Page 69 - Applied Petroleum Geomechanics

60 Applied Petroleum Geomechanics

spherical grains, the change of grain radius due to effective stresses changes of
Ds x , Ds y ,and Ds z can be obtained from the following equations:
( ( " #)1)
2 2 2 2 3
1 9ð1 n Þ pDs x pDs y pDs z
R ¼ R 0 1
2 2 E E E
(2.46)

where the positive sign is for the compressive stress, and negative sign is for
the tensile stress.
Substituting Eq. (2.46) to Eq. (2.45), the change in permeability by
normal stresses can be expressed as follows:
1 2
2

1 9p ð1 n Þ 2 2 2 3
2
k ¼ k 0 1 Ds Ds Ds z (2.47)
x
y
2 2E 2
where k 0 is the initial permeability. This relationship may be applicable for
analyzing the stress-dependent permeability of the proppants in hydraulic
fractures as described in the previous section. Hydraulic fracture conductiv-
ity changes induced by stress changes are primarily caused by the following
reasons:
(1) Reduction in pore pressure due to production depletion increases the
effective overburden stress, causing the formation compaction, fracture
deformation, even closure of small fractures.
(2) Proppant deformation, embedment, crushing, and fines migration cause
reduction in the pore spaces and permeability of the proppants in the
hydraulic fractures.

2.5 Young’s modulus

Young’s modulus is an important parameter to define the relationship
between stress and strain in a material in the linear-elastic deformation. It
generally refers to the static Young’s modulus, which can be obtained from
laboratory core tests, e.g., uniaxial or triaxial compression test. The dynamic
Young’s modulus can be calculated from theoretical equations using the
acoustic velocity data. However, for geomechanics analysis, static Young’s
modulus is needed. That is, in the stressestrain constitutive equations (refer
to Chapter 1), static Young’s modulus is required for the calculations.
Therefore, if only dynamic modulus is available, then it needs to be con-
verted to the static modulus. Some converting correlations will be intro-
duced in the following sections.