Microseismic Monitoring and Geomechanical Modelling of CO2 - bris

3.2. INVERSION METHOD
the additional compliance approach of Schoenberg and Sayers (1995), where the compliance of the
fractures, ∆S, is added to the rock frame compliance, S r , to give the overall compliance,
S = S r + ∆S. (3.1)
Symmetry arguments reduce the number of independent terms in the 3×3×3×3 stiffness tensor from
81 to 21 components. To simplify notation, the Voigt system can be used to contract C into a 6×6
matrix, where the i, j and k, l subscripts of the 3×3×3×3 tensor are mapped to m and n following
the convention
ij or kl 11 22 33 23 = 32 13 = 31 12 = 21
↓ ↓ ↓ ↓ ↓ ↓ ↓ .
m or n 1 2 3 4 5 6
The rock frame compliance S r can be anisotropic if horizontal layering is present. Based on
previous estimates of sedimentary fabric anisotropy in siliclastic rocks (Kendall et al., 2007) I consider
the rock fabric to have VTI symmetry. For such a system, the frame compliance tensor (in Voigt
notation) is given by
⎛
C11 r (C11 r − 2C66) r C13 r 0 0 0
(C 11 r − 2C66) r C11 r C13 r 0 0 0
S r C13 r C13 r C33 r 0 0 0
=
0 0 0 C44 r 0 0
⎜
⎝ 0 0 0 0 C44 r 0
0 0 0 0 0 C r 66
⎞
⎟
⎠
−1
(3.2)
. (3.3)
The strength of the shear wave anisotropy caused by the VTI system is given by Thomsen’s (1986) γ
and δ parameters, defined as
and
γ = Cr 66 − C r 44
2C r 44
(3.4)
δ = (Cr 13 + C r 44) 2 − (C r 33 − C r 44) 2
2C r 33 (Cr 33 − Cr 44 ) . (3.5)
The additional compliance introduced by a set of vertical, aligned fractures in a VTI medium with
normals parallel to the x 1 axis (n = [1, 0, 0]) is given by Grechka (2007) as
⎛
⎞
B N 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
∆S =
. (3.6)
0 0 0 0 0 0
⎜
⎟
⎝ 0 0 0 0 B T v 0 ⎠
0 0 0 0 0 B T h
B N
is the normal compliance of the fracture, and B T h and B T v are the shear compliances in the
vertical and horizontal planes. Having computed the stiffness tensor for fractures aligned in the x 2 :x 3
plane I rotate this tensor to give the stiffness tensor for fractures with the desired strike. B T h and
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