numerical modeling of rock deformation - eth z€¦ · numerical modeling of rock deformation: ......

Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich

Numerical modeling of rock deformation:

01 Introduction

Stefan [email protected], NO E 61Assistant: Sarah Lechmann, NO E 69

AS 2009, Thursday 10:15-12:00, NO D 11

mailto:[email protected]


Structures due to rock deformation


Structures in rocks on all scales

Kilometer scaleCentimeter scale

Meta-sediments from Indus Suture Zone, Northern Pakistan (picture courtesy of Pierre Bouilhol) Dietrich & Casey,

1989

Folds



picture courtesy of Jean-Pierre Burg picture courtesy of Chris Wilson

picture courtesy of Chris Wilson

Parasitic folds


Multilayer folding in the Alpstein

Heierli, 1984

Säntis



from University Lausanne home page

Multilayer foldsMountain range scale


Structures in rocks on all scalesBoudins, pinch-and-swellCentimeter scale

Analogue model (Jean-Pierre Burg)

Kilometer scale

Lithospheric extension - rifting

O’Reilly et al., 2006


Motivation - Examples

•What mechanism generates pinch-and-swell structures?•What rheologies generate pinch-and-swell structures?•What parameters control the geometry of the pinch-and- swell structures?•Does the geometry of the structure tell us something about:

•Amount of extension?•Material properties?



•What mechanism generates folds?•What rheologies generate single-layer folds?•What parameters control the geometry of the single-layer folds?•Does the geometry of the structure tell us something about:

•Style of deformation?•Amount of shortening?•Material properties?



•When do fractures form during folding?•What is the fracture orientation?•What is the impact of pre-existing fractures on folding?•Does the geometry of the structure tell us something about:

•Style of deformation?•Amount of shortening?•Material properties?



•Application of buckling in tunnel constructions.

Pictures from Teddy Burton


Motivation – Examples Master thesis of Marcel Frehner

•How do parasitic folds form?•What information can we extract from parasitic fold shapes?•What is the influence of

•Rheology?•Style of deformation?•Amount of shortening?•Material properties?


Quantify kinematics and dynamics with

numerical experiments


3D deformation Zagros mountains, SW Iran


Deformation of the lithosphereThermo-mechanical model with viscoelastoplastic rheology including viscous shear heating, gravity and erosion.


Reconstructing sedimentary basinsNumerical model of the evolution of sedimentary basins.Real stratigraphy is reproduced with the numerical model with good accuracy.

3 thinning phases

Rüpke

et al. (2008)

Result of full lithospheric model.Only sediments are shown.


Thermo-tectonic history modeling


Why model rock deformation numerically?

• To understand why observed patterns formed (e.g. folds, pinch-and-swell)?

• To reconstruct deformation: How much strain is necessary to generate an observed structure?

• To quantify deformation: How much force is necessary to generate an observed structure?

• Explain observed structures based on well established mechanical principles (no “arm waving”)

• To predict future deformations: Stability, Natural resources


Web page

The lectures are on the web under:

http://www.structuralgeology.ethz.ch/education/teaching_material/numerical_modeling

New lectures will be uploaded next week, lectures online are from last year.

http://www.structuralgeology.ethz.ch/education/teaching_material/numerical_modeling


Literature• Geodynamics, Turcotte, D.L. and Schubert, G.• Continuum mechanics, Mase, G.E.• Rheology

of the Earth, Ranalli, G.

• There are many books on finite elements. Have a look at several of them and find out yourself which style you like best. Classical books are from Hughes, T.J.R. (The finite element method), Bathe, K.-J. (Finite element procedures) and Zienkiewicz, O.C. and Taylor, R.L. (The finite element method).

• Internet: there are many scripts and pages on certain topics (e.g., Wikipedia).


The big pictureMechanical framework•Continuum mechanics•Quantum mechanics•Molecular dynamics

Governing equations•Differential equations•Integral equations•System of linear equations

Constitutive equations(Rheology)•Elastic•Viscous•Plastic

Closed system of equationsBoundary and initial conditions•Navier-Stokes equations•Euler equations•Wave equation•Heat equation

Solution technique•Analytical solution

•Linear stability analysis•Fourier transform•Green’s function

•Numerical solution•Finite element method•Finite difference method•Spectral method

Solution: valid for the applied•Boundary conditions•Rheology•Mechanical framework•etc.


