Page 351 - Numerical Analysis and Modelling in Geomechanics

332 RESERVOIR COMPACTION, SUBSIDENCE AND WELL DAMAGE

(11.7)

where σ' is the effective stress tensor, α is a material parameter referred to as the
effective stress coefficient, p is the fluid pressure in the pores, and I is the second
order identity tensor.
In considering finite, elastoplastic deformations and using the Kirchhoff total
stress tensor, defined by in the balance of mechanical energy of a mixture of
56
incompressible solids and fluids, Borja and Alarcón showed that the effective
stress, is defined as
(11.8)

f
where τ =pI is a rational concept for effective stress. They showed that, since the
fluid is assumed to be incompressible and has no shear strength, the component
of stress in the fluid phase, does not perform mechanical work on the system,
f
which is a clear distinction from the partial stress τ . Thus, the effective stress is
intuitive and provides a rational and convenient framework for describing the
stress within a saturated solid. Since the Kirchhoff stress will be used below,
note that it is defined with reference to the undeformed configuration, while the
Cauchy stress is defined with respect to the deformed configuration.
61
Nur and Byerlee showed that for linear, isotropic theory of the mechanics of
porous media, the effective stress can be defined and written exactly as:

(11.9)

where the term in parentheses is identified by the effective stress coefficient α of
Equation (11.7). The form of Equation (11.9) adopts the notation proposed by
62
Zimmerman, et al., in which C is the compressibility of the rock matrix and
r
C is the compressibility of the bulk rock under the action of the confining stress
bc
in a drained test. In an analysis of the linearized theory of Biot 63–67 and the
development of solutions to problems using that theory, Rice and Cleary 68
showed that while the effective stress law of Equation (11.9) occurs naturally in
the equations of equilibrium, it arrives from the constitutive assumption, not from
considering the pore pressure p in an analysis of equilibrium.
In the development that follows for finite deformation, elastoplastic finite
element analysis, the effective stress law is taken to be:

(11.10)

where χ is an effective stress coefficient dependent upon saturation and surface
tension between the pore fluid and solid. The coefficient χ is assumed to be 1.0
for a fully saturated medium. It may be noted that the effective stress coefficient