A Linear Domain Decomposition Method for Two-Phase Flow in Porous Media

Seus, David; Radu, F.; Rohde, Christian

doi:10.1007/978-3-319-96415-7_55

Cited by 6 publications

(11 citation statements)

References 14 publications

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“…For new coupling conditions between domains with different two‐phase flow models we developed an approach combining an

L

‐type linearization of the nonlinearities with a generalized nonoverlapping alternating Schwarz method, the LDD‐TP–R solver. This formulation unifies the work of both, [1] and [2] on homogeneous two‐phase models and allows for the treatment of complex modeling situations involving very heterogeneous soil parameters. The LDD‐TP–R solver has been analyzed rigorously on the time‐discrete level.…”

Section: Discussionmentioning

confidence: 80%

“…Convergence of a Schwarz waveform relaxation method is established in [15] for the transport equation in the fractional flow formulation of two‐phase flow. 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.…”

Section: Introductionmentioning

confidence: 99%

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Towards hybrid two‐phase modelling using linear domain decomposition

Seus

Radu

Rohde

2022

Numerical Methods Partial

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The viscous flow of two immiscible fluids in a porous medium on the Darcy scale is governed by a system of nonlinear parabolic equations. If infinite mobility of one phase can be assumed (e.g., in soil layers in contact with the atmosphere) the system can be substituted by the scalar Richards model. Thus, the porous medium domain may be partitioned into disjoint subdomains where either the full two-phase or the simplified Richards model dynamics are valid. Extending the previously considered one-model situations we suggest coupling conditions for this hybrid model approach. Based on an Euler implicit discretization, a linear iterative (L-type) domain decomposition scheme is proposed, and proved to be convergent. The theoretical findings are verified by a comparative numerical study that in particular confirms the efficiency of the hybrid ansatz as compared to full two-phase model computations.

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“…For new coupling conditions between domains with different two‐phase flow models we developed an approach combining an

L

Section: Discussionmentioning

confidence: 80%

Section: Introductionmentioning

confidence: 99%

Towards hybrid two‐phase modelling using linear domain decomposition

Seus

Radu

Rohde

2022

Numerical Methods Partial

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“…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.…”

Section: Presentation Of the Problemmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Ahmed

Japhet

Kern

2020

Computer Methods in Applied Mechanics and Engineering

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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.

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“…In [2], a space-time domain decomposition method formulated using Robin and Ventcell type coupling conditions was applied to a simple two-phase flow model between different rock types. One can also mention the linear domain decomposition method presented in [35] where two-phase flow equations are given with the physical variables. The problem is then decomposed into a set of subproblems in different subdomains, and solved in each time step in a fixed point type iteration based on the L-scheme linearization method [32].…”

Section: Introductionmentioning

confidence: 99%

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

Ahmed

2019

Advances in Water Resources

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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

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A Linear Domain Decomposition Method for Two-Phase Flow in Porous Media

Cited by 6 publications

References 14 publications

Towards hybrid two‐phase modelling using linear domain decomposition

Towards hybrid two‐phase modelling using linear domain decomposition

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

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

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