Microseismic Monitoring and Geomechanical Modelling of CO2 - bris

CHAPTER 3.
INVERTING SHEAR-WAVE SPLITTING MEASUREMENTS FOR FRACTURE PROPERTIES
Fully connected fractures
An alternative approach is to consider the low frequency limit (Hudson et al., 2001). In this limit,
there will be enough time for the pressure gradient to be completely equalised between fractures
and pores. In this limit I treat the fractures as being drained. The widely known formulations of
Gassmann (1951) can then be used to compute the effects of fluid saturation on the overall system.
The Gassmann equations have been generalised for anisotropic rocks by Brown and Korringa (1975),
who find that the fluid saturated compliance is given by
S sat
ijkl = Sijkl d − (Sd ijαα − Sm ijαα )(Sd klαα − Sm klαα )
( 1
K
) + Φ( 1
m K fl
− 1
K
) m
K d − 1
(3.9)
where Sijkl m and Km are the compliance tensor (in uncontracted 3×3×3×3 form) and bulk modulus
of the minerals making up the rock, Sijkl d and Kd are the compliance tensor and bulk modulus of the
dry rock frame, and Φ is the porosity.
Restricted fluid flow
I have now described the two endmembers, high and low frequency, that correspond to a fully connected
pore space and an isolated pore space. To model between the low and high frequency endmembers
then frequency dependence must be factored into the calculations. Hudson et al. (1996) present
an extension to the Hudson (1981) model which can account for flow between fractures and equant
porosity. As discussed previously, fluid flow will affect the normal compliance of the fracture, so a
correction is made to the Hudson (1981) term for B N , such that K is now
K = K fl (λ r + 2µ r ) 1
πaµ r (λ r + µ r ) 1 + (3(1 − i)J/2c)
(3.10)
where
J 2 = K flΦκ
2ηω . (3.11)
κ is the permeability of the rock, η is fluid viscosity, ω is the frequency of the incident wave, and c is
the average fracture aperture. Note that, because strain parallel to a fracture does not cause a volume
change, the tangential compliance of the fracture is not affected by fluid flow, and does not need to
be modified.
K can now be considered in terms of two parameters, a fluid incompressibility factor P i and an
equant porosity factor P ep (Pointer et al., 2000), such that
where
K =
(
Pi
π
λ r + 2µ r ) ( ) −1
3(1 − i)
λ r + µ r 1 +
2 √ (3.12)
P ep
P i = 1 K fl
a µ r ,
P ep = ( c J )2 = 2ωη fl
ΦK fl κ c2 . (3.13)
32