Fast computation of multiphase flow in porous media by implicit discontinuous Galerkin schemes with optimal ordering of elements

Natvig, Jostein R.; Lie, Knut–Andreas

doi:10.1016/j.jcp.2008.08.024

Cited by 117 publications

(71 citation statements)

References 31 publications

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“…This study focuses on establishing the suitability of nonlinear multilevel preconditioning for porous media. Further improvements can be expected by combining this preconditioning with dedicated approaches for multiphase flow in porous media, such as reordering (Kwok and Tchelepi 2007;Natvig and Lie 2008) and CPR (Wallis et al 1985). It is also natural to consider a posteriori error estimators along the lines presented in Vohralík and Wheeler (2013) in order to optimize stopping criteria for the different iterative processes involved.…”

Section: Discussionmentioning

confidence: 99%

“…Existing strategies that target the nonlinear solver in order to lower the number of Newton iterations include cascade and reordering methods (Appleyard and Cheshire 1982;Kwok and Tchelepi 2007;Natvig and Lie 2008) and similar adaptive localization methods (Younis et al 2010) and nonlinear preconditioning . Our interest is in the applicability of domain decomposition type strategies for preconditioning the Newton iterations.…”

Section: Introductionmentioning

confidence: 99%

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Two-Scale Preconditioning for Two-Phase Nonlinear Flows in Porous Media

2015

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Solving realistic problems related to flow in porous media to desired accuracy may be prohibitively expensive with available computing resources. Multiscale effects and nonlinearities in the governing equations are among the most important contributors to this situation. Hence, developing methods that handle these features better is essential in order to be able to solve the problems more efficiently. Focus has until recently largely been on preconditioners for linearized problems. This article proposes a two-scale nonlinear preconditioning technique for flow problems in porous media that allows for incorporating physical intuition directly in the preconditioner. By assuming a certain dominant physical process, this technique will resemble upscaling in the equilibrium limit, with the computational benefits that follow. In this study, the method is established as a preconditioner with good scalability properties for challenging problems regardless of dominant physics, thus laying the foundation for further studies with physical information in the preconditioner.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Two-Scale Preconditioning for Two-Phase Nonlinear Flows in Porous Media

2015

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“…3 The extension to non-orthogonal grids is not unique, and the TPFA method does not give a consistent discretisation in this case. Because of its simplicity, the TPFA method is strictly monotone if T ii > 0, which implies that the fluxes form a directed acyclic graph, a property that can be used to accelerate the solution of the transport equations considerably, as discussed in [28].…”

Section: Two-point Type Methodsmentioning

confidence: 99%

Open-source MATLAB implementation of consistent discretisations on complex grids

et al. 2011

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Accurate geological modelling of features such as faults, fractures or erosion requires grids that are flexible with respect to geometry. Such grids generally contain polyhedral cells and complex grid-cell connectivities. The grid representation for polyhedral grids in turn affects the efficient implementation of numerical methods for subsurface flow simulations. It is well known that conventional two-point flux-approximation methods are only consistent for K-orthogonal grids and will, therefore, not converge in the general case. In recent years, there has been significant research into consistent and convergent methods, including mixed, multipoint and mimetic discretisation methods. Likewise, the so-called multiscale methods based upon hierarchically coarsened grids have received a lot of attention. The paper does not propose novel mathematical methods but instead presents an open-source Matlab® toolkit that can be used as an efficient test platform for (new) discretisation and solution methods in reservoir simulation. The aim of the toolkit is to support reproducible research and simplify the development, verification and validation and testing and comparison of new discretisation and solution methods on general unstructured grids, including in particular corner point and 2.5D PEBI grids. The toolkit consists of a set of data structures and routines for creating, manipulating and visualising petrophysical data, fluid models and (unstructured) grids, including support for industry standard input formats, as well as routines for computing single and multiphase (incompressible) flow. We review key features of the toolkit and discuss a generic mimetic formulation that includes many known discretisation methods, including both the standard two-point method as well as consistent and convergent multipoint and mimetic methods. Apart from the core routines and data structures, the toolkit contains addon modules that implement more advanced solvers and functionality. Herein, we show examples of multiscale methods and adjoint methods for use in optimisation of rates and placement of wells.

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“…Because nothing was done to speed up the transport solve, the overall speed-up for the two-phase flow simulation (high-dimensional vs. basis construction and reduced simulation) is five. To also speed up the transport solve significantly, one could replace the explicit temporal discretization by a backward Euler scheme and utilize the fact that the resulting nonlinear system share the same unidirectional flow properties as the time-of-flight equation and hence can be computed in a per-element fashion with local control over the nonlinear iterations; see [53] for details. Note that in this case, the slope limiter presented in Section 3 needs to be replaced by a different kind of stabilization as it is not compliant with implicit time stepping schemes.…”

Section: Numerical Experimentsmentioning

confidence: 99%

The localized reduced basis multiscale method for two‐phase flows in porous media

Kaulmann¹,

Flemisch

Haasdonk³

et al. 2014

Numerical Meth Engineering

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In this work, we propose a novel model order reduction approach for twophase flow in porous media by introducing a formulation in which the mobility, which realizes the coupling between phase saturations and phase pressures, is regarded as a parameter to the pressure equation. Using this formulation, we introduce the Localized Reduced Basis Multiscale method to obtain a low-dimensional surrogate of the high-dimensional pressure equation. By applying ideas from model order reduction for parametrized partial differential equations, we are able to split the computational effort for solving the pressure equation into a costly offline step that is performed only once and an inexpensive online step that is carried out in every time step of the two-phase flow simulation, which is thereby largely accelerated. Usage of elements from numerical multiscale methods allows us to displace the computational intensity between the offline and online step to reach an ideal runtime at acceptable error increase for the two-phase flow simulation.

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Fast computation of multiphase flow in porous media by implicit discontinuous Galerkin schemes with optimal ordering of elements

Cited by 117 publications

References 31 publications

Two-Scale Preconditioning for Two-Phase Nonlinear Flows in Porous Media

Two-Scale Preconditioning for Two-Phase Nonlinear Flows in Porous Media

Open-source MATLAB implementation of consistent discretisations on complex grids

The localized reduced basis multiscale method for two‐phase flows in porous media

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