Sequential Methods for Coupled Geomechanics and Multiphase Flow

164 CHAPTER 6. COUPLED MULTIPHASE FLOW AND GEOMECHANICS
where Jfc,m, Km, and Lm are the Jacobian, stiffness, and coupling matrices, respectively.
F m = Q m + ∆tT m is the flow matrix, where Q m is the compressibility matrix, and T m is
the transmissibility matrix. The subscript m, (·)m, implies multiphase.
Similar to our treatment of single-phase flow in the previous chapters, we split the
original operator of coupled multiphase flow and mechanics. When only a single pass
strategy is used, in which the two problems are solved implicitly and in sequence, we refer
to the scheme as a staggered method.
We can also apply sequential methods to the fully coupled problem by adopting a stag-
gered Newton scheme (Schrefler et al., 1997), where full iterations are performed as shown
for single phase with elastoplasticity in Chapters 3 and 5. In this scheme, we linearize the
governing equations first, and then we solve the linearized equations sequentially with full
iterations. Then, Jfc,m is approximated by Jsq,m, the Jacobian matrix of the sequential
method of interest.
6.1.1 Drained split
In the drained split, we freeze the pressure of all the fluid phases when solving the mechanical
problem. The drained split approximates the operator A m as
⎡
⎤
⎡
⎣ un
pn ⎦
J
Au,m
dr
−→ ⎣ un+1
pn J
⎤
⎦ Ap,m
dr
−→
⎡
⎣ un+1
p n+1
J
⎤
⎦, where
⎧
A
⎪⎨
u,m
dr : Div σ + ρbg = 0, δpJ = 0
⎪⎩
A p,m
dr : mJ ˙ + Div wJ = (ρf)J,
˙ε : prescribed,
(6.3)
where the variation of fluid pressure is frozen during the mechanics step (i.e. δpJ = 0).
When a staggered Newton scheme is used with the drained split, the constraint δpJ = 0 for
the linearized mechanical problem becomes p k+1
J = pk J .