Page 80 - Reservoir Geomechanics

64 Reservoir geomechanics

Table 3.1. Relationships among elastic moduli in an isotropic material

K E λ ν G M
2G 3λ + 2G λ
λ + G – – λ + 2G
3 λ + G 2 (λ + G)
K − λ λ K − λ
– 9K – 3 3K − 2λ
3K − λ 3K − λ 2
9K − G 2G 3K − 2G G
– K − – K + 4
3K − G 3 2(3K + G) 3
εG E − 2G E 4G − E
– G − 1 – G
3(3G − E) 3G − E 2G 3G − E
3K − E 3K − E 3KE 3K + E
– – 3K 3K
9K − E 6K 9K − E 9K − E
1 + ν (1 + ν)(1 − ν) 1 − 2ν 1 − ν
λ λ – – λ λ
3ν ν 2ν ν
2 (1 + ν) 2ν 2 − 2ν
G 2G (1 + ν) G – – G
3 (1 − 2ν) 1 − 2ν 1 − 2ν
ν 1 − 2ν 1 − ν
– 3K(1 − 2ν) 3K – 3K 3K
1 + ν 2 + 2ν 1 + ν
E Ev E E (1 − ν)
– –
3 (1 − 2ν) (1 + ν)(1 − 2ν) 2 + 2ν (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 = 0ina fluid.
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:
4G
2
M = V ρ = K +
p
3
Poisson’s ratio can be determined from V p and V s utilizing the following relation

2
V − 2V s 2
p
ν = (3.6)
2
2 V − V 2
p s
Because we are typically considering porous sedimentary rocks saturated with water,
oilorgasinthisbook,itisimportanttorecallthatporoelasticeffectsresultinafrequency
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.