Page 81 - Reservoir Geomechanics

65 Basic constitutive laws

Elasticity anisotropy

A number of factors can make a rock mass anisotropic – aligned microcracks (Hudson
1981), high differential stress (Nur and Simmons 1969), aligned minerals (such as
mica and clay) (Sayers 1994) along bedding planes (Thomsen 1986), macroscopic
fractures and faults (Mueller 1991). Elastic anisotropy can have considerable effects
on seismic wave velocities, and is especially important with respect to shear wave
propagation. Although a number of investigators have argued that stress orientation
can be determined uniquely from shear velocity anisotropy (e.g. Crampin 1985), in
many cases it is not clear whether shear velocity anisotropy in a volume of rock is
correlative with the current stress state or the predominant orientation of fractures in
a rock (e.g. Zinke and Zoback 2000). Because of this, measurements of shear velocity
anisotropy must be used with care in stress mapping endeavors. That said, shear wave
velocity anisotropy measured in vertical wellbores often does correlate with modern
stress orientations (Yale 2003; Boness and Zoback 2004). We discuss this at greater
length in Chapter 8.
Withrespecttoelasticanisotropy,thegeneralformulationthatrelatesstresstostrainis

(3.7)
S ij = c ijkl ε kl

wherec ijkl ,theelasticstiffnessmatrix,isafourth-ranktensorwith81constantsandsum-
mationisimpliedoverrepeatedsubscriptskandl.Itisobviouslynottractabletoconsider
wave propagation through a medium that has to be defined by 81 elastic constants. For-
tunately, symmetry of the stiffness matrix (and other considerations) reduces this tensor
to 21 constants for the general case of an anistropic medium. Even more fortunately,
media that have some degree of symmetry require even fewer elastic constants. As men-
tioned above, an isotropic material is defined fully by two constants, whereas a material
with cubic symmetry is fully described by three constants, and a material characterized
by transverse isotropy (such as a finely layered sandstone or shale layer) is characterized
by five constants, and so on. Readers interested in wave propagation in rocks exhibiting
weak transverse anisotropy are referred to Thomsen (1986) and Tsvankin (2001).
Elasticanisotropyisgenerallynotveryimportantingeomechanics,although,asnoted
above, shear wave velocity anisotropy can be related to principal stress directions or
structural features. On the other hand, anisotropic rock strength, due, for example, to
the presence of weak bedding planes, has a major affect on wellbore stability as is
discussed both in Chapter 4 and Chapter 10.

Poroelasticity and effective stress

In a porous elastic solid saturated with a fluid, the theory of poroelasticity describes
the constitutive behavior of rock. Much of poroelastic theory derives from the work of