data-

driven.

5.8 Numerical application

The general thermodynamic framework described above is applied to a conceptual

model of carbon dioxide degassing which demonstrates the practicality and possi-

bilities of this modelling approach. Several processes of carbon degassing from the

lithosphere are now well recognised (Ciotoli et al., 1999; Mrner and Etiope, 2002;

Zhang et al., 2008) and the focus of this application is specifically on non-volcanic

degassing originating within the upper mantle. This process occurs mainly within

faults and fractures networks. The proposed thermodynamics framework which also

includes continuum damage mechanics is suitable to model these processes since it

takes into consideration the direct diffusive feedback in the energy balance. The


continuum damage mechanics description is based on the framework of Karrech

et al. (2011c) which takes into account the effect of multiple creep mechanisms on

void nucleation and growth.

A literature review shows that a full numerical thermodynamic treatment of cou-

pled chemical-deformation feedback does not exist yet(see also (Rambert et al.,

2007)). Consequently, numerical codes capable of handling this problem have

not yet been benchmarked. It is therefore chosen to adapt a fully benchmarked

solver for the equivalent problem of thermal feedback (Regenauer-Lieb and Yuen,

2004). The “coupled-temperature displacement” analysis of the commercial finite

element code ABAQUS/Standard (2008) is applied to the diffusion problem as well

as the user material subroutine UMAT to integrate the incremental relationships

between stress, deformation and diffused properties with respect to the applied

loads. ABAQUS implicit formulation is used with an elasto-visco-plastic constitu-

tive model with Von Mises yield criterion.

The conceptual model is defined by a vertical cross-section of 16 km width by 6

km depth modelling a sedimentary basin positioned above a carbon dioxide reser-

voir. The model is discretised on a regular mesh with square elements of 100m

and the normalised concentration c of carbon dioxide is tracked, relative to its

value in the reservoir (0 ≤ c ≤ 1). The ideal gas case is considered. The bound-

ary conditions consist of a free horizontal slip on the bottom edge, an extension

velocity of 3.15 mm/year on the right hand side and a fixed left edge in the hori-

zontal direction. The concentration of carbon dioxide is fixed on the bottom edge

following a bell curve centered on the middle of the model with a maximum con-

centration of 1. The temperature is initialised as a geothermal gradient of 30 K/km

with a surface temperature of 300 K. The model comprises of a single sedimen-

tary rock in the whole cross-section containing impurities which are distributed

following the uniform random distribution U(a, b) where a and b are the mini-

mum and maximum values. Random elements are selected in space with indices

Ne = floor [U(0, 1)×Nmax], where Nmax is the number of elements in the model.


The material properties (Young’s modulus and strength) of the selected element Ne

are reduced in accordance with the uniform random distribution U(0.35, 0.7). All

parameters used for the simulation are summarised in table 8.

Table 8: Simulation parameters used with names, units of measure and values.

Mass Density ρ (kg.m−3) 2730.Elasticity Young’s modulus E (GPa) 78.

Poisson’s ratio (−) 0.25Visco-Plasticity Yield strength (MPa) 100.Gravity Gravitational acceleration g (m.s−2) 9.8Creep Viscosity (Pa.s) 1.e19(T − T0)Damage constants, see (Karrech et al., 2011c)

Critical damage Dcr 0.85Damage threshold Dth (equivalent plasticdeformation)

1.e− 2

CO2 diffusivity Diffusion coefficient k0(m2.s−1)

for undamaged material see (Boving andGrathwohl, 2001)

0.0011

Damage multiplying coefficient λ 100.

Many studies can be found in the literature on carbon dioxide diffusion in rocks

(Lai et al., 1976; Penman, 1940). Rock permeability and porosity are the major

parameters controlling the diffusion process and in (Boving and Grathwohl, 2001)

for instance the diffusion coefficient k was described by the following relationship

k = kaqφm (142)

where φ denotes the porosity, m a material dependent fitting parameter ranging

from 1.3 to 2 and kaq the aqueous diffusion coefficient in pure water. By taking

advantage of the implicit relationships between damage and both porosity and tor-

tuosity, a similar description of diffusion is proposed as:

k = k0 + λD (143)

The coefficient k0 represents the diffusion coefficient in an undamaged structure

5.9. RESULTS AND DISCUSSION 103

and λ is a multiplying factor which emphasises the accelerated diffusion process

in damaged rocks. Faults can allow high-flux flow (Sibson, 2000) and a constant

numerical value for λ is arbitrarily chosen in this conceptual example (see table 8).

5.9 Results and discussion

fig. 22 shows a sequence of values for the damage parameter in the model for up to

229,000 years and 4.5% extension. fig. 23 shows the corresponding time frames for

the normalised concentration of carbon dioxide. Damage accelerates the localisation

phenomenon and distinct shear bands are formed that cross the whole model across

its height, enabling therefore carbon dioxide degassing to reach the surface through

the more diffusive damaged areas. This generic model is not calibrated on any

particular geological example but shows clearly how damage can be considered as a

tool to break impermeable seals at depth and create channels for the carbon dioxide

to reach the surface.

This numerical example also demonstrates how the framework presented allows to

simplify the problem by considering thermodynamic fluxes only without having to

resolve the modelling of the underlying physical processes. The same example ap-

plies for both chemical diffusion and Darcian flow by changing the numerical values

for the diffusivity. It is important to note however that the framework described is

nonetheless very general and can be used as a starting point for modellers with a

more specific problem at hand. To study for example fluid flow in a porous medium

one can consider separately the fluid and solid part of the RVE and re-derive

Coussy’s framework (Coussy, 2004).

The goal of this chapter is to provide a framework for linking structural geology

observations to geodynamic modelling. In order to do so one needs to go through

vastly different length scales, beginning with micro structural observations, linking

them to field observations and interpreting the results by the largest hierarchical

driver which is plate tectonics. As in any thermodynamic problem these scales are


Figure 22: Damage parameter after (a) 77,000 years and 1.5% extension, (b)150,000and 3% extension, (c) 229,000 years and 4.5% extension. Damage zones havelocalised which connect the reservoir at the bottom of the model to the surface.


Figure 23: Relative CO2 concentration after (a) 77,000 years and 1.5% extension, (b)150,000and 3% extension, (c) 229,000 years and 4.5% extension. Damage zones havelocalised which increased the CO2 diffusivity and allowed degassing at the surface.


defined by the sum of products of thermodynamic forces and their corresponding

fluxes which identifies the rate of change of entropy production. It is thus possible

to link that rate to the underlying basic physics. Likewise the internal energy can

also be linked to the underlying physics by solving either for the Gibbs free energy

or the Helmholtz free energy. This choice predicts thermodynamic length scales

and time scales. For instance those scales are linked in one-dimensional diffusive

processes by the equation

l ≡ 2√κt (144)

where l is the diffusion length, t the relaxation time and κ the diffusive constant

for the specific process modellers chose to consider. Note that eq. (144) only lists

diffusion length scales as examples and is not meant to be comprehensive. In a gen-

eralised thermodynamic system other metrics will be encountered such as reactive

and convective length scales. In order to illustrate this concept further, these scales

are related to physical processes in geodynamics and structural geology to give the

reader a better understanding of the mesh size and time steps involved. Here are

three examples for an arbitrarily time step of 3e10s ( 1000 years):

• Carbon dioxide diffusion is controlled by Darcy’s law. For the example se-

lected κ = 1.1e−3m2s−1 and then l = 11.5 km.

• Heat conduction in a solid is driven by Fourier’s law. For a granite buried at

10km depth κ = 7.5e−7m2s−1 and then l = 300m.

