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56 Applied Petroleum Geomechanics

s y s y
y y

x x
b y b y
z z
s x s x
b x b x
Figure 2.15 Simplified multiple fracture system for two mutually orthogonal sets of
parallel fractures in the z-direction.

where k z is the permeability change due to aperture increments of Db x and
Db y ; the compressive displacement is positive and tensile displacement is
negative; k 0z is the original permeability in the z-direction under the initial
stress condition; b 0x is the initial average normal aperture of the original
fracture in the x-direction; b 0y is the initial average normal aperture of
the original fracture in the y-direction.
Assuming an idealized three-dimensional regularly spaced fracture-
matrix system, as illustrated in Fig. 2.16, the change of the fracture aper-
ture along the x-direction due to the normal stress changes can be obtained
(Zhang, 2002; Zhang et al., 2007). Similarly, the change of the fracture
aperture along the y-direction can also be obtained. Therefore, the change
of permeability in the z-direction in two mutually orthogonal sets of
fractures caused by 3-D stress changes can be obtained (Zhang et al., 2007):

1 1 1
k z ¼ k z0 1 þ þ ½Ds x nðDs y þ Ds z ފ
K nx b x K nx s x E r
(2.43)
3
1 1 1
þ þ ½Ds y nðDs x þ Ds z ފ
K ny b y K ny s y E r
Δσ
z

Δσ Δσ x
x matrix
Δσ Kn
K n y
b b
s x x s x x s x z
y
x
Figure 2.16 Coupled fractureematrix system with 3-D stresses for a set of parallel
fractures (Zhang, 2002).