Page 79 - Reservoir Geomechanics

63 Basic constitutive laws

Vutukuri 1978). Static means that measurements were made in a manner similar to the
idealized experiments illustrated in Figure 3.3. Such measurements are distinguished
from dynamic measurements of elastic moduli utilizing laboratory seismic velocity
6
measurements at ultrasonic frequency (∼10 Hz), which are discussed below. As seen
in Figure 3.4, relatively low-porosity sandstones (0–6%) have Young’s moduli around
50 MPa and Poisson’s ratios of ∼0.125, whereas relatively high-porosity sandstones
(>16%) have appreciably lower Young’s moduli and higher Poisson’s ratios. Similarly,
relatively low porosity limestones have very high Young’s moduli (∼80 MPa), whereas
higher porosity limestones have appreciably lower stiffness. The relation between
porosity and Poisson’s ratio for limestones is less clear than for the sandstone sam-
ples because most values are 0.2–0.3, irrespective of porosity. So few shale samples
are in this data set that it is difficult to generalize about any relation between porosity
and Poisson’s ratio.
An important aspect of the theory of elasticity in homogeneous, isotropic material is
that only two elastic moduli are needed to describe fully material behavior. Strain can
be expressed in terms of applied stress utilizing the following relation

1 1
ε ij = (S ij − δ ij S 00 ) + δ ij S 00 (3.4)
2G 3K
Because only two elastic moduli are needed, it is often convenient to express elastic
moduli with respect to each other. For example, if one has knowledge of the bulk
modulus and Young’s modulus, one can compute the shear modulus from
3KE
G =
9K − E
Afairlycompletetableofsuchequivalencies(afterBirch1961)ispresentedinTable3.1.
It should be noted that the crack closure associated with initial loading of a sample
at very low stress (as illustrated in Figure 3.2)isanexample of non-linear elasticity.
As the cracks close with increased load the sample becomes stiffer, and as the stress
on the sample decreases so does the stiffness of the sample, in a repeatable manner.
Hence, stress and strain are proportional and deformation is fully reversible, but stress
and strain are not linearly proportional.

Elastic moduli and seismic wave velocity

In an elastic, isotropic, homogeneous solid the elastic moduli also can be determined
from the velocity of compressional waves (V p ) and shear waves (V s ) using the following
relations

K + 4G/3 G
V p = V s = (3.5)
ρ ρ