Microseismic Monitoring and Geomechanical Modelling of CO2 - bris

6.5. COMPARISON OF ROCK PHYSICS MODELS
that the response of velocities to stress change is in fact nonlinear, so a linear interpolation is in fact
a poor approach to take.
The approach most commonly used in industry is the R-factor model of Hatchell and Bourne (2005).
This model assumes that the fractional change in vertical P-wave velocity is proportional to the vertical
strain (with the R-factor being the constant of proportionality in the equation dV P z /V P z = −Rε zz ).
Vertical P-wave velocities are the most commonly measured property in conventional seismic surveys,
and changes in vertical stress (and strain) will be largest geomechanical effect above a compacting
reservoir. The R-factor approach was developed to address this particularly relevant subset of geomechanical
scenarios. Hatchell and Bourne (2005) found that R-factors were reasonable consistent
across a number of sites, although R-factors for rocks experiencing compressive strain were found to
be 5 times smaller than for rocks under extensional strain. However, more recent studies have shown
order-of-magnitude variations in R-factors depending on lithology (e.g., Staples et al., 2007; De Gennaro
et al., 2008). More importantly, R-factors have been found to be dependent on the stress path
(Holt et al., 2008), and on the magnitude of the applied stress (Pal-Bathija and Batzle, 2007).
The different R-factors required for extension and compaction, for different applied stress magnitudes
and for different triaxial stress states, all for the same rock, means that this approach does
not lend itself to model scenarios where the stress changes during production are not known in advance.
This is because the R-factor model does not adequately describe the full, triaxial, anisotropic,
nonlinear response of seismic velocities - it is limited to vertical strain and vertical P-wave velocities.
Therefore it is not capable of dealing with the observed phenomena that fall beyond its remit. Unfortunately,
these issues are very much within my remit when dealing with the geomechanical models
developed in Chapter 5, so I do not use the R-factor approach. Nevertheless, because it cuts through
a complicated system to leave one of the key 4D seismic parameters (vertical timeshift) correlated via
one parameter (the R-factor) to one geomechanical observable (vertical strain) it remains as attractive
approach within the industrial sector.
Like many rock physics models, the R-factor approach attempts to fit the observed nonlinear stressvelocity
response with a linear model, which means that it is limited in its applicability. Furthermore,
the model only considers the vertical P-wave velocity response to vertical strain. Observations show
that vertical P-wave velocity is also modulated by horizontal deformation, although this is a second
order effect (this result can be derived from equation 6.11). More importantly, even in reflection surveys,
seismic waves do not travel vertically, but through a range of inclinations about the subvertical.
Therefore the R-factor approach tends to break down for longer offset arrivals (Herwanger, 2007).
The third-order elasticity model developed by Prioul et al. (2004) includes the effects of triaxial
stress changes on the full anisotropic stiffness tensor. This means that the effects of stress-induced
anisotropy, and variations in larger offset timeshifts, can be incorporated. This model takes a mathematical
approach, including the third-order terms that are usually discarded in deriving conventional
linear elastic theory. Therefore, this model does not directly address the microstructual properties of
the rock, although it is capable of modelling the full stiffness tensor. However, to facilitate inversion
from ultrasonic data, Prioul et al. (2004) have to assume an isotropic third order tensor, which in
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