Equations of continuum mechanicsConservation of mass

Conservation of linear momentum

Conservation of angular momentum

Conservation of energy

0ii

vx

ijii

j

dv Fdt x

ij ji

v pi ij ij ij

i i i

T Tc T v k Ax x x

1 1K p

1 12 2

ijv e p nij ij ij ij ij

ij

D QG Dt

Equation of state

Constitutive equations (rheology)


Material parameters and rheologyTABLE 1: APPLIED ROCK-PHYSICAL PARAMETERS AND RHEOLOGICAL EQUATIONS (MACKWELL ET AL., 1998; AFONSO AND RANALLI, 2004 AND REFERENCES THEREIN)

Upper crust (dry granite)

Mantle (wet olivine)

Low. crust, weak (diabase)

Low. crust, strong (Columbia diabase)

Density (kg/m3) 2700 3300 2900 2900 Shear modulus (Pa) G 1 x 1010 1 x 1010 1 x 1010 1 x 1010 Power-law exponent n 3.3 4.0 3.0 4.7 Coefficient (Pa-n/s) A 3.16 x 10-26 2.0 x 10-21 3.2 x 10-20 1.2 x 10-26 Activ. energy (kJ/mol) H 190 471 276 485 Specific heat (J/kg/K) 1050 1050 1050 1050 Heat production (W/m3) 1.4 x 10-6 0 0.4 x 10-6 0.4 x 10-6 Thermal exp. coeff. (1/K) 3.2 x 10-5 3.2 x 10-5 3.2 x 10-5 3.2 x 10-5 Conductivity (W/m/K) 2.5 3.0 2.1 2.1 Internal friction (º) θ 30 30 30 30 Cohesion (MPa) C 10 10 10 10

Visco-elastic rheology: 1 1 1

/ 2 exp / 2n nII

HA E GnRT

ε τ τ

Mohr-Coulomb criterion: sin cos 0n C

Low-temperature plasticity: 0 0 0/ 3 1 / ln 3 / 2II IIE RT H E

τ ε 0 = 5.7 x 1011 (1/s)

0 = 8.5 x 109 (Pa)

0H = 525 (kJ/mol)

, ε τ = strain rate-, deviatoric stress-tensor, τ = objective time derivative of τ , T = temperature,

, IIE = second invariant of strain rate-, stress-tensor, R = gas constant, n = mean stress. Note: Low temperature plasticity is applied for the upper mantle for stresses larger than 200 MPa. (Goetze and Evans, 1979; Molnar and Jones, 2004).


Analytical and numerical solutions and scientific programming

Folding

% 9 NODE ELEMENTfor j=1:2:ny-2;

for i=1:2:nx-2;nel = nel+1; NODES(1,nel) = NUMNODE(i ,j ); NODES(2,nel) = NUMNODE(i+2,j ); NODES(3,nel) = NUMNODE(i+2,j+2); NODES(4,nel) = NUMNODE(i ,j+2); NODES(5,nel) = NUMNODE(i+1,j ); NODES(6,nel) = NUMNODE(i+2,j+1); NODES(7,nel) = NUMNODE(i+1,j+2); NODES(8,nel) = NUMNODE(i ,j+1); NODES(9,nel) = NUMNODE(i+1,j+1);Phase(nel) = 1;if j>round(ny/2)-3 & j<round(ny/2)+1

Phase(nel) = 2;end

endend

Finite element methodMatlab, Fortran, CTeaching and learning!

Schmalholz

(2008)


Diffusion: different models

Continuum mechanics model

2

2

( , ) ( , )T x t T x tt x

( 1, ) 0

( 60, ) 0

T x tx

T x tx

( [28 : 32], 0) 0.2( [1: 27,33: 60], 0) 0

T x tT x t

Equation describing diffusion in 1D

Boundary conditions

Initial condition


Diffusion: different models

Cellular automata model


Continuum mechanics

• Physical processes are described by a set of partial differential (or integral) equations.

• Variables (e.g. velocities) are continuous in space and time (i.e. differentiable).

• Set of conservation equations.• Closed system of equations.• Well established and successfully applied in

fluid dynamics, elasticity theory, etc.


Dimensional analysis

• Find simplest equation• Making an educated guess (order of

magnitude)• Check and test equations• Scaling• Minimize number of parameters and therefore

number of experiments• Determine controlling parameters (e.g.,

Reynolds number)


Atomic explosionThe British physicist G.I.Taylor estimated the energy of the first atomic explosion in 1945 based on a series of pictures that where published in a popular magazine. At that time the energy of the atomic explosion was considered top secret and Taylor’s estimate caused “much embarrassment” in American government circles, because the series of pictures was not classified.

2 2 3

2 115 55

1 2 15 5 5

5 514

2 2

,[ ] ,[ ]

[ ] [ ] [ ]

, assume 1

80 1.2 10 25 kilo tons of TNT0.006

E ML T t T ML

R E tR const const

E tREt

Dimensional analysis is a powerful tool!


Analytical solutions

• Relationship between involved parameters in equation form

• Testing of numerical code• Best insight into the physics of a process


The dominant wavelength theory•Natural single-layer folds seem to have a ratio of arc length to thickness which is relatively constant.

•What controls the wavelength of the folds?

•Is there a mechanical explanation for this phenomenon?

•Does the geometry depend on the rheology?

•How are these folds generated anyway?

•Why do we care?

A = amplitudeL = arc length

= wavelength

= max. limb dip

RayFletcher

Maurice Biot

HansRamberg


Folding: Linear stability analysis

Is a layer with small sinusoidal perturbations stable under compression?Do small perturbations grow very fast?

Thin-plate equation for viscous folding.

12

3 5 21

1 24 2

24 4 03H w w wH

tx t x

H

w

2Partial differential equation. x

0, exp cos 2 /w x t w t x


Folding: Linear stability analysis


Is a layer with small sinusoidal perturbations stable under compression?Do small perturbations grow very fast?

Homogeneous pure shear thickening of the layer is:

stable if

< 0 shortening/thickeningneutrally stable if

= 0 shortening/thickeningand unstable if

> 0 folding/buckling

Thin-plate equation for viscous folding.

12

3 5 21

1 24 2

24 4 03H w w wH

tx t x

H

w

2

Assuming exponential growth with time.

Partial differential equation. x


Folding: Linear stability analysisThin-plate equation for viscous folding.

The layer is:

< 0 shortening/thickening

= 0 shortening/thickening

> 0 folding/buckling

12

Wavelength,

Dispersion curve


3 5 21

1 24 2

24 4 03H w w wH

tx t x

H

w

2

Viscosity ratio

Gro

wth

rate

,

Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich1

1

Dispersion curve

13

11

2

26

H

23

11

2

46

~1 ~1

Biot, 1957Ramberg, 1963

Folding: Numerical verification

Viscosity ratio

Initial random perturbation –

all wavelengths present.


The dominant wavelength theory


Numerical modeling of rock deformation

The steps to study rock deformation and the relating structures:

• Determine the physical mechanism which generates a certain structure (e.g. folds)

• Develop an analytical solution for a simple model setup• Determine the parameters which control the mechanism and

the geometry of the structure• Perform numerical simulations to verify the analytical solution

for more complex and realistic geometries and more complex settings.

• Apply the numerical model to predict and reconstruct rock deformation.


Historical note on foldsFirst observations of folds in the Swiss Alps by Marsili & Scheuchzer ~1700

Sir James Hall proposed in 1815 that the mechanism generating folds is layer- parallel compression

Analytical solutions for viscous folding by Biot and also Ramberg ~1960

Numerical simulation of folding using finite elements by Dietrich and Carter in 1969


Kinematic models - strain ellipses


Numerical methods

• There are different numerical methods, e.g.:– Finite element method– Finite difference method– Finite Volume method– Spectral method

• Numerical methods transform ordinary or partial differential equations into a linear system of equations, which can be solved by a computer.


Finite Element Method (FEM)

• We will use the FEM• The FEM is powerful:

– Handling of complex geometries– Very flexible– Many parts of a finite element code need to be

programmed only once and are then widely applicable

Richard Courant


Finite Element Method



Taken from MIT lecture, de Weck and Kim



Taken from unknown web script


Finite Element MethodGoals• To understand the basics of the FEM• To program your own finite element code in Matlab• To use the FEM to model and study rock

deformation


The FEM Matlab code2

2

( ) 0u xA Bx


Finite Element Method - ExamplesRayleigh-Taylor instability – FEM with remeshing


Finite Element Method - ExamplesSubduction zone – FEM with remeshing


Finite Element Method - Examples

Large deformations with free surfaces to minimize the computational domain.

Viscous beam under gravity


Finite Element Method - ExamplesII

Ductile: Power-law, n=5

Elastoplastic: von Mises pII.

FEM: total stress formulation with consistent tangent

Distributed folding

Localized shear bands


Current research with FEM Shortening of continental lithosphere


32 GB RAM workstation.~60^3

Superposed folding Bachelor/Master thesis of Jaqueline Reber


Stress evolution in 3D folding Master thesis of Jaqueline Reber



Work of Boris Kaus, multigrid parallel solver.

Gonzales supercomputer


Finite Element Method Be patient when programming

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