Sequential Methods for Coupled Geomechanics and Multiphase Flow

20 CHAPTER 2. FORMULATION
coupling between mass-flow and mechanics, and we write
Equation 2.21 can be rewritten as
δm
δσ = Cud : δε − MJKbK1, (2.21)
ρ J
δm
δpJ = MJK −bKδεv + . (2.22)
ρ
K
δσ = Cdr : δε − bJδpJ1, (2.23)
where Cdr represent the drained bulk moduli. In the case of single-phase flow, the coefficients
are obtained from drained and undrained experiments. For example,
b = 1 − Kdr
,
Ks
1
M = φcf +
b − φ
, (2.24)
where cf is the fluid compressibility, and Kdr and Ks are the drained bulk modulus and the
solid grain stiffness, respectively. b and M for single-phase correspond to bJ and MJK for
multiphase systems. In the case of multiphase flow, the Biot moduli M = {MJK} and Biot
coefficient b = {bJ} are not given explicitly by Coussy (1995). The subsequent sections
provide appropriate expressions of M and b.
2.4 Derivation of the Coupling Coefficients
2.4.1 Coupling coefficients for single-phase flow
The governing mass conservation equation for single-phase flow is
Ks
∂(ρfφ)
∂t + Div(ρfφ ˜vp) = 0, (2.25)
where ˜vp is the interstitial fluid velocity. Note that Equations 2.2 and 2.25 are equivalent.
Equation 2.2 is expressed using the material derivative (i.e., Lagrangian description), while