  In this paper, we are concerned with the global pressure formulation of
immiscible incompressible two-phase flow between different rock types. We
develop for this problem two robust schemes based on domain decomposition (DD)
methods and operator-splitting techniques. The first scheme follows a
sequential procedure in which the (global) pressure, the saturation-advection
and the saturation-diffusion problems are fully decoupled. In this scheme, each
problem is treated individually using various DD approaches and specialized
numerical methods. The coupling between the different problems is explicit and
the time-marching is with no iterations. To adapt to different time scales of
problem components and different rock types, the novel scheme uses a multirate
time stepping strategy, by taking multiple finer time steps for
saturation-advection within one coarse time step for saturation-diffusion and
pressure, and permits independent time steps for the advection step in the
different rocks. In the second scheme, we review the classical Implicit
Pressure--Explicit Saturation (IMPES) method (by decoupling only pressure and
saturation) in the context of multirate coupling schemes and
nonconforming-in-time DD approaches. For the discretization, the
saturation-advection problem is approximated with the explicit Euler method in
time, and in space with the cell-centered finite volume method of first order
of Godunov type. The saturation-diffusion problem is approximated in time with
the implicit Euler method and in space with the mixed finite element method, as
in the pressure problem. Finally, in a series of numerical experiments, we
investigate the practicality of the proposed schemes, the accuracy-in-time of
the multirate and nonconforming time strategies, and compare the convergence of
various DD methods within each approach.