• Chemical diffusion in a crystal grain follows Fick’s law. For the case of oxygen

in quartz κ = 1.1e−19m2s−1 and then l = 115µm.

For a given geodynamic problem, the formulation can hence be anchored on the

basis of the real underlying physics and the problem can be solved with the full

thermodynamic framework.

The approach by Houlsby and Puzrin (2007) is more theoretical than practical


because they are dealing with a large number of degrees of freedom. In their formu-

lation for porous continua for example they define their system with 57 equations

without considering any chemical reaction. Gradient theory approaches require

even more equations as they consider in addition to the above approach one or

more gradients of the total strain (Rambert et al., 2007), where the first order gra-

dient is a 3rd order tensor and the second order gradient is a fourth order tensor.

These approaches are arguably not practical for realistic simulations with the cur-

rent computing power available. Consequently, simplifications are introduced in

order to fulfil the postulate that present advances in computational power allow

the extension of far-from-equilibrium thermodynamic theories into the non-linear

regime. These simplifications are particularly suited to geodynamic systems with

vastly different length and time-scales.

The first simplification makes use of the inherent multi-scale nature of the prob-

lem. The complexity of the problem lies embedded in the multitude of physical

dissipation processes that control the dimensionality of the thermodynamic system

and hence the emergence of dissipative structures. A multi-scale problem lends

itself to reduction of dimensions of the underlying partial differential equations if a

given pre-set time and length scale are considered. For this pre-set scale, emergent

properties can be derived as spatial and temporal averages. Physical processes that

cannot be derived as average properties and hence need to be considered for the

explicit modelling of dissipative structures can be derived from the rate of change

of entropy production as discussed in eq. (144). A large plate tectonic length scale

of tens of kilometres and million of years time scale implies, for instance, a criti-

cal length scale governed by the thermal diffusion process. At this scale far from

equilibrium solutions to chemical or fluid problems need not be resolved. This is

because their inherent time and length scales are very much shorter (see time scales

for chemical diffusion or fluid flow discussed for eq. (144)). They are expected to

experience small perturbations from equilibrium according to Onsager’s regression

hypothesis. For instance, by assuming for the plate tectonic scale the physics of a


“creeping flow” and by using a flow potential that is normal to the second invariant

of the deviatoric stress tensor, only two critical quantities need to be considered

in the energy equation: the Peclet number describing the diffusion process and the

dissipation number describing the shear heating process (Regenauer-Lieb and Yuen,

2004). For general flows a third number would normally be considered (Cherukuri

and Shawki, 1995) which describes inertial processes.

The second simplification is to differentiate the different feedback processes in terms

of direct and indirect feedbacks. Direct feedbacks are defined as all couplings that

are solved explicitly by the numerical formulation of all partial differential equa-

tions. For example the energy equation allows to take into account shear heating

feedback (Regenauer-Lieb et al., 2006) or chemical processes (Regenauer-Lieb et al.,

2009). Indirect feedbacks are defined all couplings that are introduced by making

material properties dependent on other primary variables without solving for these

dependencies explicitly in the system of equations. However these material proper-

ties are calculated self-consistently from Gibbs minimisation techniques and stored

in a database for interpolation purposes (Siret et al., 2009), as presented in chap-

ter 2. The scale of observations provides the best mechanism for identification of

direct and indirect feedbacks.

Chapter 6

Reactive transport with damage

mechanics

6.1 Introduction

The natural complexity of geological systems has motivated researchers to consider

the processes of thermal transfer (T), hydraulic flow in porous media (H), mechan-

ical deformation (M) and fluid-rock chemical interactions (C) in a coupled manner.

These processes when considered individually rarely explain observed geological

features. Different combinations of those THMC processes have previously been

studied and are based on theoretical frameworks like the one developed by Coussy

(2004). Most THMC applications; however, occur in engineering domains such as

nuclear waste disposal, gas and oil recovery, hot-dry-rock geothermal systems, or

contaminant transport (Lanru and Xiating, 2003), and it is still rare to see THMC

analyses of larger scale geological systems such as those involved in ore body for-

mation (e.g. Lanru and Xiating, 2003; Tsang et al., 2004; Shao and Burlion, 2008).

This observation can be explained by two main reasons. The first one is that cou-

pled numerical simulation of all processes still represents a significant computational

challenge and cannot currently be solved within weeks, the time-scale of a mineral

109

110 CHAPTER 6. REACTIVE TRANSPORT WITH DAMAGE MECHANICS

exploration programmes (Potma et al., 2008). The second reason, which follows,

is that geo-scientists understand the importance of conceptualising their problems

(Andersson and Hudson, 2004) and are often able to identify a subset of relevant

processes which represent the first-order controls in their studies. These restrictions

often disregard the complex feedback interactions between processes, which might

not always be negligible.

There are many different potential feedbacks between all processes (see Lanru and

Xiating, 2003, for example) and the complexity of this set of checks and balances

justifies coupled THMC simulations for various geological scenarios. Numerical

simulations are specifically adapted to understand and isolate particular feedbacks

as they allow scenarios to be investigated methodically under changing conditions,

including the sets of processes considered. They often provide an indispensable

tool to analyse the relative importance of various feedback mechanisms and the

competition of rates of processes. Coupling mechanisms are critical factors for the

localisation of geological structures such as folds or shear zones, which are often

vital to the formation ore bodies. Shear heating in thermo-mechanical simulations,

for example, has helped to understand shear zone formations at the kilometre scale

(Regenauer-Lieb and Yuen, 2004). Additional coupling mechanisms can inhibit or

accentuate those behaviours, and damage mechanics is now a recognised means to

enhance localisation (Regenauer-Lieb, 1998; Bercovici and Ricard, 2003; Karrech

et al., 2011a), as shown in chapter 4.

This chapter presents a new THMC code designed to increase the understanding

of hydrothermal and geothermal geological systems. This coupled code is based

on escriptRT (Poulet et al., 2012a, see chapter 3) for the thermal, hydraulic

and chemical processes, as well as an Abaqus (ABAQUS/Standard, 2008) user

material implementation (Karrech et al., 2011a, see chapter 4) for the mechanical

deformation, including continuum damage mechanics. The chapter is composed of

three parts. Firstly, the constitutive models for the THMC processes are introduced,

including the important feedbacks considered (Section 6.2). These include a link

6.2. CONSTITUTIVE MODEL 111

between the damage parameter and the evolution of porosity, affecting in turn the

rock permeability. The numerical implementation is then presented which details

the coupling between the two codes used, escriptRT and Abaqus (Section 6.3).

Finally, Section 6.4 illustrates the importance of considering all THMC processes

for mineral exploration by simulating a generic unconformity-related albitisation

deposit scenario.

6.2 Constitutive model

The constitutive model presented in this chapter applies to a fully saturated porous

medium, where a fluid phase interacts with the host rock mechanically, energet-

ically and chemically. This model integrates several processes, that are coupled

sequentially. A Representative Volume Element (RVE) Ω of a material specimen is

considered, which contains a solid skeleton and a fluid saturating the porous space,

and undergoes non-uniform inelastic deformation from an arbitrary fixed reference

configuration.

