Sequential Methods for Coupled Geomechanics and Multiphase Flow

6.4. STABILITY ANALYSIS VIA THE ENERGY METHOD 183
dζm 2
Tm
1
=
2
= 1
2
+
Ω
′ −1
dσ : Cdr dσ′ + dκ · H −1
dκ + dpJNJKdpK dΩ,
(dεe : Cdrdεe + dξ · Hdξ
Ω
dm
− bJdεe,v
ρ
J,e
MJK
dm ρ
K,e
− bKdεe,v
= dχm 2
, (6.75)
Nm
where we define the norm of χm Nm as
χm 2
Nm
1
= (εe : Cdrεe + ξ · Hξ
2 Ω
m
+ − bJεe,v
ρ
J,e
MJK
m ρ
K,e
− bKεe,v
Nm := χ := (εe, ξ, mJ,e) ∈ S × R nint × R np : εeij ∈ L2 (Ω),
dΩ,
dΩ, (6.76)
ξi ∈ L 2 (Ω), mJ,e ∈ L 2 (Ω) , (6.77)
which originates from the Helmholtz free energy (Coussy, 1995).
Since the corresponding solutions from two arbitrary initial conditions satisfy the govern-
ing equations and boundary conditions, the fully coupled method yields from Equation 6.1
⎡
⎣ dun
dp n J
⎤
⎦ Am fc
−→
⎡
⎣ dun+1
dp n+1
J
⎤
⎦, where A m fc :
⎧
⎪⎨ Div dσ = 0,
⎪⎩ dmJ
˙ + Div dwJ = 0,
(6.78)
where maximum plastic dissipation is assumed for elasto-plasticity. Note that homogeneous
boundary conditions are obtained. Then the coupled problem is contractive relative to the
norms, · Nm and · Tm . Specifically,