Microseismic Monitoring and Geomechanical Modelling of CO2 - bris

6.3. A MICRO-STRUCTURAL MODEL FOR NONLINEAR ELASTICITY
Diffraction) were used to asses the preferred orientation of anisotropic minerals, or crystal preferred
orientation (CPO), and mineral modal proportions were measured using QXRD (Quantitative X-Ray
Diffraction). The left panels show the best fit crack density α values normalised by h i , and the right
panels compare back-calculated velocities to the observed velocities. The back-calculated velocities
in general show a reasonable fit with observed velocities, especially for the P-waves (V P ). The fit for
V P 45 is poor. Hall et al. (2008) suggest that this may be a result of difficulties in cutting and analysing
the core at 45 ◦ .
6.3.3 Joint inversion for α and β
For the scalar crack assumption to be appropriate, the rocks must have a low Poisson’s ratio, which is
generally acceptable for reservoir rocks, and the cracks must be flat, poorly bonded features. If there
are significant amounts of diagenetic clay or debris within the cracks then equation 6.15 may not be
valid. In order to model how cracks are influenced by pressure, I make the assumption that they are
planar, penny shaped features without any fill (see next section). By analysing the contribution of β,
I can assess how appropriate this assumption is.
Hall et al. (2008) provide a method for estimating β from ultrasonic velocity measurements, based
on Sayers (2002); however, this method assumes that the contribution from β is small, and is responsible
solely for the misfit between observed and back-calculated velocities from the inversion for α. I
wish to test the assumption that β is small, and so I develop an inversion procedure where β is not
required a priori to be small.
I assume that the cracks are disc-shaped, identical and that β is isotropic (to do otherwise introduces
impractical complexity given that my principle aim is to evaluate the magnitude of β rather
than its orientation distribution). I note that for such a distribution, I can rewrite equation 6.9 for α
and β (Sarout et al., 2007)
α ij = πNr2
3V
B T δ ij
The non-vanishing components of α and β are
β ijkl = πNr2
15V (B N − B T )(δ ij δ kl + 2(δ ik δ jl + δ il δ jk )). (6.16)
α 11 = α 22 = α 33 = 1 3 α mm,
β 1111 = β 2222 = β 3333 , (6.17)
β 1122 = β 1133 = β 2233 = β 1212 = β 1313 = β 2323 = 1 3 β 1111,
where
α 11 = πNr2
3V B T , (6.18)
( )
β 1111 = πNr2
3V
B BN
T − 1 , (6.19)
B T
111