6.2.1 Continuum damage mechanics

The mechanical model used for the skeleton is based on a damaged visco-plasticity

model for frictional geomaterials under the assumption of small deformation, as pre-

sented in chapter 4 (Karrech et al., 2011a). This model is described using a classical

Helmholtz free energy ψ(ǫij,αij, T,D) where ǫ represents the total strain tensor,

α the inelastic strain tensor, T the temperature and D a scalar damage parame-

ter. This free energy function along with the second principle of thermodynamic

results in a Clausius-Duhem inequality which allows to obtain some constitutive

relationships of the form:

σ = ρ∂ψ

∂ǫe, s = −∂ψ

∂T, and Y = ρ

∂ψ

∂D, (145)


where σ the Cauchy stress tensor, s is the entropy, ρ the density, and Y the ther-

modynamic force associated to the damage parameter D. In case of isotropic linear

behaviour, the involved thermodynamic forces can be expressed as (see chapter 4

or (Karrech et al., 2011a) for details):

σ = (1−D)(K − 3

2G)tr(ǫe)1+ 2G(1−D)ǫe , (146)

s = c(T − T0) + 3αK(1−D)tr(ǫe) , (147)

Y =−σ2

eq

2(1−D)2

[

1

3G+

1

K

(

σHσeq

)2]

, (148)

where K and G are respectively the bulk and shear moduli, c is the heat capacity,

α is the thermal expansion coefficient, σH = 13tr(σ) the hydrostatic pressure, and

σeq =√

32s : s the equivalent stress, with s = σ − σH1 representing the deviatoric

stress. This description is based on the definition of effective stress as introduced

by Kachanov (1958) and used extensively by Lemaıtre and Chaboche (2001): (1−D)σ = σ. A custom dissipation function is also postulated (see chapter 4 or Karrech

et al. (2011a) for details) to account for shear dissipation, volumetric change, rate

sensitivity, and damage. Combined with the principle of maximum dissipation

(Ziegler, 1963), this dissipation function is used to relate the thermodynamic forces

to their respective forces (Karrech et al., 2011b). The assumed damage potential

considered herein defines the evolution of the isotropic damage parameter with

time and takes into account void nucleation through a linear relationship involving

strain rate and damage. This potential gy is based on the theory of limit analysis.

It accounts for the nucleation and growth of voids and defects in the dissipative

regime and reads:

gy =

(

1

(1−D)n+1− 1

)

Y +H

κ+ 1(Y

H)κ+1 (149)


where H and κ are material constants used to describe damage nucleation (Karrech

et al., 2011a), and n is the exponent of the dislocation power law1. This potential

is used to calculate the incremental variation of damage with respect to loading

parameters; deriving the potential with respect to the thermodynamics force of

damage Y results in a flow direction which orients the damage evolution:

D = λ

(

1

(1−D)n+1− 1 + (

Y

H)κ)

(150)

where λ is a Lagrange multiplier which is proportional to the equivalent inelastic

deformation. The usage of this recent model is an important component of this

study as (Karrech et al., 2011a, see chapter 4) show that the frictional behaviour

of geomaterials highly influences fault orientations.

6.2.2 Porosity

The evolution of the rock porosity φ is considered from its initial value φ0 due to

mechanical and chemical processes (Kuhl et al., 2004)

φ− φ0 = φem + φpm + φc , (151)

where φem and φpm represent the elastic and plastic partitions of the porosity varia-

tion due to mechanical deformation (Armero, 1999), and φc represents the porosity

evolution due to mineral dissolution and precipitation.

The calculation of φpm, the porosity update due to continuum damage, is based

on an elementary interpretation of damage as spherical void growth in rocks. The

scalar damage parameter D for a given RVE can be interpreted as the proportion of

void surface intersecting the RVE boundary surface over the whole surface (Cocks

and Ashby, 1980). Considering a RVE of characteristic length R containing voids

of radius r lead to the relationship D ∝ ( rR)2. The porosity φ is itself defined as

1The rheology used herein combines dislocation and diffusion mechanisms in series.


the volume ratio of the void compared to the whole RVE, hence φ ∝ ( rR)3. The

relationship φ ∝ D32 can then be deduced. Considering a maximum porosity value

of φmax for a totally damaged element, the following damage-porosity evolution can

then be postulated:

φpm = (φmax − φ0)D32 (152)

6.2.3 Effective stress

All pores are assumed to be fully saturated with an interstitial ideal fluid which

exerts a static pressure p on solid grains. Following Biot’s approach (Biot, 1941)

this pressure term is accounted for to calculate the effective or equivalent stress as

σijeq = σij + bpδij . (153)

b is the Biot coefficient defined in (Coussy, 2004) as

b = 1− K

ks, (154)

where K and ks are respectively the bulk moduli of the empty porous solid and of

the solid matrix forming the solid part of the porous solid2. The elastic part of the

evolution of porosity can be expressed (Coussy, 2004) as

dφem = bdǫ− αφdT +dp

N, (155)

where ǫ represents the volumetric strain, αφ is the thermal expansion coefficient of

the porosity, and N is the Biot modulus defined as

1

N=b− φ0

ks. (156)

2Note that if the solid matrix is incompressible ks becomes infinite and b = 1. Biot’s effectivestress σij

eq then reduces to the classical Terzaghi’s effective stress σij′ of soil mechanics defined by

σij′ = σij + pδij .


6.2.4 Fluid transport

Fluid transport through the porous medium is described using the classical Darcy

law under the assumption of quasi–static fluid flow and neglecting the tortuosity

effect:

q = − k

µf(∇p− ρfb). (157)

where q denotes the Darcy flux vector, k the rock permeability, µf the fluid viscosity

and b the body force. The pore pressure equation is derived from the fluid mass

conservation law linking the fluid density ρf , porosity φ, Darcy flux q and the solid

velocity vs in the reference configuration. Its local form (see for example (Coussy,

2004)) can be written as

d(φρf )

dt+ φρf∇ · vs +∇ · (ρfq) = 0. (158)

The elastic part of the porosity φem is considered to be dependent on pore pressure

p, temperature T and volumetric strain ε, which can be expressed as

dφemdt

=∂φem∂p

dp

dt+∂φem∂T

dT

dt+∂φem∂ε

dε

dt(159)

This relation leads to

d(φρf )

dt= φρf

[

βrdp

dt+ γf

dX

dt− αr

dT

dt

]

+ bρfdε

dt

−φρf∇ · vs + ρf (dφpmdt

+dφcdt

) , (160)

where αr and βr are respectively the thermal expansivity and compressibility of

the porous rock, and γf is the fluid chemical expansivity defined as γf = 1ρf

∂ρf∂X

which depends on the salinity X (wrt CNaCl). The pressure equation can then be


rewritten as

βrφρfdp

dt−∇ · (ρf

k

µf∇p) = −∇ · (ρ2f

k

µfb)− φρf∇ · vs

+αrφρfdT

dt− γfφρf

dX

dt− bρf

dε

dt(161)

The fluid properties (density, viscosity, specific heat, compressibility and thermal

expansivity) are calculated from one of the three equations of state (EOS) for pure

water available in the escriptRT code (Poulet et al., 2012a, see chapter 3). This

choice allows to account for the important variation of fluid properties (especially

density) in a realistic way for low concentrations of chemical species, and with a

first order approximation for salinity.

6.2.5 Heat transport

The solid and liquid components of the medium are assumed to be in thermal

equilibrium. Heat is then transported via the standard advection diffusion equation

(Nield and Bejan, 2006) with an additional shear heating source term

ρ Cp∂T

∂t+ (ρ Cp)fv.∇T −∇.(D ∇T ) = Q+ χσeq ǫdiss , (162)

where ρ Cp = φ(ρ Cp)f + (1− φ)(ρ Cp)s is an average specific heat Cp for the rock,

D = φDf +(1−φ)Ds is the averaged thermal diffusion coefficient, Q is a radiogenic

heat source, ǫdiss is the dissipative strain rate from creep and plastic deformations,

and χ is the nondimensional Taylor-Quinney heat conversion efficiency coefficient.

