Sequential Methods for Coupled Geomechanics and Multiphase Flow

22 CHAPTER 2. FORMULATION
where Div vs ≡ dεv/dt, and we define the Biot coefficient b for single phase flow as
b = 1 − Kdr
. (2.32)
Ks
Then, from Equations 2.30 and 2.31, Equation 2.29 can be expressed as
ρf
b − φ dp
φcf + + bdεv + Div(ρfvf) = 0. (2.33)
dt dt
Ks
Since the first term of Equation 2.33 corresponds to the mass variation, δm, we obtain
δm b − φ
= φcf + δp + bδεv. (2.34)
ρf
Hence, the Biot coefficient b and the Biot modulus M are obtained as
b = 1 − Kdr
,
Ks
Ks
1
M = φcf +
2.4.2 Coupling coefficients for multi-phase flow
b − φ
. (2.35)
In Equation 2.2, the flux term, Div wJ, is easily expressed in terms of fluid pressures using
Darcy’s law, Equation 2.11. Hence, we focus on the expression of the accumulation term,
dmJ/dt. We apply the same procedure used for single-phase flow. Equation 2.29 for single-
phase flow yields
Ks
δm = δ(ρfφ) + ρfφδεv. (2.36)
The Biot moduli for multiphase flow can be obtained by expanding the accumulation term
in the flow equation (i.e., δmJ) as follows:
δmJ = δ ((ρS) J φ) + (ρS) J φδεv
= (ρS) J δφ + φ(ρδS) J + φ(Sδρ) J + (ρS) J φδεv. (2.37)