Page 356 - Numerical Analysis and Modelling in Geomechanics

RESERVOIR COMPACTION, SUBSIDENCE AND WELL DAMAGE 337
Constitutive theories for fluid-filled rock
Linear and nonlinear models for the constitutive behavior of fluid-filled rocks are
reviewed in this section. An elastoplastic, Drucker-Prager model modified to
incorporate a Cap surface is applied in later sections in a computational
application.

Linear constitutive theory
It is useful to review the general theory of poroelasticity, or Biot theory, which is
a formulation of a theory of the stress and strain relations for geomaterials.
Biot 63 presented a constitutive equation for fluid filled, porous material (e.g.,
rock or soil) assumed to behave linearly and isotropically under isothermal
conditions. It is assumed that the pores are filled completely, or saturated, with a
fluid under a pressure p, and that the volume of the pores is characterized by the
specific volume v=V /V where V is the volume of the pore fluid and V is the
b
b
f
f
volume of the bulk sample of porous, saturated solid. If the rock is saturated with
pore fluid and the fluid is incompressible, the specific volume may be recognized
as the porosity, ф. Biot’s linear constitutive equations may be written in the
following form:
(11.26)

and

(11.27)

where ε and σ are the components of the infinitesimal strain and stress tensors,
ij
ij
respectively, and E and v are the Young’s modulus and Poisson’s ratio
determined from a “drained” laboratory test of a sample of rock. It may be
recalled that the shear and bulk moduli can be computed from Young’s modulus
and Poisson’s ratio by G=E/2(1+v) and K=2G(1+v)/3(1–v), respectively. The
drained test provides the moduli of the porous solid. The two material properties
H and R are known as Biot’s constants. H and R are often referred to as coupling
constants, since they relate the deformation of the porous solid to changes in the
pore pressure or pore volume changes. It can be shown that: 71,72

(11.28)