This study fixes the χ constant in time with a value of 0.9 in agreement with most

material values (Chrysochoos and Belmahjoub, 1992).


6.2.6 Chemistry

In the mass transport, consider the set of chemical species in the aqueous phase

that are transported with the fluid and diffused using Fick’s law. Following (Poulet

et al., 2012a, see chapter 3) geological time scales are considered and those species

are assumed to be in chemical equilibrium with the solid minerals of the host rock

matrix at the end of every discrete time step. In other words, chemical reactions are

assumed to be infinitely faster than the advective and diffusive transport processes

for the geological scenarios considered. The transport problem is then solved firstly

for the chemical elements which constitute these aqueous species:

φ∂Ca

i

∂t+ φu · ∇Ca

i −∇ · (φD∇Cai ) = 0 , (163)

where Cai is the molar concentration of the ith chemical element in aqueous phase,

D is the mass dispersion coefficient tensor, u is the fluid velocity (q = φu) with

respect to the solid matrix.

The chemical equilibrium between the fluid and host rock is then computed sepa-

rately using theGibbsLib solver (viaGibbsLib.dll) from theHCh package (Shvarov

and Bastrakov, 1999). This solver uses a Gibbs free energy minimisation technique

at prescribed temperature and pore pressure.

6.2.7 Permeability evolution

Permeability is one of the most critical parameters in the calculation of transport

equations and there exist many empirical relationships linking it to porosity under

different assumptions (see Kuhn (2009) for example). The Blake-Kozeny equation

presented in McCune et al. (1979) was selected:

k = k0

(

φ

φ0

)(

1− φ0

1− φ

)2

, (164)


Figure 24: Comparison of permeability evolution due to increasing damage. Resultsshow an acceptable fit with a reference law presented by Pijaudier-Cabot et al. (2009),which matches two analytical solutions for very low and very high damage.

0.0 0.2 0.4 0.6 0.8 1.0Damage

0

2

4

6

8

10log(k/k0)

Reference

max=0.9

max=1

and validated its usage in combination with the damage-porosity relationship intro-

duced in eq (152) by comparing the porosity evolution as a function of the damage

parameter with a matching law presented in Pijaudier-Cabot et al. (2009). This

law was developed to model the interaction between material damage and transport

properties of concrete. However it was derived from two asymptotic cases where

theoretical modelling exists, for low and high values of damage, and can therefore be

applied to a wider range of geomaterials. Figure 24 shows good agreement between

this matching law and two results obtained with different values of the maximum

porosity φmax. φmax = 1 represents the case where the fully damaged rock (D = 1)

is completely dissolved. It reproduces very well the asymptotic behaviour of the

Poiseuille flow for large values of damage, where permeability is controlled by a

power function of the crack opening. This model however leads to excessive values

of permeability in that case (D → 1) as it can also exaggerate the value of porosity

for a fully damaged rock. The maximum value φmax = 0.9 was chosen to overcome

this problem as shown on Figure 24. The results obtained still match closely the

reference law visually, including for large values of damage. There is a loss of the

asymptotic behaviour for D → 1, which is non critical as the common practise of

capping the value of damage to a maximum of 0.9 is used.

6.3. NUMERICAL APPROACH 119

Figure 25: Simplified flow of information between escriptRT and Abaqus

6.3 Numerical approach

In order to solve the model described in Section 6.2 the capabilities of two simulation

codes, escriptRT and Abaqus, were coupled. The resulting code is based on the

modular architecture of escriptRT, which provides a suitable framework to link

to Abaqus easily.

6.3.1 escriptRT, a modular architecture

escriptRT was designed to provide a high level interface and a modular archi-

tecture to build simulations using various components implemented with different

programming languages (Poulet et al., 2012a, see chapter 3). For that purpose it

uses python, a high level object oriented language, as a binding tool for optimised

algorithms from different packages. The modularity of escriptRT mainly comes

from its usage of the escriptmodelling library and its modelframe approach (Gross

et al., 2008). This flexible approach provides an efficient mechanism to combine var-

ious processes sequentially, with different levels of coupling possible by recursively


using this framework when needed (Poulet et al., 2012a, see chapter 3). This re-

cursive method is useful for coupling two processes in a tight manner, where the

joint convergence of both codes at every time step is required. escriptRT is then

extended to account for mechanical processes using Abaqus as another external

component.

6.3.2 Mechanical modelling with Abaqus

The visco-elasto-plastic mechanical model was implemented numerically follow-

ing (Karrech et al., 2011a, see chapter 4) through the user material subroutine

UMAT of ABAQUS/Standard (2008). This implementation uses relatively new

techniques of thermal and rate dependency considerations. It also includes a predictor-

corrector algorithm of radial return mapping characterised by high convergence

rates. This subroutine implements a large deformation formulation and allows for

high strain simulations that can reproduce realistic geological structures (Zhang

et al., 2012). It also implements a tight thermo-mechanical coupling but in this

study only mechanical deformation part of the Abaqus implementation was used

while the temperature equation was solved using escript.

6.3.3 Connecting escriptRT and Abaqus

The high level interface from escriptRT is well suited to connect to other packages

as it uses python and its existing python threading and socket modules. The

master process of the coupling code is therefore escriptRT, which controls all

time steps and starts the Abaqus simulation on a thread during the initialisation

step, along with two other threads to monitor and communicate with Abaqus (see

Figure 25). The communication between both simulation codes is handled through

sockets and data files, for convenience in this initial version of the coupling code.

Communication on the Abaqus side is done through a C interface, which facilitates

the usage of threads and can exchange data easily with fortran, the language

6.4. APPLICATION TO ALBITISATION 121

Figure 26: Initial geometry and boundary conditions for the generic 2D modelused for the numerical application. The fault and lower sandstone layers are 50mthick. Boundary conditions involve the rock pressure P s, stress σ, pore pressure P ,temperature T and chemical composition c

used to implement the Abaqus subroutines. The temperature, pore pressure and

porosity values computed by escriptRT are used as input by Abaqus, which in

turn updates the damage, equivalent stress and dissipative strain rates that are used

by escriptRT to calculate the new porosity and shear heating source term. Both

simulation codes use specific mesh file formats and a converter was implemented to

convert from one file to another, keeping track of the mapping between indices for

the mesh nodes and elements.

6.4 Application to albitisation

The application of a numerical simulator which takes into account deformation,

fluid and heat transport as well as geochemical reactions is well suited to the study

of alteration and mineralisation. To demonstrate the capability of this simulator,

a simple geologic scenario is modelled which involves shear zone development and

albitisation. Albitisation is a common alteration process in which albite forms by

the replacement of primary feldspars, both K-feldspar and plagioclase. It can occur


Table 9: Initial material properties and mineralogy of the various units in the modelshown in Figure 26

upper layer quartz layer basement faultMineralogy k-feldspar (60%) quartz (100%) albite (60%) albite (60%)(volume percentages) quartz (40%) quartz (40%) quartz (40%)Fluid composition (molarity) KCl (0.1) NaCl (0.1) NaCl (2.0) NaCl (2.0)Density (kg.m−3) 2400. 2400. 2600. 2600.Young’s modulus (GPa) 32. 32. 72. 48Poisson’s ratio (-) 0.29 0.29 0.20 0.20Cohesion (MPa) 20. 20. 30. 30.Friction angle () 30. 30. 30. 30.Thermal expansion 2.4 2.4 2.4 2.4(×10−5K−1)Porosity (%) 15 5 5 10Log permeability (m2) 16 18 18 17Specific heat 850. 850. 850. 850.(J.kg−1.K−1)Thermal conductivity 3.1 3.1 3.1 3.1(W.m−1.K−1)

over a wide range of pressure and temperature conditions, from a diagenetic envi-

ronment (Saigal et al., 1988) to a current geothermal environment (Cavarretta and

