Sequential Methods for Coupled Geomechanics and Multiphase Flow

2.3. COUPLING AND CONSTITUTIVE RELATIONS 15
Figure 2.1: Ωt and Ω f
t
are the geometrical volumes of the solid and fluid at time t, respectively.
Even though the elementary solid system has the same fluid mass content at
, the solid system does not have the same fluid mass content after time dt,
time t, Ωt = Ω f
t
Ωt+dt = Ω f
t+dt . wJ is the mass flux of fluid phase J relative to the solid skeleton (Coussy,
1995).
2.3 Coupling and Constitutive Relations
We adopt a classical continuum representation, where the fluid and solid are viewed as over-
lapping continua. We summarize the general coupling among mass, energy, and mechanical
equilibrium based on the approach by Coussy (1995) and Coussy et al. (1998). For more
generality, we include the effect of the source term in Equation 2.2.
2.3.1 Coupling of fluid, heat flow, and mechanics
The variation of internal energy of an open system with respect to time is described by the
first law of thermodynamics. The infinitesimal transformation is adopted, and that allows
for using the linearized strain tensor ε, which can be written as
where u is the displacement vector.
ε = Grad s u = 1
2 (Grad u + Gradt u) (2.3)