Microseismic Monitoring and Geomechanical Modelling of CO2 - bris

6.3. A MICRO-STRUCTURAL MODEL FOR NONLINEAR ELASTICITY
and α mm is the trace of α (α mm = α 11 + α 22 + α 33 ). By rearranging equations 6.18 and 6.19 I can
rewrite β in terms of α and the ratio B N /B T , such that
β 1111 = 1 ( )
BN
− 1 α mm . (6.20)
3 B T
Substitution of this relationship into equations 6.11 and 6.12 yields the overall stiffness as a function
of both the second and fourth order crack density tensors, by way of the ratio B N /B T ,
[
] ⎫
C ii = (Sjk r + f 9 αm ) 2 − (Sjj r + α jj + f 3 αm )(Skk r + α kk + f 3 αm ) /D ⎬
[
]
C ij = (Sij r + f 9 αm )(Skk r + α kk + f 3 αm ) − (Sik r + f 9 αm )(Sjk r + f 9 αm ) /D ⎭ i ≠ j ≠ k ≤ 3
C ii = (S r ii + α jj + α kk + 4f 9 αm ) −1 i ≠ j ≠ k ≥ 4 (6.21)
where
D = (S r 11 + α 11 + f 3 αm )(S r 23 + f 9 αm ) 2 + (S r 22 + α 22 + f 3 αm )(S r 13 + f 9 αm ) 2
+(S r 33 + α 33 + f 3 αm )(S r 12 + f 9 αm ) 2
−2(S r 12 + f 9 αm )(S r 13 + f 9 αm )(S r 23 + f 9 αm )
−(S r 11 + α 11 + f 3 αm )(S r 22 + α 22 + f 3 αm )(S r 33 + α 33 + f 3 αm ) , (6.22)
f =
( )
BN
− 1 . (6.23)
B T
Having defined a set of equations for the overall stiffness with the fourth order crack density tensor
written in terms of the second order crack density tensor and B N /B T
ratio, the inversion for crack
density and B N /B T can then be performed using an iterative Newton-Raphson approach with model
update:
The vector model misfit is evaluated
δb l = C obs
l
where the matrix Jacobian
is inverted to evaluate the vector model update
α ii = α ii + δm i , i = 1, 3. (6.24)
− C model
l l = 11, 22, 33, 44, 55, 66, 12, 13, 23 , (6.25)
J il = ∂Cmodel l
(6.26)
∂α i
δm i = J −1
il
δb l . (6.27)
The matrix Jacobian of derivatives with respect to the second order crack density tensor α ii is
evaluated using the following set of equations below. The partial derivatives of the denominator term
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