Microseismic Monitoring and Geomechanical Modelling of CO2 - bris

CHAPTER 6.
GENERATING ANISOTROPIC SEISMIC MODELS BASED ON GEOMECHANICAL SIMULATION
where
D = (S r 11 + α 11 + β 1111 )(S r 23 + β 2233 ) 2 + (S r 22 + α 22 + β 2222 )(S r 13 + β 1133 ) 2
+(S r 33 + α 33 + β 3333 )(S r 12 + β 1122 ) 2 − 2(S r 12 + β 1122 )(S r 13 + β 1133 )(S r 23 + β 2233 )
−(S r 11 + α 11 + β 1111 )(S r 22 + α 22 + β 2222 )(S r 33 + α 33 + β 3333 ). (6.12)
Based on previous work by Gueguen and Schubnel (2003), Hall et al. (2008) introduce an anisotropic
normalising factor h i , where
h i = 3E i(2 − ν i )
32(1 − ν 2 i ) . (6.13)
When α is multiplied by this factor, a non-dimensional discontinuity density tensor is returned, which
is a function only of discontinuity number density, radius cubed, and orientation distribution,
3∑
h i α ii = ξ c , (6.14)
i=1
where ξ c = N V r3 . N is the number of discontinuities in a volume V , and ξ c is equivalent to the
non-dimensional crack density term used in many effective medium theories, such as Hudson (1981),
Hudson et al. (1996), and Thomsen (1995).
6.3.2 Inversion for scalar cracks
From equation 6.10, we can express the ratio
B N /B T = (1 − ν i /2). (6.15)
Sayers and Kachanov (1995) define the scalar crack as B N /B T ≈ 1 to simplify various expressions
and make elasticity estimates more treatable. In making this simplification, they assume a rock with
low Poisson’s ratio (ν o < 0.2), where β will be at least an order of magnitude smaller than α, and so
can be neglected. In this limit any crack set can be described by considering its contribution to the 3
orthogonal components of the 2nd order tensor, α 11 , α 22 , and α 33 . For instance, a random isotropic
distribution can be described by α 11 = α 22 = α 33 , and transverse symmetry by α 11 > α 22 = α 33 .
Later in this chapter I will discuss the effects of including β in the inversion procedure.
Hall et al. (2008) develop an inversion procedure to determine α based on observed velocity measurements,
assuming β = 0. Equations 6.11 and 6.12 are used to relate the observed stiffness tensor
as determined from velocity measurements to the background stiffness and α. An iterative Newton-
Raphson approach is used, where a Jacobian matrix describes the variation of the modelled stiffness
tensor with α. It is assumed that the velocity measurements are aligned with the principle axes of α.
Figure 6.2 shows the results of this inversion procedure for several samples from the Clair reservoir,
a sandstone reservoir sited on the UK continental shelf. The cores have been taken from depths of
1784, 1788, 1909, 1950, and 2194 meters. The individual samples will be referred to by their depths
hereafter. The background compliances were determined using the geomathematical method described
by Kendall et al. (2007). XRTG (X-Ray Texture Goniometry) and EBSD (Electron Back Scattering
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