Mineral Physics Tutorial 2Physical Properties of Composite Materials: Effective Medium Models

Effective Elastic Moduli

´ Rocks are polymineralic, heterogeneous

´ To estimate effective elastic moduli of rocks, we need:´ Elastic moduli of each constituent (minerals, interstitial

fluid etc.)

´ Volume of each constituent

´ Distribution of all constituents

´ The geometric details of the distribution are the hardest to measure

3-dimensional (3D) Imaging

Dvorkin et al., 2008

Digital Rock Physics

Digital Rock Physics I

´ Obtain Darcy velocity (q)

´ Apply Darcyʼs Law

´ Virtual permeability experiments (Miller, Zhu, Montesi, and Gaetani, 2014)

´ Mimics an actual permeability experiment.

´ Imposing pressures, P1 > P2, induces fluid flow.

´ Solve Stokes Equations for velocity (finite-volume method)

v : velocity [m s-1]µ : viscosity [Pa s-1]p : pressure [Pa]

q : Darcy velocity [m s-1]k : permeability [m2]L : length of sample [m]

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∇⋅ v = 0µ∇2v −∇p = 0

⎧ ⎨ ⎩

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q = −kµP1 − P2L

Melt Transport

´ Miller, Zhu, Montesi and Gaetani, 2014

Digital Rock Physics II

´ Apply Ohm’s Law for volume-averaged quantities:

´ Virtual direct current experiment (Miller, Montesi, and Zhu, 2015)

´ Assign conductivities to materials.

´ Melt: 7.53 S/m (ten Grotenhuis et al., 2005)

´ Olivine: 0.009 S/m (Constable, 2006)

´ Imposing voltage gradient, V1 > V2, induces electric current.

´ Solve Laplace Equation (in-house finite-difference code).

1Ωtotal

J|| dΩ =σ bulkV1 −V2L

J is current density [A m-2]σ is conductivity [S/m]V is voltage [V]

where,

Where Ω is volume

∇2V = 0J = −σ∇V

Radius-dependence of fluid & current

´ Stronger radius-dependence of fluid flux.

´ Weaker radius-dependence of electric flux.

L = 70 μm

R =10 μm

P,Voutlet`

P,Vinlet

Schematic

Velocity

Electric Field

Φ = πR2 ΔVLElectric Flux:

Q = πR4

8µΔPLFluid Flux:

Elastic Modulii

State-of-the-Art

´ 3D-Atom Probe (LEAP 5000)´ Both 3D imaging

and chemical composition at atomic scale

´ Evaporate atoms by field effect (near 100% ionization), projected onto a position sensitive detector

(Madonna et al., 2012)

Voigt-Reuss-Hill Average

´ Voigt bound (upper bond)

´ Reuss bound (lower bound)

´ Voigt-Reuss-Hill Average

KVoigt = f1K1 + f2K2 µVoigt = f1µ1 + f2µ2

KReuss = f1K1−1 + f2K2

−1−1( )−1

µReuss = f1µ1−1 + f2µ2

−1( )−1

KVoigt-Reuss-Hill =12KVoigt +KReuss( ) µVoigt-Reuss-Hill =

12µVoigt +µReuss( )

Geometric Interpretations(From G. Mavko)

Hashin-Shtrikman Bounds´ HS upper bound: Stiff material is shell, K1, µ1

´ HS lower bound: soft material is shell, K1, µ1

Hashin-Shtrikman Bounds´ Shell: K1, µ1

´ Sphere: K2, µ2

KH-S 1 = K1 +f2

1K2 −K1

+f1

K1 + 43 µ1

µH-S 1 = µ1 +f2

1µ2 −µ1

+2 f1 K1 + 2µ1( )5µ1 K1 + 4

3 µ1( )

KH-S 2 = K2 +f1

1K1 −K2

+f2

K2 + 43 µ2

µH-S 2 = µ2 +f1

1µ1 −µ2

+2 f2 K2 + 2µ2( )5µ2 K2 + 4

3 µ2( )

Transport properties

´ Permeability: k

´ Parallel connection (arithmetic mean): Upper Bound

´ Serial connection (harmonic mean): Lower Bound

´ Random connection (geometric mean): average

´ Hashin-Shtrikman bounds

kParallel = f1k1 + f2k2

kSerial = f1k1−1 + f2k2

−1−1( )−1

kRandom = k1f11 k2

f2

kH-S 1 = k1 +f2

1k2 − k1

+f1

3k11

kH-S 2 = k2 +f1

1k1 − k2

+f2

3k2

Other elastic properties

´ Estimate bulk and shear moduli from Young’s modulus and Poisson’s ratio

´ Mixing theory on K or µ

´ Compute Young’s modulus, P-wave and S-wave velocity

E = 9K µ3K +µ

µ =E

2 1+ν( )K =

E3 1− 2ν( )

VP =K + 4

3 µρ

VS =µρ