Sequential Methods for Coupled Geomechanics and Multiphase Flow

28 CHAPTER 3. STABILITY OF THE DRAINED AND UNDRAINED SPLITS
equations take the following form:
1
ρf,0
σ − σ0 = Cdr : ε − b(p − p0)1, (3.2)
(m − m0) = bεv + 1
M (p − p0), (3.3)
where the linearization is applied from the reference state to the current state for single-
phase flow of a slightly compressible fluid. The subscript 0 means reference state, Cdr is
the rank-4 drained elasticity tensor, 1 is the rank-2 identity tensor, p is fluid pressure, m is
fluid mass per unit bulk volume, M is the Biot modulus, and b is the Biot coefficient. Note
that we use the convention that tensile stress is positive. Here, ε is the linearized strain
tensor under the assumption of infinitesimal transformation:
Note that we also have (Coussy, 1995)
ε = Grad s u = 1
2 (Gradu + Gradt u). (3.4)
1
M = φ0cf +
b − φ0
, (3.5)
Ks
b = 1 − Kdr
, (3.6)
Ks
where cf is the fluid compressibility (1/Kf), Kf is the bulk modulus of the fluid. Again, the
subscript 0 means reference state. It is convenient to express the strain and stress tensors
in terms of their volumetric and deviatoric parts,
ε = 1
3 εv1 + e, (3.7)
σ = σv1 + s, (3.8)
where εv = trε is the volumetric strain (the trace of the strain tensor), e is the deviatoric
part of the strain tensor, σv = 1
3trσ is the volumetric (mean) total stress, and s is the
deviatoric total stress tensor.