Sequential Methods for Coupled Geomechanics and Multiphase Flow

50 CHAPTER 3. STABILITY OF THE DRAINED AND UNDRAINED SPLITS
Adding Equations 3.77 and 3.78,
From Equation 3.67, we have
dζ 2
T
1
=
2
= 1
2
Ω
Ω
Ω
= dχ 2
N ,
where we define the norm of χ N as
χ 2
N
dσ ′ : dεp ˙ + dκ · d ˙
ξ dΩ ≥ 0 . (3.79)
dσ ′ : C −1
dr dσ′ + dκ · H −1 dκ + 1
M dp2
dΩ, (3.80)
2
dme
dεe : Cdrdεe + dξ · Hdξ + M − bdεe,v dΩ,
2
1
me
= εe : Cdrεe + ξ · Hξ + M − bεe,v dΩ, (3.81)
2 Ω
ρf,0
ρf,0
N := χ := (εe, ξ, me) ∈ S × R nint × R : εeij ∈ L2 (Ω),
ξi ∈ L 2 (Ω), me ∈ L 2 (Ω) , (3.82)
where εeij and ξi are the components of εe and ξ, respectively. Equation 3.81 originates
from the Helmholtz free energy (Coussy, 1995). Let us denote dχ 2
N by Ψd for convenience.