reservoir geomecanics

64 Reservoir geomechanics
Table 3.1. Relationships among elastic moduli in an isotropic material
K E λ ν G M
λ + 2G 3
3λ + 2G
G
λ + G
– 9K K − λ
3K − λ
–
εG
3(3G − E)
9K − G
3K − G
–
–
K − 2G 3
– G E − 2G
3G − E
– – 3K 3K − E
9K − E
λ 1 + ν
3ν
2 (1 + ν)
G
3 (1 − 2ν)
(1 + ν)(1 − ν)
λ
ν
2G (1 + ν) G 2ν
1 − 2ν
– 3K (1 − 2ν) 3K ν
1 + ν
E
3 (1 − 2ν)
–
λ
2 (λ + G)
λ
3K − λ
3K − 2G
2(3K + G)
– λ + 2G
3 K − λ
2
3K − 2λ
– K + 4 G 3
E
2G − 1 – G 4G − E
3G − E
3K − E
6K
3KE
9K − E
– – λ 1 − 2ν
2ν
Ev
(1 + ν)(1 − 2ν)
3K 3K + E
9K − E
λ 1 − ν
ν
– – G 2 − 2ν
1 − 2ν
– 3K 1 − 2ν
2 + 2ν
–
E
2 + 2ν
3K 1 − ν
1 + ν
E (1 − ν)
(1 + ν)(1 − 2ν)
It is obvious from these relations that V p is always greater than V s (when ν = 0.25,
V p /V s = √ 3 = 1.73) and that V s = 0inafluid.
It is also sometimes useful to consider relative rock stiffnesses directly as determined
from seismic wave velocities. For this reason the so-called M modulus has been defined:
M = V 2
p ρ = K + 4G 3
Poisson’s ratio can be determined from V p and V s utilizing the following relation
ν = V p 2 − 2V s
2
( ) (3.6)
2 Vp 2 − V s
2
Because we are typically considering porous sedimentary rocks saturated with water,
oil or gas in this book, it is important to recall that poroelastic effects result in a frequency
dependence of seismic velocities (termed dispersion), which means that elastic moduli
are frequency dependent. This is discussed below in the context of poroelasticity and
viscoelasticity.