Microseismic Monitoring and Geomechanical Modelling of CO2 - bris

6.3. A MICRO-STRUCTURAL MODEL FOR NONLINEAR ELASTICITY
The total additional strain within a volume V due to the presence of a set of discontinuities x is
written
ε ij = S r ijklσ kl + 1
2V
∑
∫
([u i ]n j + [u j ]n i )ds, (6.5)
x
where S r is the background compliance of the rock matrix in the absence of discontinuities. This can
be estimated by calculating the Voigt average moduli based upon individual mineral elasticities and
their relative modal proportions (Kendall et al., 2007) using
( N
) −1
∑
S r = (C r ) −1 = f i C m i , (6.6)
where C r is the effective or average stiffness, f i is the volume fraction of mineral constituent i, and
C m i
s
i=1
is the mineral stiffness. In the absence of mineral stiffness data, the stiffness C r can be estimated
from the behaviour of the rock at high pressures (Sayers, 2002). Equation 6.5 can be rewritten as
ε ij = (S r ijkl + ∆S ijkl )σ kl , (6.7)
where ∆S is the additional compliance caused by the presence of the displacement discontinuities.
For displacement discontinuities that are considered as planar features rotationally invariant around
n, ∆S is given by Sayers and Kachanov (1995) as
∆S ijkl = 1 4 (δ ikα jl + δ il α jk + δ jk α il + δ jl α ik ) + β ijkl , (6.8)
where δ ij is the Kronecker delta. The second and fourth order tensors α and β are given by
α ij = 1 ∑
BT x n x i n x j s x
V
x
β ijkl = 1 ∑
(BN x − BT x )n x i n x j n x
V
kn x l s x . (6.9)
x
BN x and Bx T characterise the normal and tangential compliances across an individual discontinuity
surface. For a planar, penny shaped crack with radius r, in a drained, anisotropic rock with Young’s
modulus E i and Poisson’s ratio ν i (e.g., Turley and Sines, 1971), B N and B T in the direction i normal
to the surface are given by Sayers and Kachanov (1995) as
B N = 16(1 − ν2 i )r
3πE i
, B T = 32(1 − ν2 i )r
3πE i (2 − ν i ) . (6.10)
These equations are equivalent to those provided by Hudson (1981) for penny shaped cracks in the
limit that the infilling material has zero bulk modulus.
Sayers (2002) provides a set of equations describing the stiffness tensor of a rock in terms of the
6×6 compliance matrix S r (the 81 component tensor S r is condensed using Voigt notation), α and β.
Hall et al. (2008) extends these terms to include the presence of an anisotropic background medium
with orthorhombic symmetry, the principle axes of which are aligned with those of α, finding that
[
] ⎫
C ii = (Sjk r + β jjkk) 2 − (Sjj r + α jj + β jjjj )(Skk r + α kk + β kkkk ) /D ⎬
[
]
C ij = (Sij r + β iijj)(Skk r + α kk + β kkkk ) − (Sik r + β iikk)(Sjk r + β jjkk) /D ⎭ i ≠ j ≠ k ≤ 3
C ii = (S r ii + α jj + α kk + 4β jjkk ) −1 i ≠ j ≠ k ≥ 4 (6.11)
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