Page 32 - Applied Petroleum Geomechanics

22 Applied Petroleum Geomechanics

1.5.1 Orthotropic elastic rocks
Orthotropy (orthorhombic symmetry) is a simplification of the anisotropy.
An orthotropic rock has three mutually perpendicular symmetry planes at
each point in the rock, and these planes have the same orientation
throughout the rock (Amadei et al., 1987; Amadei, 1996). This type of rock
has nine independent material constants. For a rock that is orthotropic in a
local 1, 2, 3 Cartesian coordinate system attached to clearly defined planes
of anisotropy (refer to Fig. 1.14), the elastic compliance matrix can be
expressed as follows:
2 0 0 0 3
1=E 1 n 21 =E 2 n 31 =E 3
6 0 0 0 7
n 12 =E 1 1=E 2 n 32 =E 3
6 7
6 7
6 n 13 =E 1 n 23 =E 2 1=E 3 0 0 0 7
S ¼ 6 7 (1.37)
0 0 0 1=G 23 0 0
6 7
6 7
0 0 0 0 0
6 7
4 1=G 13 5
0 0 0 0 0 1=G 12
where E 1 , E 2 , and E 3 are Young’s moduli in the 1, 2, and 3 directions,
respectively; G 12 , G 13 , and G 23 are the shear moduli in planes parallel to
the 12, 13, and 23 planes, respectively; n ij (i, j ¼ 1, 2, 3) are Poisson’s ratios
that characterize the normal strains in the symmetry directions j when a
stress is applied in the symmetry directions i. Because of symmetry of the
compliance matrix, Poisson’s ratios n ij and n ji are such that n ij /E i ¼ n ji /E j
(Amadei, 1996).
For the orthotropic rock the thermal expansion coefficient tensor has
the following form:
2 3
0 0
a T1
a ¼ 4 0 a T2 0 5 (1.38)
7
6
0 0 a T3

3
2

1

Figure 1.14 An orthotropic rock (a layered rock formation) with three planes of
symmetry normal to the axes (the 1, 2, 3 directions).