Sequential Methods for Coupled Geomechanics and Multiphase Flow

6.2. STAGGERED NEWTON SCHEMES FOR MULTIPHASE FLOW 167
discrete form as
to
σ n+1
v − σ n v = σ n v − σ n−1
v . (6.12)
With the staggered Newton method, for the linearized flow problem, δ ˙σv = 0 is modified
σ k+1
v − σ n v = σ k v − σ n v . (6.13)
6.2 Staggered Newton schemes for Multiphase Flow
In reservoir engineering, the most common solution algorithms for multiphase flow are the
FIM (Fully Implicit Method) and the IMPES (IMplicit Pressure and Explicit Saturation).
The FIM solves the multiphase flow equations simultaneously for the pressure and satu-
ration fields. In the IMPES approach, the pressure field is obtained implicitly, and the
saturation field is computed explicitly based on the updated pressure field. Thus, the FIM
provides unconditional stability for the multiphase flow problem, but requires large systems
and high computational cost. IMPES is only conditionally stable because the saturation
field is obtained explicitly. But, IMPES yields small systems relative to FIM and saves
computational resources.
In this chapter, we perform stability analysis of multiphase flow for staggered Newton
schemes for the drained, undrained, fixed-strain, and fixed-stress splits. We first employ the
Von Neumann method, since we linearize all the equations first for the Newton-Raphson
method and solve the linearized equations sequentially. We take full iterations until the
solutions are converged. We use the energy method for nonlinear stability analysis of
staggered strategies.
6.2.1 Convergence of FIM
Oil pressure and water saturation are typically used as the primary variables for FIM in
the case of oil-water flow. The Von Neumann method is applied to the linearized equations