Page 35 - Applied Petroleum Geomechanics

Stresses and strains 25

σ σ 3
ε 1 ε 1

σ σ
Ε 3 , ν 31 1 Ε 1 , ν 12 ν 13 1
3 ε
2 2
ε 3 ε
σ 3 1 3
Figure 1.15 Two compression tests in vertical and horizontal directions to obtain all
Young’s moduli and Poisson’s ratios in the rock withvertical transverse isotropy.

axis (3-axis) are represented by h, H, and V (e.g., E h ¼ E 1 , E V ¼ E 3 ,
n h ¼ n 12 , n V ¼ n 31 , a Th ¼a T1 ), and for a porous rock the effective stresses
replace the total stresses, then the above equations can be rewritten as
the following forms for the effective in situ stresses:

E h n V E h E h n h E h a Th
0 0 DT
s ¼ s þ ε h þ ε H þ (1.46)
h V 2 2
E V ð1 n h Þ 1 n h 1 n h 1 n h
E h n V E h E h n h E h a Th
0 0 DT
s ¼ s þ 2 ε H þ 2 ε h þ (1.47)
V
H
E V ð1 n h Þ 1 n h 1 n h 1 n h
0
where s and s 0 H are the minimum and maximum effective horizontal
h
0
stresses, respectively; s V is the effective vertical stress; E h and E V are
Young’s moduli in horizontal and vertical directions, respectively; n h and
n V are Poisson’s ratios in horizontal and vertical directions, respectively;
ε h and ε H are the strains in the minimum and maximum horizontal stress
directions, respectively.
Substituting Eq. (1.9) of the effective stress law into above equations,
the minimum and maximum horizontal stresses can be obtained, i.e.,
E h n V E h E h n h E h a Th
s h ¼ ðs V a V p p Þþ a h p p þ 2 ε h þ 2 ε H þ DT
E V ð1 n h Þ 1 n h 1 n h 1 n h
(1.48)
E h n V E h E h n h E h a Th
s H ¼ ðs V a V p p Þþ a h p p þ ε H þ ε h þ DT
E V ð1 n h Þ 1 n 2 h 1 n 2 h 1 n h
(1.49)