Microseismic Monitoring and Geomechanical Modelling of CO2 - bris

CHAPTER 6.
GENERATING ANISOTROPIC SEISMIC MODELS BASED ON GEOMECHANICAL SIMULATION
This approach does have its strengths, in particular the reduction of a complex system to 3
empirically determined constants, which facilitates the population of any models we may wish to
build. For this reason the 3rd-order approach is becoming increasingly popular in the industrial
sector (Herwanger, 2007). However, in order to fully define the 3rd order tensor for isotropic samples,
velocity measurements are required at non-hydrostatic stresses (Prioul et al., 2004). As a result, there
is little experimental data of the sort required to determine the 3rd-order terms, and as they are
empirical constructs, they tell us little about the processes that lead to the development of non-linear
elasticity. For this reason I prefer an approach that considers the micro-structure of the rocks in
question.
6.3 A micro-structural model for nonlinear elasticity
A number of micro-structural models exist in the literature (e.g., Zatsepin and Crampin, 1997; Shapiro
and Kaselow, 2005) where variations in micro-structural parameters are defined as a function of stress
and related to the overall elastic properties of the rock via an effective medium model. In this section
we consider the generalised effective medium approach of Schoenberg and Sayers (1995). In this
approach the elasticity of a rock is evaluated in terms of the stiffness of its mineral components and
the presence of low volume displacement discontinuities, which serve to increase compliance. This
model is highly generalised, where few assumptions need to be made about the discontinuities. By
assuming that the discontinuities can be considered as rotationally invariant cracks, we can extend
the model by using a number of methods available in the literature to describe such discontinuities
(e.g., Sayers and Kachanov, 1995; Hudson et al., 1996; Hall, 2000) and how they might vary with
pressure (e.g., Tod, 2002). Following the approach of Tod (2002), I develop a simple model to describe
the change in elasticity of a rock as a function of the stress applied to it. This approach is capable
of considering anisotropy that develops due to both intrinsic rock properties and as a result of nonhydrostatic
stresses. It is also capable of providing a framework within which we might consider
damage due to coring or thermal effects.
6.3.1 Theoretical background
Schoenberg and Sayers (1995) introduce an effective medium approach to describe the compliance, S,
of a damaged rock. This approach is defined in terms of a matrix material and a random distribution
of low volume, poorly bonded discontinuities. When a stress is applied across such a discontinuity,
there will be a difference in displacement between the faces - a displacement discontinuity, [u i ] - that
is proportional to the traction, t i = σ ij n j , on the discontinuity surface s. Hence, for a discontinuity
with normal n, the total displacement discontinuity [u i ] is given by
∫
[u i ]ds ∝ σ ij n j . (6.4)
s
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