Page 77 - Reservoir Geomechanics

61 Basic constitutive laws

Linear elasticity

The theory of elasticity typically is discussed in terms of infinitesimally small defor-
mations. In this case, no significant damage or alteration of the rock results from an
applied stress and the assumption that stress and strain are linearly proportional and
fully reversible is likely to be valid. In such a material, stress can be expressed in terms
of strain by the following relation

S ij = λδ ij ε 00 + 2Gε ij (3.3)
where the Kronecker delta, δ ij ,isgivenby

δ ij = 1 i = j
δ ij = 0 i
= j

Upon expansion, equation (3.3) yields

S 1 = (λ + 2G) ε 1 + λε 2 + λε 3 = λε 00 + 2Gε 1
S 2 = λε 1 + (λ + 2G) ε 2 + λε 3 = λε 00 + 2Gε 2
S 3 = λε 1 + λε 2 + (λ + 2G) ε 3 = λε 00 + 2Gε 3

and λ (Lame’s constant), K (bulk modulus) and G (shear modulus) are all elastic
moduli.
There are five commonly used elastic moduli for homogeneous isotropic rock. As
illustrated in Figure 3.3, the most common is the bulk modulus, K, which is the stiffness
of a material in hydrostatic compression and given by

S 00
K =
ε 00
The physical representation of K is displayed in Figure 3.3c. The compressibility of a
−1
rock, β,isgiven simply by β = K .Young’s modulus, E,is simply the stiffness of a
rock in simple (unconfined) uniaxial compression (S 11 is the only non-zero stress)

S 11
E =
ε 00
which is shown in Figure 3.3a (and Figure 3.2). In the same test, Poisson’s ratio, ν,is
simply the ratio of lateral expansion to axial shortening

ε 33
ν =
ε 11
In an incompressible fluid, ν = 0.5. The shear modulus, G,isphysically shown in
Figure 3.3b. It is the ratio of an applied shear stress to a corresponding shear strain
1
S 13
G =
2 ε 13