Page 34 - Applied Petroleum Geomechanics

24 Applied Petroleum Geomechanics

where G 1 ¼ E 1 /[2(1þn 12 )]; Poisson’s ratios are not symmetric, but satisfy
n 31 /E 3 ¼ n 13 /E 1 .
Hooke’s law in the VTI rock can be expressed as follows:
2 3 2 3
ε 11 1=E 1 n 12 =E 1 n 31 =E 3 0 0 0
ε n 12 =E 1 1=E 1 n 31 =E 3 0 0 0
6 7 6 7
6 22 7 6 7
6 7 6 7
6 ε 33 7 6 n 13 =E 1 n 13 =E 1 1=E 3 0 0 0 7
6 7 ¼ 6 7
0 0 0 0 0
6 7 6 7
6 2ε 23 7 6 1=G 3 7
6 7 6 7
0 0 0 0 0
4 2ε 13 5 4 1=G 3 5
2ε 12 0 0 0 0 0 1=G 12
2 3 2 3
s 11 a T1
6 7 6 7
6 s 22 7 6 a T1 7
6 7 6 7
6 s 33 7 6 a T3 7
6 7 DT 6 7
0
6 7 6 7
6 s 23 7 6 7
6 7 6 7
4 0
4 s 13 5 5
0
s 12
(1.41)
The minimum and maximum horizontal stresses in the VTI rock can be
derived from Eq. (1.41). In the principal stress state, shear stresses and shear
strains are zero; therefore, the first two equations in Eq. (1.41) can be
rewritten as follows:
s 1 n 12 s 2 n 31 s 3
ε 1 ¼ a T1 DT (1.42)
E 1 E 1 E 3
s 2 n 12 s 1 n 31 s 3
ε 2 ¼ a T1 DT (1.43)
E 1 E 1 E 3
where s and ε are principal stresses and strains, respectively.
Solving Eqs. (1.42) and (1.43), the principal stresses (s 1 and s 2 ) can be
obtained:
E 1 n 31 E 1 E 1 n 12 E 1 a T1
s 1 ¼ s 3 þ 2 ε 1 þ 2 ε 2 þ DT (1.44)
E 3 ð1 n 12 Þ 1 n 1 n
12 12 1 n 12
E 1 n 31 E 1 E 1 n 12 E 1 a T1
s 2 ¼ s 3 þ 2 ε 2 þ 2 ε 1 þ DT (1.45)
E 3 ð1 n 12 Þ 1 n 12 1 n 12 1 n 12
where the principal stresses, Young’s moduli, and Poisson’s ratio are illus-
trated in Fig. 1.15. If the two horizontal axes (1 and 2) and the vertical