Puxeddu, 1998). Albitisation has also been observed to be associated with min-

eralising systems with extensive zones of albitisation associated with amphibolite

metamorphism and IOCG mineralisation in the Proterozoic Cloncurry district of

the Mount Isa Inlier in Australia (Oliver and Wall, 1987; Williams, 1994; Rube-

nach, 2005). An association between albitisation and uranium mineralisation has

also been recognised (da Silveira et al., 1991; Polito et al., 2009). Albitisation of

K-feldspar grains is strongly controlled by temperature and the K+/Na+ activity

ratio:

KAlSi3O8 +Na+ ↔ NaAlSi3O8 +K+ (165)

6.4.1 Problem description

In order to test the code presented in Section 6.3, a simple model was run which

simulates the process of albitisation. The initial model is a two-dimensional cross-

section (Figure 26) which is 1.5km wide and 1km deep. A 500m thick basal unit


comprising albite and quartz in equilibrium with an NaCl brine is overlain by a

unit comprising K-feldspar and quartz in equilibrium with a KCl brine. The two

units are separated by a 50 m thick impermeable quartz layer which ruptures during

fault development allowing fluid migration between the two units and subsequent

albitisation of the K-feldspar layer as fluids move up the fault zone. An initial fault

zone, 50 m thick with a dip of 60, is placed in the basement to localise a new shear

zone within the upper layers of the model and which deforms the quartz layer.

Mechanical, thermal and fluid properties used in the model are presented in Table 9.

Boundary conditions (see Figure 26) are applied on the top surface of the model to

simulate its burial 5km below ground, with a temperature of 150C, a pore pressure

of 49Mpa, an effective rock pressure of 76Mpa with a Biot coefficient of 0.85, and a

fixed chemical composition. A hydrostatic initial pore pressure gradient and an ini-

tial geothermal gradient of 30/km are applied through the section and the system

is equilibrated for those conditions. The model is then considered to be under com-

pression with horizontal velocity boundary conditions of 4mm/year applied on the

right hand side, no horizontal displacement on the left boundary, and no horizontal

fluid or heat fluxes on both sides. Free slip, fixed temperature and a fixed chem-

ical composition (after initial equilibrium) are applied on the bottom boundary.

Changes in porosity as a result of geochemical reactions via equation (151) were

excluded for this study, whose purpose is to demonstrate the first order importance

of porosity and permeability enhancements through damage. Full albitisation can

induce a molar volume decrease of approximately 8%. Since this simulation only

calculates the onset of albitisation, the chemical induced changes in porosity are

neglected as they are expected to be small compared to those induced by damage.

This assumption will be tested when the results are presented.

The model was first initialised without any compression to obtain an equilibrium

solution for the temperature (T ) and pore pressure (P ) fields, with fluid properties

dependent on T and P . The simulation was then run for more than 30,000 years

under compression.


(a) Without damage mechanics. (b) With damage included.

0 25 50 75 100 125 150True distance across shear zone

0.00

0.05

0.10

0.15

0.20

0.25

Equiv

ale

nt

inela

stic

str

ain


0.0

0.5

1.0

1.5

Equiv

ale

nt

inela

stic

str

ain

(c) Values across A-B in (a). (d) Values across A-B in (b).


4

3

2

1

0

log 1

0(D

)

(e) log10(Damage) across A-B in (b).

Figure 27: Effect of damage mechanics on equivalent plastic strain after30, 000 years, with the outline of the deformed quartz layer overlaid in white. Equiv-alent plastic strain values are plotted along a 150m long path (A-B) across the shearzone, showing higher and more localised strains when damage mechanics is consid-ered. A damage profile across A-B shows an exponential evolution of damage awayfrom the shear zone.


(a) After 15,000 years (3% strain). (b) After 30,000 years (6% strain).

Figure 28: Evolution of log10(permeability) over time, with temperature contourssuperimposed. Temperature contours vary between 150 at the top surface and 180

at the bottom boundary.

6.4.2 Results

Localisation and damage

Two mechanical models were tested: with and without the calculation of mechanical

damage. In both scenarios, plastic strain initiates at the intersection of the pre-

existing fault and the unconformity. This reactivates the original fault and then

propagates upwards into the upper layer. The fault is also reflected at the bottom

of the model towards the end of the simulation due to the choice of sliding boundary

condition, but this phenomenon occurs far from the zone of interest for our study at

the centre of the cross-section. The model without damage (Figure 27a) produces

a diffuse shear zone that appears to behave with more ductility than the discrete

more brittle fault formed in the model with damage mechanics (Figure 27). This is

illustrated by the basement offset at the quartz layer (overlaid in white on Figure 27)

which shows very angular block faulting when damage is considered. Two plots of

the equivalent plastic strain profile along a 150m long path across the propagated


Figure 29: Fluid flow and temperature map around the offset of the quartz layer atthe newly developed shear zone after 32,600 years, with boundaries of quartz layersuperimposed in white. Distances in metres from the bottom left corner of the modelare indicated on the axes.


(a) From damage (b) From poro-elasticity.

Figure 30: Decomposition of total porosity evolution from the initial configurationafter 32, 600 years.

shear zone (Figures 27c–d) allow a more quantitative comparison, with a more

localised shear zone and much larger equivalent plastic strain (more than 150%) in

the case when damage mechanics is considered, compared to less than 25% strain

in the other case for the same simulated time.

Damage evolution away from the new shear zone exhibits a logarithmic profile

with respect to the distance from the shear zone (Figure 27e). This behaviour is

consistent with some geological observations made by (Mitchell et al., 2011) around

the Arima-Takatsuki tectonic line in Japan, where the authors noted the same

evolution for the degree of pulverisation in the surrounding rock around the slip

zone. This observation reinforces the justification to use damage mechanics for

geological applications and link the theoretical concept of damage to porosity and

permeability evolution as done in Sections 6.2.7.


Figure 31: Subsection of model showing distribution of albite mineralisation andassociated alteration around fault zone after 32,600 years. a) K-feldspar is replacedby albite immediately above the fault zone. Number of moles are related to volumepercentages by comparison with Figure 31b. Unconformity and initial fault contoursin white dotted lines. Distances in metres from the bottom left corner of the modelare indicated on the axes; b) Plot of volume % albite as a proportion of total feldsparcontent shows the evolution of feldspar compositon over time for point C on Fig-ure 31a.


Reactive transport

The evolution of porosity and permeability with damage(see Sections 6.2.2 and 6.2.7)

along the created shear zone opens a fluid pathway with increases in permeabil-

ity by up to three orders of magnitude in the upper layer (see Figure 28). After

15,000 years the fault is fully reactivated and a zone of damage has begun prop-

agating into the upper layer (Figure 28a). Damage results in a porosity increase,

leading to a drop in pressure and fluid flow in the opposite direction of damage

propagation as a transient phenomenon at the tip of the propagating shear zone.

After 30,000 years the shear zone has reached the top boundary and opened a con-

duit through the entire model (Figure 28b), allowing hotter and more pressurised

fluid from the basement to flow through the upper layer, as illustrated by the tem-

perature contours in Figure 28. The simulation shows that after 30,000 years the

offset of the upper boundary has created an important topographical flow due to the

imposed pressure boundary condition. This causes fluids to be drawn downwards

as shown on the top of Figure 29 and therefore the simulation was terminated after

a simulated time of 32,600 years. At that time, the fluid flows downwards from

the top boundary across half the upper layer, and the upwards fluid across the rest

of the shear zones reaches Darcy velocities of 30cm/year in the lower part of the

upper layer.

