Numerical modeling of rock deformation: 04 Continuum ...€¦ · Numerical modeling of rock...
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Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich
Numerical modeling of rock deformation:
04 Continuum mechanics - Rheology
Stefan [email protected]
NO E 61
AS 2009, Thursday 10-12, NO D 11
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich
Conservation equationsThe fundamental equations of continuum mechanics describe the conservation of• Mass • Linear momentum • Angular momentum and • Energy.
There exist several approaches to derive the conservation equations of continuum mechanics:• Variational methods (virtual work)• Derivations based on integro-differential equations (e.g., Stokes theorem)• Balance of forces 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
x0 x0 +xx
p 0 2
0 0
p xp x x p x x O x
x
0p x
0p x x
00
p xp x x
x
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich
Conservation of mass in 2D
2pp x x p x x O xx
Taylor series
Net rate of mass increase
x y x yt t
must balance the net rate of flow of mass, e.g. vy, into the element
2 2
2 2
x xx x
y yy y
yx
v vx xv y v yx x
v vy yv x v xy y
vvx y y x
x y
2p xpx
x
y
2p ypy
2p ypy
x
y
2p xpx
p
2
1 kg kgx y mmt s sm
2
2
v =
v
kg mpsm
kg m kgy ms sm
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich
Conservation of mass in 2D
0
0
yx
yx
vvx y x y y x
t x yvv
t x y
divt
v
Net mass increase in element balances net flow of mass into element
0
0
0
0
yx
yx
tvv
x yvv
x ydiv
v
If we assume the density to be constant then
2p xpx
x
y
2p ypy
2p ypy
x
y
2p xpx
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich
Conservation of linear momentum
2xx
xxx
x
x
y
2yx
yxy
y
2yx
yxy
y
x
y
2xx
xxx
x
2 2
2 2
xx xxxx xx
yx yxyx yx
x x y zx x
y y x zy y
Force balance in the x-direction
0yxxx
x y
Force balance in the x-direction is fulfilled if
0
0
yxxx
xy yy
x y
x y
Force balance in two dimensions
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich
Conservation of linear momentum
0
0
yxxx
xy yy
x y
x y
General force balance in two dimensions
0
0
0
S
ji
jS V V
ji
j
dS
dS div dV dVx
x
T
σn σ
Derivation based on integro- differential equations
Gauss divergence theorem
0, 1, 2ji
j
jx
, 0, 1, 2ji j j
0div σ
Cauchy tensor
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich
Conservation of linear momentum
yxxx xx
xy yy yy
vF
x y tv
Fx y t
General force balance in two dimensions
yF g
Under gravity we use
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich
Conservation of angular momentum
yx xy
Stress tensor is symmetric
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
Conservation of energy
y yx xx y xx yx xy yy
v vv vDT T Tc k k QDt x x y y x y x y
Heat equation for two dimensions
Heat conduction-advection Heat source
Heat production due to shear heating
x yDT T T Tv vDt t x y
In Eulerian system the total time derivative is (material time derivative)
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich
Ductile rheology1D viscous (Newtonian) rheology•Time dependent•Energy is not conserved, dissipation, shear heating•Mostly incompressible
2 vx
The rheology is linear.Deviatoric stress is related to deviatoric strain rate.
2
1
vx
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich
Ductile rheology1D power-law rheology
1
1 11
2
2
2
eff
n
n
nn
eff
vx
vx
v v vx x x
The rheology is nonlinear.The effective viscosity is a function of the strain rate.Iterations are usually necessary in numerical algorithms.
1 11expn n E VA
nRT
Typical structure of rock rheology.
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich
Ductile rheology2D viscous rheology•Time dependent•Energy is not conserved, dissipation, shear heating•Incompressible
2
2
122
xxx
yyy
yxyx
vp
xv
py
vvy x
p is pressure.
is total stress.
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich
Ductile rheology2D non-Newtonian (power-law) rheology•Time dependent•Energy is not conserved, dissipation, shear heating•Incompressible
1 1
1 1
1 1
2 2
2
2
122
1 14 4
xnxx II
ynyy II
yxnyx II
y yx xII
vp
xv
py
vvy x
v vv vx y y x
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich
Closed sys. of eqns: incompressible fluid
yx xy
0
0
yxxx
xy yy
x y
x y
,,
,
,
,,
xx
yy
yx
xy
x
y
puu
Rheology,Three equations
Conservation of linear momentum,Force balance,Two equations
Conservation of angular momentum,One equation
Sevenunknowns
2
2
122
xxx
yyy
yxyx
vp
xv
py
vvy x
0yx vvx y
Conservation of mass,One equation
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich
Elastic rheology
Elastic rheology•Time independent•Energy is conserved, no dissipation, no shear heating•In 2D different for plane strain and plane stress
11 1 2 1
11 1 2 1
2 1
yxxx
yxyy
yxyx
uE ux y
uE ux y
uuEy x
2
2
yxxx
yxyy
yxyx
uuL G L
x yuu
L L Gx y
uuG
y x
E = Young’s modulus
= Poisson ratioL = Lame parameterG = Shear modulus
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich
Closed system of equations: solid
yx xy
0
0
yxxx
xy yy
x y
x y
11 1 2 1
11 1 2 1
2 1
yxxx
yxyy
yxyx
uE ux y
uE ux y
uuEy x
,,
,
,
,
xx
yy
yx
xy
x
y
uu
Rheology,Three equations
Conservation of linear momentum,Force balance,Two equations
Conservation of angular momentum,One equation
Sixunknowns
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich
Rheology reformulated
Viscous rheology1 2 0 01 0 2 00 0 0 1
x
xxy
yy
yxyx
vxv
py
vvy x
01 2 0 01 0 2 0 00 0 0 1
xxx
yyy
yx
xv
pvy
y x
2
2
122
xxx
yyy
yxyx
vp
xv
py
vvy x
σ p DBu
Dσ B up
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich
The constitutive equations
2
2
yxxx
yxyy
yxyx
uuL G L
x yuu
L L Gx y
uuG
y x
Constitutive equations for 2D plane strain elasticity
02 0
2 0 00 0
xxx
yyy
yx
xL G Lu
L L Guy
G
y x
σ D uB
σ DBu
Numerical modeling of rock deformation: Continuum mechanics - Rheology. Stefan Schmalholz, ETH Zurich
Rheology – Force balance
0T B σ
σ DBu
0
0
x
y
y x
B
Elastic rheology
0
0
yxxx
yx yy
x y
x y
Force balance
TB DBu fK u f
Substitution of rheology in force balance equations
T B σ fVector f includes the boundary conditions if no physical external forces are present.
xx
yy
yx
σ
B(1,ii ) = DHDX(1,:);B(2,ii+1) = DHDX(2,:);B(3,ii ) = DHDX(2,:);B(3,ii+1) = DHDX(1,:);
E = MATPROP(1,Phase(iel));nu = MATPROP(2,Phase(iel));prefac = E/((1+nu)*(1-2*nu));D = prefac * [ 1-nu nu 0; nu 1-nu 0; 0 0 (1-2*nu)/2];
K = K +( B'*D*B )*wtx*detjacob;
Extract from finite element code
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