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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
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
Structures due to rock deformation
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
Structures due to rock deformation
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
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
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
Structures in rocks on all scales
picture courtesy of Jean-Pierre Burg picture courtesy of Chris Wilson
picture courtesy of Chris Wilson
Parasitic folds
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
Multilayer folding in the Alpstein
Heierli, 1984
Säntis
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
Structures in rocks on all scales
from University Lausanne home page
Multilayer foldsMountain range scale
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
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
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
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?
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
Motivation - Examples
•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?
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
Motivation - Examples
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
Motivation - Examples
•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?
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
Motivation - Examples
•Application of buckling in tunnel constructions.
Pictures from Teddy Burton
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
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?
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
Quantify kinematics and dynamics with
numerical experiments
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
3D deformation Zagros mountains, SW Iran
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
Deformation of the lithosphereThermo-mechanical model with viscoelastoplastic rheology including viscous shear heating, gravity and erosion.
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
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.
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
Thermo-tectonic history modeling
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
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
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
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.
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
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).
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
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.
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
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)
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
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).
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
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)
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
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
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
Diffusion: different models
Cellular automata model
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
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.
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
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)
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
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!
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
Analytical solutions
• Relationship between involved parameters in equation form
• Testing of numerical code• Best insight into the physics of a process
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
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
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
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
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
Folding: Linear stability analysis
0, exp cos 2 /w x t w t x
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
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
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
0, exp cos 2 /w x t w t x
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.
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
The dominant wavelength theory
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
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.
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
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
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
Kinematic models - strain ellipses
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
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.
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
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
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
Finite Element Method
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
Finite Element Method
Taken from MIT lecture, de Weck and Kim
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
Finite Element Method
Taken from unknown web script
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
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
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
The FEM Matlab code2
2
( ) 0u xA Bx
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
Finite Element Method - ExamplesRayleigh-Taylor instability – FEM with remeshing
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
Finite Element Method - ExamplesSubduction zone – FEM with remeshing
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
Finite Element Method - Examples
Large deformations with free surfaces to minimize the computational domain.
Viscous beam under gravity
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
Finite Element Method
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
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
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
Current research with FEM Shortening of continental lithosphere
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
32 GB RAM workstation.~60^3
Superposed folding Bachelor/Master thesis of Jaqueline Reber
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
Stress evolution in 3D folding Master thesis of Jaqueline Reber
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
Finite Element Method
Work of Boris Kaus, multigrid parallel solver.
Gonzales supercomputer
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich
Finite Element Method Be patient when programming
Numerical modeling of rock deformation: Introduction. Stefan Schmalholz, ETH Zurich