Sequential Methods for Coupled Geomechanics and Multiphase Flow

2.3. COUPLING AND CONSTITUTIVE RELATIONS 17
entropy provided by conduction and the external volume heat source.
The Helmholtz free energy (Ψ) is defined as
Ψ = E − TS . (2.6)
Then, using Equations 2.4, 2.5, and 2.6, the fundamental inequality is obtained as
σ : ˙ε − gJ [Div(wJ) − (ρf) J ] − dΨ dT
− S −
dt dt
q
· GradT
T
Φ1
−w · [Grad(gJ) + sJ GradT − F]
Φ3
Φ2
≥ 0 , (2.7)
where gJ is the Gibbs potential per unit mass of phase J, which can be written as
gJ = eJ + (p/ρ)J − TsJ = gJ(pJ, T) . (2.8)
Equation 2.7 consists of the intrinsic dissipation (Φ1), thermal dissipation associated
with heat conduction (Φ2), and the dissipation due to mass transport (Φ3). Applying the
hypothesis that each dissipation is non-negative, we obtain
Φ1 = σ : ˙ε − gJ [Div(wJ) − (ρf) J ] − dΨ
dt
Φ2 = − q
T
− S dT
dt
≥ 0, (2.9)
· GradT ≥ 0, (2.10)
q = −kc · GradT,
w
Φ3 = − · [Grad(pJ) − ρJF] ≥ 0, (2.11)
ρ J
w
= −kp,JK · [Grad(pK) − ρKF] ,
ρ
J