Numerical modeling of rock :

04 Continuum -

Stefan Schmalholz [email protected] NO E 61

AS 2009, Thursday 10-12, NO D 11

Numerical modeling of rock deformation: - Rheology. Stefan Schmalholz, ETH Zurich Conservation equations

The fundamental equations of continuum mechanics describe the conservation of • • Linear • Angular momentum and • .

There exist several approaches to derive the conservation equations of continuum mechanics: • Variational methods (virtual ) • Derivations based on integro-differential equations (e.g., Stokes theorem) • Balance of and fluxes based on Taylor series.

We use in this lecture the balance of forces and fluxes in 2D, because it may be the simplest and most intuitive approach.

Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich Taylor series

p px 0  px x  px   x  O  x2 00x

px 0   x px  px 0  x 0 x px 0 

x x0 x0+ x

Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich in 2D

Taylor series p y p p pxxpx    xOx   2 y 2 x

Net rate of mass increase  1 kg kg xy  mm  2 tssm  x yxy   y tt p x p p x p  y x p x 2 x 2 must balance the net rate of flow of kg m x mass, e.g. vy, into the element p   v=  m2 s kg m kg vym  vvxxxx2 vyvyxx s sm p y xx22 p  y 2 vvyyyy vxvxyy   yy22 v v x xyy yx xy

Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich Conservation of mass in 2D

Net mass increase in element balances p y net flow of mass into element p y 2  v v x yxyyx x  y  txy  v v x y 0 tx  y y p x p x  p  y p divv 0 x  t x 2 x 2 x If we assume the to be constant then   0 t  p y v v p  x y 0 y 2 xy v v x y 0 xy divv  0

Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich Conservation of linear momentum

Force balance in the x-direction  y  yx xx   yx xx xx y 2 xx xx yz xx22

yx yy yx  yx yx x z yy22 y   x  x xx y  xx  xx  x xx balance in the x-direction is x 2 x 2 fulfilled if x   xx yx 0 xy  y   yx yx y 2 Force balance in two dimensions   xx yx 0 xy  xy yy 0 xy

Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich Conservation of linear momentum

General force balance in two dimensions Derivation based on integro- differential equations   xx yx 0  TdS  0 xy Cauchy tensor S  σndS divσ dVji dV 0 xy yy  x 0 SV Vj xy  ji  0 x j Gauss theorem  ji 0,j 1,2 x j

 ji, j 0,j 1,2

divσ  0

Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich Conservation of linear momentum

General force balance in two dimensions

  v xxxyx F  xyx  t  v xy yy F  y xyy  t

Under gravity we use

Fgy 

Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich Conservation of angular momentum

Stress tensor is symmetric

 yx  xy

This is the simplest version of the conservation of angular momentum and most common. Cosserat theory includes additional moments and the conservation equation becomes more complicated.

Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich

Heat equation for two dimensions

DT T  T vvxxvvyy ckx kQ y xx yx xy yy Dt x x y y  x  y  x  y

Heat conduction- Heat Heat production due to source shear heating

In Eulerian system the total time derivative is (material time derivative) DT T T T vv  Dt txy x y

Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich Ductile rheology 1D viscous (Newtonian) rheology •Time dependent •Energy is not conserved, , shear heating •Mostly incompressible v 2  2 x  1 The rheology is linear. Deviatoric is related to deviatoric .

v x

Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich Ductile rheology 1D power-law rheology v n  2 x 1 v n 2 x 1 1 1 vvn  v n 2 eff  x  xx  eff

The rheology is nonlinear. 11 The effective is a function of the strain rate. 1 E V  nnA exp Iterations are usually necessary in numerical algorithms. nRT Typical structure of rock rheology.

Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich Ductile rheology 2D viscous rheology •Time dependent •Energy is not conserved, dissipation, shear heating •Incompressible

v p 2 x xx x p is .  is total stress. vy p 2 yy y

1 vx vy yx 2  2 yx

Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich Ductile rheology 2D non-Newtonian (power-law) rheology •Time dependent •Energy is not conserved, dissipation, shear heating •Incompressible

1 1 v p 2  n x xx II x 1 1 v p 2  n y yy II y 1 1 v n 1 vx y yx2  II  2 yx 22 11vvxxvvyy II  44xy  yx

Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich Closed sys. of eqns: incompressible

  Conservation of xx yx 0 linear momentum, xy

Force balance, xy yy Seven 0 Two equations xy unknowns  , xx Conservation of

 yy , angular momentum,  yx  xy One equation  yx , v v  xy , Conservation of mass, x y 0 One equation xy p, v ux , x xx p 2 u x y v Rheology, y yy p 2 Three equations y

1 vx vy yx 2  2 yx Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich Elastic rheology

Elastic rheology E = Young’s modulus •Time independent  = Poisson ratio •Energy is conserved, no L = Lame parameter dissipation, no shear heating G = Shear modulus •In 2D different for plane strain and plane stress

u E 1 u  u ux y  x y  LG2  L xx  xx 112  x  1 y x y

ux uy E 1  ux uy  LLG2   yy  yy  x y 1121 x y ux uy E ux uy  yx G   yx  yx 21 yx

Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich of equations:

  Conservation of xx yx 0 Six linear momentum, xy unknowns Force balance,  Two equations xy yy 0 xy  xx ,  , yy Conservation of  yx  xy  yx , angular momentum,  , One equation xy E 1 u  u  x y u , xx  x 112 xy 1 u y E 1  u u Rheology,  x y yy  Three equations 1121 xy

E ux uy  yx  21 yx Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich Rheology reformulated

vx x  1 200 Viscous rheology xx v p 1020  y yy y  yx 0001 v v x  y vx yx xx p 2  x 0  1200x v xx y  vx p 2 yy p 10200  yy y v y  0001 y yx  yx 1 vx vy yx 2  2 yx σ p D B u

σ pDBu 

Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich The constitutive equations

Constitutive equations for 2D plane strain  0 x ux uy  xx LG 20 L  xx LG2  L  ux x y  yy LLG200 y uy u u  yx 00G  LLGx 2 y  yy x y yx

ux uy  yx G  yx  σ D B u

σ  DBu

Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich Rheology – Force balance

 Elastic rheology 0 Substitution of rheology in force x  xx balance equations  σ   B  0 σ  DBu yy y  yx   yx BDBufT  Ku  f Force balance Extract from finite element code

 xx  yx B(1,ii ) = DHDX(1,:);   0 B(2,ii+1) = DHDX(2,:); BT σ  0 xy B(3,ii ) = DHDX(2,:);  B(3,ii+1) = DHDX(1,:); yx yy  0 xy E = MATPROP(1,Phase(iel)); T nu = MATPROP(2,Phase(iel)); prefac = E/((1+nu)*(1-2*nu)); B σ  f D = prefac * [ 1-nu nu 0; nu 1-nu 0; 0 0 (1-2*nu)/2]; Vector f includes the boundary conditions if no physical external forces are present. K = K +( B'*D*B )*wtx*detjacob;

Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich FEM Examples - linear viscous

Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich FEM Examples – power law

Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich FEM Examples – linear viscous & gravity

Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich Next week: Matlab

• Next week we meet at 10:15 in HG E 27 • Matlab scripts are on course web page

Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich