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

Abstract

In [3] a space-time domain decomposition method was proposed for two-phase flow in a porous medium composed of two different rock types, so that the capillary pressure field is discontinuous at the interface between the rocks. For this nonlinear and degenerate parabolic problem, with nonlinear and discontinuous transmission conditions on the interface, the Optimized Schwarz waveform relaxation method (OSWR) with Robin or Ventcell transmission conditions was considered. A guaranteed and fully computable a posteriori error estimate was derived, which in particular took into account the domain decomposition error. In this paper we provide a mathematical and numerical analysis of this space-time domain decomposition method in the Robin case. Complete numerical approximation is achieved by a finite volume scheme in space and the lowest order discontinuous Galerkin method in time. We prove the existence of a weak solution of the two-phase flow subdomain problem with Robin boundary conditions by analyzing the convergence of the finite volume scheme. The domain decomposition algorithm is based on the solution of space-time nonlinear subdomain problems over the whole time interval, allowing for different time steps in different parts of the domain, adapted to the physical properties of each subdomain, and we show that such an algorithm is well-defined. Numerical experiments on three-dimensional problems with different rock types illustrate the performance of the domain decomposition method.

Non-Overlapping Schwarz Waveform-Relaxation for Nonlinear Advection-Diffusion Equations

“…Before we start with some preliminaries on error measurement and associated notations let us note that various non-overlapping domain decomposition approaches have been developed in the context of two-phase flow. Ahmed et al have studied a posteriori error estimates and stopping criteria based on space-time domain decomposition for two-phase flow between different rock types in [1,2]. Seus et al proposed robust linear domain decomposition methods of the time-discrete equations for partially saturated flow as well as for two-phase flow in porous media in [47,48].…”

Towards hybrid two‐phase modelling using linear domain decomposition

“…Recall that for a solution (( 1 ⋅ n 1 is defined by the right hand side of either ( 6) or (7), compare also (17) and ( 16). Thus, the functionals…”

“…From the numerical point of view it is therefore important to investigate the applicability of the LDD solver to the case of continuous pressures not only as a first step, but notably so as an approximation of the more realistic discontinuous case. We refer to [17] for a recent contribution in this direction.…”

“…Lunowa et al in [16] apply ideas from [1, 2] to a dynamic capillary pressure model with hysteresis on a two‐domain setting. The work [17] is concerned with two‐phase flow with discontinuous capillary pressures. None of these works address the case of a hybrid model ansatz.…”