Splitting-based domain decomposition methods for two-phase flow with different rock types

Abstract

In this paper, we are concerned with the global pressure formulation of immiscible incompressible two-phase flow between different rock types. The aim is to develop for this problem a robust algorithm based on domain decomposition methods and operator splitting techniques, in which the numerical solution is achieved by solving sequentially reduced pressure, saturation-advection and saturation-diffusion problems posed on the interfaces between the rocks. This approach makes possible the use of specialized numerical methods for each sub-problem and different time steps for diffusion and advection as well as independent time steps for the advection in the different rocks. For the discretization, the advection problem is approximated in time, with the explicit Euler method where different time grids are employed to adapt to different time scales in the rocks, and in space with hybridized cell-centered finite volume method of first order of Godunov type. That of the diffusion problem is approximated in time with the implicit Euler method and in space with a hybridized mixed finite element method as is used for the pressure problem. Numerical experiments illustrate the performance and the flexibility of our domain decomposition algorithm on different model problems in three space dimensions.Key words: Two-phase flow in porous media; continuous capillary pressure; non-overlapping domain decomposition; operator splitting; hybrid mixed finite element; hybrid finite volume; non-conforming time grids. * This work was partially funded by the Hydrinv Inria Euro Med 3+3: HYDRINV project. It has also received funding from the EPIC project (within LIRIMA: http://lirima.inria.fr) and the Tunisian Ministry

Space–time domain decomposition for two-phase flow between different rock types

“…Both simplifications were made in order to follow the set-up of [48], and because this paper is the first application of the OSWR method to this type of problems. We emphasize that both restrictions can be lifted: the multidomain coupled problem (without domain decomposition) has been treated in [30], while the OSWR method has been extended to the diffusion-advection case in [66], See also [2] for related work, and [83] for a different domain decomposition method applied to the full two-phase flow model. For simplicity, we consider only Dirichlet boundary conditions on ∂Ω.…”

“…Due to different hydrogeological properties of the different rocks, domain decomposition (DD) methods appear to be a natural way to solve efficiently two-phase flow models, see [92,93,2], and also [57,85,86,83,82]. This paper complements [3], where a global-in-time domain decomposition method for this nonlinear and degenerate parabolic problem was proposed (without analysis), using the Optimized Schwarz Waveform Relaxation algorithm (OSWR) with Robin or Ventcell transmission conditions.…”

Linearized domain decomposition methods for two-phase porous media flow models involving dynamic capillarity and hysteresis

“…In [17], a non-overlapping domain decomposition method is analyzed for nonlinear convection-diffusion equations in a time-continuous setting. Such methods can also be used after temporal discretization for porous media equations, as proposed in [18,19] for a simplistic setting, while Richards' equation and the two-phase flow equations are considered in [20,21], where a-posteriori error estimates and multirate time stepping methods are derived. In [22,23], the domain decomposition is integrated in the linearization process for both Richards' equation and two-phase flow, and the convergence is proven rigorously.…”