Microseismic Monitoring and Geomechanical Modelling of CO2 - bris

6.3. A MICRO-STRUCTURAL MODEL FOR NONLINEAR ELASTICITY
6.3.4 Effects of stress on crack density
We know from micro-structural analysis (e.g., Batzle et al., 1980) that cracks and discontinuities are
complex features, with rough walls, nonlinear geometry, irregular intersections, and clay or diagenetic
infill; however, in the previous section I have shown that by modelling them as highly simplified,
rotationally invariant, smooth (penny-shaped), empty features we can still approximate the effective
rock properties to a reasonable degree of accuracy. I use this observation to my advantage in order to
predict how the effective properties will be influenced by an applied stress field. In effect, this model
assumes that the microcracks within the rock will respond to the long duration, high strain, finite
strain-rate deformation imposed by geomechanical effects in the same manner as they do to the short
duration, infinitesimal strain, high strain-rate deformation imposed by the passage of a seismic wave.
Therefore this assumption will only be appropriate as long as the rock doesn’t undergo any plastic or
brittle failure during geomechanical deformation.
Hudson (2000) and Tod (2002) present analytical models where the aspect ratio and number
density of cracks is dependent upon applied stress and fluid pressure. This model considers elastic
deformation only, where the permanent deformation of pores into cracks and the development of new
cracks is not considered; hence, when the stress state is returned to its original magnitude, the material
will relax to its reference state.
Hudson (2000) derives an expression for the change in aspect ratio, δa, of a penny-shaped crack
due to a change in applied stress and/or fluid pressure,
δa = − 2(1 − νr )
πµ r (δσ ij n i n j − β w δP fl ) − a K r β wδP fl , (6.33)
where δσ and δP fl are the change in applied stress tensor and fluid pressure, β w is the Biot-Willis
parameter, assumed here for simplicity to be unity, n is the crack normal, and µ r , ν r and K r are
respectively the shear modulus, Poisson’s ratio and bulk modulus of the matrix in the absence of
compliant porosity. The right-hand term
a K r
β w δP fl of equation 6.33 is small in comparison with the
other terms and can be neglected (Hudson, 2000). Integrating equation 6.33 gives
a = a 0 − 2(1 − νr )
πµ r σ c(n) , (6.34)
where a 0 is the aspect ratio in the absence of an applied stress (or at a pre-defined reference stress).
The effects of applied stress and pore pressure combine to give the effective crack normal stress
σ c(n) = σ ij n i n j − β w P fl . (6.35)
For a crack with an initial aspect ratio a 0 , there will be a critical stress where
a 0 = 2(1 − νr )
πµ r σ c(n) , (6.36)
and the crack can be considered as closed. Tod (2002) assumes an exponential distribution of initial
aspect ratios. If this is the case, then crack density will decrease exponentially with pressure due to
crack closure (van der Neut et al., 2007), so that
ξ(σ c ) = ξ 0 exp (−c r σ c(n) ), (6.37)
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