Figure 30 shows the separate contributions of poro-elasticity and damage on the

total porosity evolution at the end of the simulation. This figure highlights the

major importance of damage in opening fluid pathways with porosity changes by

up to nearly 47 percentage points (Figure 30a). By comparison the contribution of

poro-elasticity is relatively minor with up to 0.3 percentage points porosity increase

in some dilation zones and 1.2 percentage points decrease in most of the shear

zone due to the compression boundary conditions. Those zones of dilation and

contraction do play a slight role on the transient evolution of fluid flow in the shear

zone, but their effect remains negligible compared to the effect of damage.


Due to the nature of the model and the initial chemical equilibrium fluid-rock

reactions are not initiated until the fault zone has been fully reactivated (after

10,000 years), allowing fluids to migrate from the albite-quartz basement into the

overlying K-feldspar-quartz unit. When the propagating fault reaches the bottom

of that top layer, the permeability at this location (point C on Figure 31b) jumps by

more than two orders of magnitudes. It reaches a maximum value of 10−13.83m2 after

11,800 years and stays at this level until the end of the simulation. The compression

boundary conditions also cause a slight reduction in permeability through the Biot

poro-elastic behaviour (see Section 6.2.3), but this effect is minor compared to the

damage impact and only reduces the permeability to 10−13.85m2 at the end of the

simulation. As the NaCl brine reacts with the K-feldspar progressive alteration to

albite is observed after 15,000 years (Figure 31b). Once started, the albitisation

process accelerates and reaches a constant rate after 28,000 years. At the end of the

simulation one can see a maximum of 12% albite formed (as a percentage of total

feldspar content), which would represent less than 1% porosity decrease based on

a 8% molar volume decrease for full albitisation. This result shows that chemical

porosity evolution, in this particular example, is indeed negligible compared to the

change of porosity attributed to damage. This validates the initial assumption not

to consider chemically induced porosity changes (equation (151)) in this application.

6.5 Conclusion

A new THMC simulation code is presented by linking escriptRT to Abaqus,

thus adding a thermodynamically consistent visco-elasto-plastic deformation im-

plementation to an existing reactive transport code. The mechanical model used

in this study includes a continuum damage mechanics formulation which is linked

to permeability through the evolution of porosity. This significant feedback allows

fluid pathways to be dynamically generated from the localisation of shear zones in

particular. The importance of this phenomenon is illustrated by a numerical study

6.5. CONCLUSION 131

of a generic albitisation scenario involving a pre-existing fault and an unconformity.

While a full geochemical study of this example is beyond the scope of this paper,

a significant conclusion from this study is that the coupling of THMC processes

and the evolution of permeability mechanism can allow to examine the progression

of geochemical reactions in comparison with the rates of other processes from the

fluid motion within the host rock. They can be used as well to predict the distribu-

tion of alteration assemblages which in term can provide vectors to mineralisation.

This information can also be used to recognise prospective geophysical responses in

mineral exploration (e.g., Chopping and Cleverley, 2008). The features and results

of the simulation presented may be applicable to a wide range of hydrothermal ore

deposits as they show that: (i) the inclusion of damage in the code produces narrow

discrete shear zone which match more closely those observed in nature, (ii) dam-

age mechanics reproduces the logarithmic pattern of rock pulverisation which can

be observed in nature on the sides of shear zones, (iii) permeability creation as a

function of damage greatly enhances flow rates and focusing, and (iv) elevated flow

rates can affect geochemical reactions by increasing the supply of reactants as well

as changing the temperature through heat advection. In this example the effects of

damage on permeability are shown to be significantly greater than those caused by

poro-elasticity. While the chemical/dissolution induced porosity changes were not

specifically addressed, the effects of damage are also interpreted to be significantly

greater in this particular example where the changes in mineralogy resulted in a

minor change in molar volume. Simulating scenarios where the fracturing of an im-

permeable seal promotes fluid flow from deep reservoirs into overlying sequences can

be used to examine processes in many mineral systems such as unconformity-related

uranium, Archean gold or others.

Chapter 7

Conclusions and perspectives

7.1 Conclusions

In this thesis, a novel numerical framework for THMC coupling has been formu-

lated, based fully on the thermodynamical potential functions. It distinguishes itself

from the many other THMC formulations in its emphasis on how mechanical de-

formation can create permeable pathways. Classical mechanics has previously been

used to close the system of equations to define the mechanical problem. This thesis

formulates an explicit fully coupled mechanical framework that considers entropy

production.

A particularly important aspect of the new approach presented consists of model-

ling explicitly the various feedbacks which couple all processes involved such as

mechanical deformation, heat transport, fluid flow, chemical transport, and fluid-

rock chemical reactions. Those feedbacks are indeed critical to explain geological

localisation phenomena such as folding or faulting, for example. Both the number of

processes considered as well as the quality of their coupling improve our geological

understanding. This thesis provides a unified framework based on thermodynamics

to tightly couple geological processes, as well as a scalable approach to combine

133

134 CHAPTER 7. CONCLUSIONS AND PERSPECTIVES

processes sequentially and include more feedbacks. Various couplings were investi-

gated successively and their importance was shown through rigorous benchmarks

and illustrative geological simulations. These couplings appear in two forms, di-

rect and indirect, as they can be directly explicited in the constitutive equations

or indirectly introduced through the material properties dependencies on the state

variables. For instance, the shear heating term in the heat equation represents a

direct feedback. An example of indirect coupling is the dependency of mechani-

cal properties on chemical compositions. The key to separate direct and indirect

coupling is the diffusion length and time scales of the feedback processes consid-

ered. For example, for geological problems at the kilometre scale and for times

of hundreds of thousands of years, chemical feedbacks can be incorporated to the

material properties through indirect coupling, while thermal feedbacks need to be

resolved explicitly. At those scales, most chemical processes can indeed be assumed

to have reached equilibrium. Therefore, temperature and pressure dependence of

these properties can be pre-calculated and considered as indirect couplings.

Chapter 2 demonstrates the importance of considering the link between thermo-

chemistry and mechanics, where thermodynamic potential functions can be used to

calculate reversible material properties such as thermal expansion coefficient, spe-

cific heat, elastic shear modulus, bulk modulus, and density. Physical properties

of terrestrial rocks and minerals derived from thermodynamic potentials efficiently

complement geophysical observations and experimental measurements. They may

even record more precise discontinuities due to thermodynamically predicted phase

transitions. Such indirect feedbacks are incorporated numerically in escriptRT, a

new reactive transport simulation code for fully saturated porous media presented

in chapter 3. This sequential coupling code also simulates direct fluid-rock chem-

ical interactions through another Gibbs minimisation solver, as well as fluid flow,

variation of fluid properties through equations of state of pure water, heat transfer

and chemical transport. This flexible code incorporates those numerous feedbacks

to produce more realistic geochemical simulations of hydrothermal and geothermal

7.1. CONCLUSIONS 135

systems.

A formulation is presented in chapter 4 which takes into account thermo-mechanical

coupling, continuum damage mechanics (CDM) and viscoplastic rheology with mul-

tiple creep mechanisms. Continuum damage contributes significantly to the mate-

rial softening and represents an even more efficient weakening mechanism than

shear heating to be considered for the obtention of a more realistic strength of the

lithosphere. The loading-rate control shown in chapter 4 also operates as a switch

between brittle and ductile responses for associated damaged materials.

