Sequential Methods for Coupled Geomechanics and Multiphase Flow

2.4. DERIVATION OF THE COUPLING COEFFICIENTS 23
For multiphase flow, we introduce the total fluid pressure, pt (i.e., the equivalent pore
pressure in Coussy (2004)). Then, the true porosity variation δφ in multiphase flow can be
extended from that for single-phase using the total fluid pressure. Specifically, we can write
δφ = ∂φ
δpt + (b − φ) δεv
∂pt
= b − φ
Ks
SJδpJ + (b − φ)δεv, (2.38)
where δpt is assumed to be SJδpJ (recall that summation is implied), the validity of which
will be examined in Remarks 2.2 and 2.3. Since we can define the fluid compressibility of
phase J in multiphase flow, cJ, as
cJ = 1
then, by using Equation 2.38, we can express Equation 2.37 as
ρJ
δρJ
, (2.39)
δpJ
δm b − φ
= SJ SJδpJ + bδεv + φδSJ + φ(Scδp) J . (2.40)
ρ J
Ks
In order to express δSJ in terms of δpJ, we can use the capillary relations. Typically,
oil, gas, and water systems are considered. Each saturation can be expressed as
δSo = −δSg − δSw
= − dSw
(δpo − δpw) −
dpco
dSg
(δpg − δpo)), (2.41)
dpcg
δSg = dSg
(δpg − δpo), (2.42)
dpcg
δSw = dSw
(δpo − δpw) , (2.43)
dpco
where pco is the capillary pressure between oil and water, and pcg is the capillary pressure