Rock Physics
Elasticity
Hooke’s law
Strain ε (dimensionless : ε=∂uu, where u is a distance). Stress σ (Pa, equivalent to N.m−2) Modulus of elasticity or Young’s Modulus E (Pa, equivalent to N.m−2) σ=E.ε
Tensors
σxy=(σ11σ12σ13 σ21σ22σ33 σ31σ32σ33 ),εxy=(ε11ε12ε13 ε21ε22ε33 ε31ε32ε33 )Symmetry implies σij=σji and εij=εji which leaves 6 unique terms in both tensors.
Cauchy Stress Tensor. By Sanpaz (Own work) CC BY-SA 3.0 or GFDL], via Wikimedia Commons
Elastic material constants
Young’s Modulus , (Pa⇔N.m−2⇔kg.m−1.s−2).
E=σ11ε11As a function of seismic velocities and density: E=ρV2s(3V2p−4V2s)V2p−V2s As a fonction of P and S Impedances: Eρ=I2s(3I2p−4I2s)I2p−I2s
Poisson’s Ratio (dimensionless : ratio of strains).
ν=ε33ε11As a function of seismic velocities and impedances ν=12V2pV2s−2V2pV2s−1=12I2pI2s−2I2pI2s−1
Shear Modulus μ , (Pa⇔N.m−2⇔kg.m−1.s−2).
μ=12.σ13ε13Bulk Modulus K , (Pa⇔N.m−2⇔kg.m−1.s−2).
K=σ00ε00=σ11+σ22+σ33ε11+ε22+ε33Seismic velocities as a function of Lame parameters
P-Wave velocity (m.s−1)
Vp=√K+43μρ=√λ+2μρS-Wave velocity (m.s−1)
Vs=√μρLame parameters (Pa⇔N.m−2⇔kg.m−1.s−2):
μ=V2sρ λ=ρ(V2p−2V2s)Stiffness pmatrix - Voigt notation
In the anisotropic case, the relation between stress and strain is a 6 by 6 tensor. (σ1 σ2 σ3 σ4 σ5 σ6 )=(c11c12c13c14c15c16 c21c22c23c24c25c26 c31c32c33c34c35c36 c41c42c43c44c45c46 c51c52c53c54c55c56 c61c62c63c64c65c66 ).(ε1 ε2 ε3 ε4 ε5 ε6 )
where (σ1 σ2 σ3 σ4 σ5 σ6 )=(σ11 σ22 σ33 σ23 σ13 σ12 ),and,(ε1 ε2 ε3 ε4 ε5 ε6 )=(ε11 ε22 ε33 ε23 ε13 ε12 )
Stiffness matrix - Isotropic Material
In the isotropic case, the relation between stress and strain depends only on 2 parameters : c11 and c44. The isotropic solid has no preferred direction regarding elastic properties. C=(c11c11−2c44c11−2c44000 c11−2c44c11c11−2c44000 c11−2c44c11−2c44c11000 000c4400 0000c440 00000c44 ) C=(λ+2μλλ000 λλ+2μλ000 λλλ+2μ000 000μ00 0000μ0 00000μ )
Stiffness matrix - VTI Material
Material with a vertical axis of symmetry (layer cake), 5 parameters in the stiffness matrix. C=(c11c11−2c66c13000 c11−2c66c11c13000 c13c13c33000 000c4400 0000c440 00000c66 ) VPH=√c11ρ,VPV=√c33ρ,VSH=√c44ρ,VSV=√c66ρ;
Stiffness matrix - HTI Material
Material with an horizontal axis of symmetry, this is the transposed of the VTI case, 5 parameters. C=(c11c13c13000 c13c33c33−2c44000 c13c33−2c44c33000 000c4400 0000c550 00000c55 ) VPH=√c11ρ,VPV=√c33ρ,VSH=√c44ρ,VSV=√c55ρ;
Stiffness matrix - Orthorhombic Material
Material with three mutually orthogonal axis of symmetry : 9 parameters in the stiffness matrix. C=(c11c12c13000 c12c22c23000 c13c23c33000 000c4400 0000c550 00000c66 ) VPH1=√c11ρ,VPH2=√c22ρ,VPV=√c33ρ; VSH1=√c44ρ,VSH2=√c55ρ,VSV=√c66ρ;
Olivier SEISMIC_RESERVOIR
anisotropy rock physics elasticity modeling