Some effects of chemical feedbacks can be directly linked to continuum damage

through the consideration of diffusive processes, as shown in chapter 5. Here, the

thermodynamic fluxes exchanged by a representative volume element with its sur-

rounding are taken into account. This thermodynamic framework represents the

base used to derive the thermo-mechanical coupling with damage presented in chap-

ter 4, as well as other couplings published in some subsequent work not included in

this thesis (Karrech et al., 2012, 2011b). An important feature of this framework is

that it allows deducing the constitutive relationships, yielding limits and flow rules

directly from suitable free energy and dissipation functions, instead of introducing

them on an ad hoc basis as commonly accepted in the classic theories of soil and

rock mechanics. It provides therefore a convenient mechanism to study processes

of different natures without violation of the basic first and second principles of

thermodynamics.

Continuum damage also provides an important link between mechanical deforma-

tion and reactive transport, and the developed framework presented in chapter 6

builds a strong bridge between structural geology and geochemistry. Damage can

indeed be seen as a distribution of cracks and voids at the micro-scale, which allows

the connection of damage progression with the evolution of porosity and permeabi-

lity. These properties affect fluid flow and fluid-rock chemical interactions in turn.

The software framework connecting escriptRT with Abaqus provides great flexi-

bility to take advantage of both the mechanical model presented in chapter 4 and the


reactive transport features of escriptRT. The final example presented in the last

chapter 6 illustrates the power of this novel approach to investigate the competition

of rates of thermal, hydraulic, mechanical and chemical processes.

7.2 Perspectives

This new framework represents a step forward in numerical modelling and provides

an extensible platform whereby more geological processes can be added through

the expression of their internal energy and dissipation functions. Describing the

underlying physics and chemistry of all processes in such a way enables the rigorous

derivation of the full system of equations, including all coupled feedbacks through

cross-terms which are generally neglected in sequential coupling approaches. Some-

times those feedback terms can play an important role and should be considered

(see Karrech et al. (2012) for example), but they can also be neglected for perfor-

mance reasons when it is safe to do so. In any case this decision should be taken

after evaluation of the analytical description of the terms considered in the final

system of equations, rather than made at the start of the analysis when the effect

of those terms is not clear yet on the final result.

Future challenges consist of adding additional descriptions of geological processes

such as grain size evolution, and connect analytically in the framework the reactive

terms of chemical equilibrium presented in chapter 3 that were coupled sequen-

tially. The current framework can also lead directly to a novel formulation of the

system where material properties are described in the initial configuration, in the

same way the bulk and shear moduli appear in Karrech et al. (2012) for exam-

ple. The temperature and pressure dependency of those properties can then be

derived analytically using finite strain theory and incorporated adequately in the

system of equations. This coherent approach will represent a major change from

current descriptions, infinitesimal or finite strain, which might sometimes consider

those material properties evolutions as derived by petrologists but include them

7.2. PERSPECTIVES 137

artificially in their frameworks. Those frameworks might therefore duplicate the ef-

fects of those properties variations as they consider independently the same reasons

which lead to the evolution of those properties in the first place. By comparison,

the considerations of a single unified framework are truly exciting.

As long as some sequential couplings remain, sensitivity issues will always arise

from the choice of coupling sequence. Future implementations will harness the full

potential of the thermodynamic approach by inverting the whole system of THMC

equations at once, following theoretical approaches as presented by Coussy (2004)

for example. This solution method would remove the need for a sequential resolution

of some processes and the overlook of some possible feedback mechanisms. Coussy

(2004) only looked at THM couplings and concluded on page 58 that it is sufficient to

solve 28+N equations with 28+N unknowns to obtain a mathematically rigorous

solution to the problem. The beauty of this mathematical formalism is hiding

however the daunting complexity of the corresponding numerical problem and the

work presented in this dissertation only represents a step on a long path. Many

more efforts will still be required to allow geomodellers to take advantage of the

increasing computing power available and simulate geological scenarios in their full

complexity to account for 3D realistic geometry, heterogeneity, scale dependency,

etc., not to mention the uncertainty analysis on all those processes and associated

parameters. True challenges still lie ahead!

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Appendix A

Derivations used in chapter 4

A.1 Local form of the first principle of thermo-

dynamics

There are three laws of conservation which lead to the principle of thermodynamics

in its local form. The conservation of mass in a given domain Ω can be written as:

d

dt

∫

Ω

ρdΩ = 0 (166)

where ρ is the material density. It means that time derivative of the mass of the

domain is equal to zero. In its local form, the material derivative (166) results in

the continuity equation:∂ρ

∂t+ div(ρv) = 0 (167)

where v is the material velocity. The second important law that is needed is the

conservation of momentum which can be expressed as follows:

d

dt

∫

Ω

ρvdΩ−∫

ω

T dω =

∫

Ω

fdΩ (168)

163

164 APPENDIX A. DERIVATIONS USED IN CHAPTER 4

where T = σ.n is a surface traction, ω is the domain surface, n is an outward

normal to it, and f is a body force. Using the local equation of conservation of

mass (167) as well as the theorem of divergence, the conservation of momentum

results:

divσ + f = ρdv

dt(169)

It is the local equation of motion. The third necessary law is the conservation of

energy which can be written as:

d(K + U)

dt= Pext +Q (170)

where K denotes the kinetic energy, U is the internal energy, Pext is the power of

the eternal forces and Q is the rate of heat supply to the domain. These quantities

can be written in integral forms. The quantities in the right hand side are:

Pext =

∫

Ω

f .vdΩ +

∫

ω

T .vdω and Q =

∫

Ω

rdΩ−∫

ω

q.ndω (171)

where r is the volume density of heat production and q is the heat flux. The

quantities in the left hand side are:

K =1

2

∫

Ω

ρv.vdΩ and U =

∫

Ω

ρudΩ (172)

where u is the specific internal energy. Using the local equation of conservation

of mass (167) as well as the integrals (172), the right hand side of equation (170)

reduces to:d(K + U)

dt=

1

2

∫

Ω

ρdv.v

dtdΩ +

∫

Ω

ρdu

dtdΩ (173)

Using the theorem of divergence as well as the integrals (171), the left hand side of

equation (170) reduces to:

Pext +Q =

∫

Ω

(σ : D + (divσ + f).v)dΩ +

∫

Ω

(r − divq)dΩ (174)

A.2. LOCAL SECOND PRINCIPLE 165

where 2D = gradv + gradtv represents the rate of deformation. Substituting for

equation (169) into (174) and for equations (173) and (174) into (170) results in:

∫

Ω

ρdu

dtdΩ =

∫

Ω

σ : DdΩ +

∫

Ω

(r − divq)dΩ (175)

Since D can be approximated by ǫ, the local form of the above equation can be

written as:

ρu = σ : ǫ+ r − div(q) (176)

A.2 Local second principle

The second principle of thermodynamics can be expressed as follows:

dS

dt≥∫

Ω

r

TdΩ−

∫

ω

q.n

Tdω (177)

where S is the entropy of the domain. Using the local equation of conservation of

mass (167) as well as the theorem of divergence, the above equation reduces to:

∫

Ω

(

ρds

dt+ div

q

T− r

T

)

dΩ ≥ 0 (178)

Substituting for divq

T=

divq

T− q.gradT

T 2 in the above equation, the local form of the

second principle can be obtained:

ρT s+ div(q)− r −q

T.grad(T ) ≥ 0 (179)


