Sequential Methods for Coupled Geomechanics and Multiphase Flow

2.4. DERIVATION OF THE COUPLING COEFFICIENTS 25
Remark 2.2. Note that δpt = δ (SJpJ). If δpt = δ (SJpJ), which is used in Lewis and
Sukirman (1993) and Lewis and Schrefler (1998), the constitutive relations given in Equa-
tions 2.22 and 2.23 cannot be obtained in the presence of capillarity. δpt = δ (SJpJ) cannot
guarantee thermodynamic stability and well-posedness, as shown in Appendix B.4. Com-
pare Chapter 6.3.3 with Appendix B.4 regarding the contractivity properties of coupled
flow and mechanics. When pt is assumed to be SJpJ, this implies that we assume physical
linearization of the total-stress from the initial (reference) state to the current state. But
this assumption is not appropriate because the saturation field varies across the entire range
of possible values, and that precludes linearization from the initial state to the current state.
Remark 2.3. δpt = SJδpJ, which leads to bJ = bSJ, honors thermodynamic stability as
well as the form of the constitutive equations given by Equations 2.22 and 2.23. Only the
incremental total-pressure, δpt, is defined, since the linear model is relevant within a narrow
range of variation around a reference state (Coussy, 1995). The Biot coefficient bJ = bSJ
matches the derivation of Coussy (1995) and Coussy et al. (1998).
Appendix C shows the formulation for the constitutive relations relevant to reservoir
simulation based on the standard black-oil model, where there are two general choices for
the primary variables: namely, the phase pressures po, pw, and pg, or one phase pressure
and two saturations, such as po, Sw, and Sg.