Page 72 - Reservoir Geomechanics

3 Basic constitutive laws

InthischapterIbrieflyreviewanumberoftheconstitutivelawsgoverningrockdeforma-
tion. Fundamentally, a constitutive law describes the deformation of a rock in response
to an applied stress (or vice versa). Because of the breadth of this subject, the material
below is restricted to covering key principles that will be referred to later in the text.
One unconventional topic discussed at some length at the end of this chapter is the
viscous compaction of uncemented sands. As explained below, the presence of water
or oil in the pores of a rock will result in time-dependent deformation of any porous
elastic (poroelastic) solid. The topic I discuss below is the viscous deformation of dry
uncemented sands. In other words, in addition to the poroelastic deformation, there is
also time-dependent deformation of the dry matrix. There are two main reasons for
this. First, many oil and gas reservoirs in the world occur in such formations. Thus, it
is important to accurately predict: (i) how they will compact with depletion (especially
as related to compaction drive); (ii) what the effects of compaction will be on reservoir
properties (such as permeability); and (iii) what the effects will be on the surrounding
formations (such as surface subsidence and induced faulting). The basic principles
of viscous compaction of uncemented sands are outlined in this chapter, whereas the
compaction of reservoirs composed of such materials is addressed in more detail in
Chapter 12. Second, while there are several exellent texts on the subject of constitutive
laws applicable to rocks (e.g. Charlez 1991;Paterson and Wong 2005; Pollard and
Fletcher 2005), the subject of viscous deformation in very weak formations, while
dealt with extensively in soil mechanics and geotechnical engineering, has not been
discussed extensively in the context of hydrocarbon reservoirs.
The schematic diagrams in Figure 3.1 illustrate four generic types of constitutive
laws for homogeneous and isotropic materials. Even though each of these constitutive
laws is described in greater detail later in this chapter, the following introduction may
be useful.
A linearly elastic material (Figure 3.1a) is one in which stress and strain are linearly
proportional and deformation is reversible. This can be conceptualized in terms of a
force applied to a spring where the constant of proportionality is the spring constant,
k.An ideal elastic rock strains linearly in response to an applied stress in which the
stiffness of the rock is E,Young’s modulus. An actual rock mechanics test is presented
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