A.3 Derivation of flow rules from a dissipation

potential

The flow rule used in this chapter are based on the principle of maximum dissipa-

tion which was introduced by Ziegler (1977).This allows treating the elasto-visco-

plasticity as an optimisation problem under an equality constraint:

supF∈S

(

σ : ˙ǫin + Y D)

φ(σ, Y, T ) = 0(180)

where F = (σ, Y ) contains the thermodynamic forces affecting the intrinsic dis-

sipation, S = (E , R) contains the domains of definitions. E = σ, f(σ) ≤ 0 is the

elasticity envelope which can be identified experimentally and R is the ensemble

of real numbers. Bearing these considerations in mind, the optimisation problem

(180) can be rewritten as follows:

infF∈S′

−Di

φ(σ, Y ) = 0

f(σ, D, T ) ≤ 0

(181)

where S ′ = (R6, R). If F is an optimum for the problem (181), then there exist

real constants λ and λ such that:

−∂D∂F

+ λ φ∂F

+ λ f∂F

= 0

Non-negativity : λ ≥ 0

Complementary slackness : λf(σ, D, T ) = 0

Feasibility condition :

φ(σ, Y ) = 0

f(σ, D, T ) ≤ 0

(182)

The above expressions are known as the Karush-Kuhn-Tucker necessary conditions

of optimality. In the first term of the above equation, deriving Di = σ : ˙ǫin + Y D

A.3. DERIVATION OF FLOWRULES FROMADISSIPATION POTENTIAL167

with respect to the thermodynamic forces results in:

˙ǫin = λ ∂φ∂σ

+ λ ∂f∂σ

D = λ ∂φ∂Y

(183)

The conditions of non-negativity, complementary slackness, and feasibility hold

with the above flow rules. In case of associated material behaviour, the condition

of yielding becomes automatically verified when plastic deformation takes place.

Hence the flow rule becomes:

˙ǫin = λ ∂φ∂σ

D = λ ∂φ∂Y

(184)

Note that the constraints need to satisfy only the common conditions of regularity

(continuity and derivability). They do not need to be convex. In the text, λ is

replaced by λ for convenience of notation.

Appendix B

Derivations used in chapter 5

B.1 Dissipation equation

Using the definitions of Q and dSdt

from eqs. (110) and (120), the Second Law of

thermodynamics eq. (123) can be rewritten as

∫

Ω

[(ρ+ ρdiv(u))s+ ρs] dV +

∫

δΩ

sρkvk.n da ≥∫

Ω

r

TdV −

∫

δΩ

q

T.n da (185)

The surface integrals can be transformed in volumetric integrals using the divergence

theorem, leading too the equation in its local form as

(ρ+ ρdiv(u))s+ ρsdiv(sρkvk) ≥r

T− div

( q

T

)

(186)

Using eq. (114) and the fact that T > 0 lead to

(ρ+ ρdiv(u))Ts+ ρT sdiv(sρkvk)T ≥ r − div(q) +q.grad(T )

T(187)

169

170 APPENDIX B. DERIVATIONS USED IN CHAPTER 5

Subtracting eq. (122) from eq. (187) allows to write the dissipation equation as

(ρ+ ρdiv(u))(Ts− e) + ρ(T s− e) + div (ρkvk(Ts− e))− sρkvk.grad(T )

≥q.grad(T )

T− σ : ǫ− µkαk (188)

Identifying the specific Helmholtz free energy eq. (105) and its time derivative

eq. (124) in the previous equation lead to the dissipation equation eq. (125).

B.2 State equations

By developing ψ with the chain rule eq. (125) and using eq. (127), the dissipation

equation eq. (125) can be written as

− (ρ+ ρdiv(u))ψ − div(ρkψvk)− ρ

(

s+∂ψ

∂T

)

T +

(

σ − ρ∂ψ

∂ǫe

)

: ǫe − ρ∂ψ

∂DD

+

(

µk − ρ∂ψ

∂αk

)

αk − ρksvk.grad(T )−q.grad(T )

T+ σ : ǫp ≥ 0 (189)

Using the conservation of mass eq. (121) then leads to

− ρkvk. (grad(ψ) + s grad(T ))− ρ

(

s+∂ψ

∂T

)

T +

(

σ − ρ∂ψ

∂ǫe

)

: ǫe − ρ∂ψ

∂DD

+

(

µk − ρ∂ψ

∂αk

)

αk −q.grad(T )

T+ σ : ǫp ≥ 0 (190)

From this form, some specific thermodynamic mechanisms as considered, as de-

scribed in (Coleman and Gurtin, 1967) for example, to cancel all terms but one in

that inequality. For example, a transformation where the temperature is constant

(T = 0) and uniform (grad(T ) = 0) is firstly considered. That transformation

is also considered not to modify any of the plastic deformation (ǫp = 0), internal

variables αk (αk = 0) or damage D (D = 0). Finally that transformation is con-

sidered not to involve any mass transfer to or from the RVE (vk = 0). In order

B.3. HEAT EQUATION 171

to respect the inequality appendix B.2 the state equation eq. (128) must then be

verified. Similarly other specific thermodynamic mechanisms can be considered to

obtain the other state equations eqs. (129) and (130).

From those expressions and the definition eq. (131), the dissipation equation ap-

pendix B.2 can be simply rewritten in the form of eq. (132).

B.3 Heat equation

From the state equations eqs. (128) to (130), equation eq. (125) can be written as

ρψ = σ : ǫe − ρsT + Y D + µkαk (191)

Using eqs. (121) and (122) the energy balance becomes

ρe+ ρkvk.grad(e) = σ : ǫµkαk + r − div(q) (192)

With eqs. (124), (191) and (192) can then be rewritten as

ρsT + Y D + ρkvk.grad(e) = σ : ǫp + r − div(q) (193)

Using eqs. (106) and (129) s gets rewritten as

s = −1

ρ

∂σ

∂T: ǫe +

C

TT − 1

ρ

∂Y

∂TD − 1

ρ

∂µk∂T

αk (194)

Note that the assumption of small transformations was used to neglect the second

order terms in displacement. This includes the damage term which is equivalent,

see (Karrech et al., 2011c). Equation (194) also used the definition of specific heat

C

C = T∂s

∂T(195)

172 APPENDIX B. DERIVATIONS USED IN CHAPTER 5

eq. (193) then becomes

ρCα− µk∂T

αkT + Y D + ρkvk.grad(e) = σ : ǫp + r − div(q) (196)

The definition of the specific Helmholtz free energy eqs. (105) and (133) leads to

grad(e) =µkρkgrad(αk) + T grad(s) (197)

Combining eqs. (134), (137), (196) and (197), the second balance equation is finally

obtained: the heat equation eq. (137).

B.4 Continuity equations

Using the principle of virtual power eq. (108) and the definition of internal eq. (111)

and external power eq. (139) provide the equality

−∫

Ω

σ : ǫdV −∫

Ω

µkαkdV +

∫

Ω

f.udV +

∫

δΩ

T .nda (198)

Using eq. (136) and neglecting the second order terms, eq. (198) can be rewritten

as

−∫

Ω

σ : ǫdV +

∫

Ω

f.udV −∫

Ω

αkgrad(µk).(u+ vk)dV +

∫

Ω

µkαkdV

+

∫

δΩ

µkαk(u+ vk).nda+

∫

δΩ

T .nda = 0 (199)

The first term of this last equation can be rewritten as

−∫

Ω

σ : ǫdV = −∫

Ω

σ : grad(u)dV =

∫

δΩ

σ.u.nda+

∫

Ω

div(σ).udV (200)

and finally identify the continuity equation eq. (140) and its boundary conditions

eq. (141).

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