Page 75 - Reservoir Geomechanics

59 Basic constitutive laws

Figure 3.2 illustrates why a strength of materials approach to rock failure, that will
be utilized to consider wellbore failure in the chapters to follow, is frequently a useful
simplification of the inherently complex process of rock failure. Using a strength of
materials approach assumes that one can consider rock deformation to be elastic until
the point of failure. It is evident in Figure 3.2 that a small degree of inelastic deformation
precedes failure such that this assumption is not strictly correct. Nonetheless, for well-
cemented rocks, a strength of materials approach does a good job of approximating
failure. In weak, poorly cemented formations, the applicability of this approach is more
questionable.
Figure 3.1b schematically illustrates the profound effect that water (or oil) in the pores
of a rock has on its behavior. A porous rock saturated with fluid will exhibit poroelastic
behavior. One manifestation of poreoelasticity is that the stiffness of a fluid-saturated
rock will depend on the rate at which external force is applied. When force is applied
quickly, the pore pressure in the rock’s pores increases because the pore fluid is carrying
some of the applied stress and the rock behaves in an undrained manner. In other words,
if stress is applied faster than fluid pressure can drain away, the fluid carries some of the
applied stress and the rock is relatively stiff. However, when an external force is applied
slowly, any increase in fluid pressure associated with compression of the pores has time
to drain away such that the rock’s stiffness is the same as if no fluid was present. It is
obvious that there is a trade-off between the loading rate, permeability of the rock and
viscosity of the pore fluid, which is discussed further below. For the present discussion,
it is sufficient to note that the deformation of a poroelastic material is time dependent, a
property shared with viscoelastic materials, as illustrated in Figure 3.1d and discussed
further below.
Figure 3.1c illustrates elastic–plastic behavior. In this case, the rock behaves elasti-
callytothestresslevelatwhichityieldsandthendeformsplasticallywithoutlimit.Upon
unloading the rock would again behave elastically. Although some highly deformable
rocks behave this way in laboratory testing (right panel), we discuss in Chapter 4 how
this type of behavior is also characteristic of deformation in the upper crust being taken
up by fault slip. In this case, appreciable deformation (i.e. fault slip) can occur at a
relatively constant stress level (i.e. that required to cause optimally oriented faults to
slip).
As alluded to previously, a viscoelastic rock (Figure 3.1d) is one in which the defor-
mation in response to an applied stress or strain is rate dependent. The stress required
to cause a certain amount of deformation in the rock depends on the apparent viscosity,
η,of the rock (center panel). One can also consider the stress resulting from an instan-
taneously applied deformation (right panel) which will decay at a rate depending on
the rock’s viscosity. The conceptual model shown in the left panel of Figure 3.1d cor-
responds to a specific type of viscoelastic material known as a standard linear solid.A
variety of other types of viscous materials are described below. A viscous material that
exhibits permanent deformation after application of a load is described as viscoplastic.