Page 28 - Applied Petroleum Geomechanics
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18 Applied Petroleum Geomechanics
and from Eq. (1.24) the stressestrain relations are as follows:
E Ea T DT
s x ¼ ½ð1 nÞε x þ nε y þ
ð1 þ nÞð1 2nÞ ð1 2nÞ
E Ea T DT
s y ¼ ½ð1 nÞε y þ nε x þ
ð1 þ nÞð1 2nÞ ð1 2nÞ
(1.32)
s z ¼ nðs x þ s y Þþ Ea T DT
Enðε x þ ε y Þ Ea T DT
s z ¼ þ
ð1 þ nÞð1 2nÞ ð1 2nÞ
s xy ¼ 2Gε xy
if the thermal effect is not considered, then s z ¼ n(s x þs y ).
Plane strain state is often applicable to very long or thick structures,
where the length of the structure is much greater than the other two di-
mensions (e.g., Zhang et al., 2018). It is applicable to boreholes, hydraulic
fractures, and two-dimensional openings. For instance, Fig. 1.12 shows a
classic hydraulic fracture model (the PKN model), where the fracture is very
long in y-direction. The PKN model assumes a plane strain deformation in
the vertical plane, i.e., each vertical cross section acts independently; i.e., the
z y
Fracture tip
L
H w(x)
x
Figure 1.12 Plane strain wellbore model of the PKN fracture for simplifying the 3-D
problem.
