Sequential Methods for Coupled Geomechanics and Multiphase Flow

Chapter 2
Formulation
2.1 Background
The coupling between flow and mechanics has been studied by many authors (e.g., Biot
(1941); Geertsma (1957); Biot and Willis (1957); Coussy (1995); Lewis and Schrefler (1998);
Borja (2006)). Biot (1941), Geertsma (1957), and Biot and Willis (1957) develop the
constitutive equations in the case of single-phase flow with a slightly compressible fluid,
and they proposed appropriate laboratory tests that can determine the various coupling
coefficients.
For multiphase flow, the approach by Lewis and Schrefler (1998) provides explicit ex-
pressions of the physical quantities of interest (i.e., coefficients in the governing equations),
even though the formulation does not necessarily ensure complete consistency with the ther-
modynamic constraints when capillarity is present. Borja (2006) shows that the coupling
is thermodynamically stable, examining different definitions of the effective stress, but the
bulk modulus used is neither the drained, nor the undrained bulk modulus. Thus, the Biot’s
coefficient in Borja (2006) is different from that used by other investigators. Coussy (1995)
and Coussy et al. (1998) provide thermodynamically consistent constitutive relations. Nev-
ertheless, the coefficients of the constitutive equations (e.g., Biot’s moduli) for multiphase
flow are not explicitly given. For these reasons, the purpose of this chapter is to describe the
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