Page 350 - Numerical Analysis and Modelling in Geomechanics

RESERVOIR COMPACTION, SUBSIDENCE AND WELL DAMAGE 331
where
The linear momentum of a saturated body can be derived by considering it as a
mixture of solid and fluid phases. In this context, define the Cauchy partial stress
tensors on the solid and fluid arising from inter-particle and fluid stresses as σ g
f
and σ , respectively. 56,58,59 Also, denote by n the unit normal vector to the surface
(U) of the deformed body. Then, the linear momentum balance of the solid phase
in the current (deformed) configuration is:

(11.2)

The momentum balance for the fluid phase is:

(11.3)

In the above momentum balance equations, g is the gravitational acceleration
g
vector, ρ and ρ are the solid and fluid mass densities, h is a vector of force per
f
g
unit volume arriving from the frictional force of the flowing pore fluid on the
g
solid, and h f is the reaction on the fluid arising from h . Since h g and hf are
internal forces, their sum is zero. Summing Equations (11.1) and (11.2) gives the
linear momentum balance for the solid and fluid mixture, as follows:
(11.4)

where the Cauchy total stress tensor has been defined as
(11.5)

and the mass density of the mixture, referred to as the saturated mass density, is
(11.6)

The concept of effective stress
Terzaghi 60 is generally credited with first recognizing that the deformation of
fluid-filled, porous media is governed by an effective stress. The effective stress
has been used as an alternative to the total stress decomposition of Equation (11.
5) based on the physical intuition of Terzaghi, experimental validation by many
investigators in soil and rock mechanics, and more recently has been shown to
have a rational interpretation in solid mechanics. The fundamental concept is
that given an idealized volume of water-saturated soil, on the boundary of which
is imposed a state of stress, given by the Cauchy total stress tensor, σ, that stress
is shared by the soil and water in the following way: