Accelerating multiscale simulation of complex geomodels by use of dynamically adapted basis functions

Klemetsdal, Øystein; Møyner, Olav; Lie, Knut–Andreas

doi:10.1007/s10596-019-9827-z

Cited by 12 publications

(9 citation statements)

References 46 publications

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“…Conservation of mass in each block, as defined in Eq. 17, is then enforced by multiplying the saturation by a saturation discrepancy (see Klemetsdal et al 2019b for details). Altogether, this ensures that the mapping is both mass conservative and consistent with the flow model.…”

Section: Dynamic Coarseningmentioning

confidence: 99%

“…However, careful examination of several numerical examples conducted on real field models indicates that even in cases with strong gravity effects, cells belonging to a connected component only amount to a small fraction of the total number of grid cells. Some of these results are reported in Klemetsdal et al (2019bKlemetsdal et al ( , 2019d. One can also use a nonlinear Gauss-Seidel solver Lie et al 2014) in LRNTS, in which the nodes in a connected component are solved sequentially in a predefined order with variables in all other nodes fixed.…”

Section: Localized Nonlinear Solversmentioning

confidence: 99%

“…This is admittedly not sufficient to describe real reservoir fluids but greatly simplifies the interpretation of numerical results. More complex cases involving compositional simulations have been demonstrated for an early version of the dynamic coarsening method in Klemetsdal et al (2019b), and the efficiency of LRNTS has been demonstrated for black oil and compositional simulations in Klemetsdal et al (2019bKlemetsdal et al ( , 2019d. All results reported in the following are based on a proofof-concept implementation in the MRST (Lie 2019).…”

Section: Numerical Examplesmentioning

confidence: 99%

“…However, whereas dynamic refinement usually introduces a structured subdivision of cells in a structured grid, coarsening methods can easily form nonuniform and more irregular blocks to adapt to flow patterns or features in the reservoir model (Durlofsky et al 1997;Aarnes et al 2007;Hauge et al 2012;Durlofsky 2014, 2018;Guion et al 2019;Klemetsdal et al 2019a). In particular, Hauge et al (2012) discussed an example of dynamic coarsening with nonuniform, flow-adapted coarse blocks, whereas Klemetsdal et al (2019b) introduced a two-level version of the framework we develop further herein. The dynamic multiscale operator method for transport equations introduced recently in and Cusini and Hajibeygi (2018) also has some similarities with our approach but has so far only been developed on rectilinear grids.…”

Section: Introductionmentioning

confidence: 99%

“…Likewise, one can permute the transport equations into a block-triangular form that can be solved efficiently by a (nonlinear) back-substitution method by reordering the grid cells according to fluid-phase potential (Appleyard and Cheshire 1982;Kwok and Tchelepi 2007;Hamon and Tchelepi 2016), or based on the total flux from the pressure equation (Aarnes et al 2006;Natvig and Lie 2008). This has been demonstrated for a wide range of scenarios, including linear transport equations (Natvig et al 2007;Eikemo et al 2009; Rasmussen and Lie 2014), two-phase flow (Natvig and Lie 2008), and polymer flooding (Lie et al 2014;Raynaud et al 2016), and more recently for compressible black oil (Klemetsdal et al 2019d) and compositional problems (Klemetsdal et al 2019b).…”

Section: Introductionmentioning

confidence: 99%

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Dynamic Coarsening and Local Reordered Nonlinear Solvers for Simulating Transport in Porous Media

Klemetsdal

Lie

2020

SPE Journal

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Summary We present a robust and flexible sequential solution approach in which the flow equation is solved on the original grid, whereas the transport equations are solved with a new dynamic coarsening method that adapts the grid resolution locally to reduce the number of cells as much as possible. The resulting grid is formed by combining precomputed coarse partitions of an underlying fine model. Our approach is flexible and makes very few assumptions on cell geometries and the topology of the grid. To further accelerate the transport step, we combine dynamic coarsening with a local nonlinear solver that permutes the discrete transport equations into an optimal block-triangular form so that these can be solved very efficiently using a nonlinear back-substitution method. Efficiency and utility of the overall approach are assessed through a number of conceptual test cases, including the Olympus field model.

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Section: Dynamic Coarseningmentioning

confidence: 99%

Section: Localized Nonlinear Solversmentioning

confidence: 99%

Section: Numerical Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Dynamic Coarsening and Local Reordered Nonlinear Solvers for Simulating Transport in Porous Media

Klemetsdal

Lie

2020

SPE Journal

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Block constrained pressure residual preconditioning for two-phase flow in porous media by mixed hybrid finite elements

2023

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This work proposes an original preconditioner that couples the Constrained Pressure Residual (CPR) method with block preconditioning for the efficient solution of the linearized systems of equations arising from fully implicit multiphase flow models. This preconditioner, denoted as Block CPR (BCPR), is specifically designed for Lagrange multipliers-based flow models, such as those generated by Mixed Hybrid Finite Element (MHFE) approximations. An original MHFE-based formulation of the two-phase flow model is taken as a reference for the development of the BCPR preconditioner, in which the set of system unknowns comprises both element and face pressures, in addition to the cell saturations, resulting in a $$3\times 3$$ 3 × 3 block-structured Jacobian matrix with a $$2\times 2$$ 2 × 2 inner pressure problem. The CPR method is one of the most established techniques for reservoir simulations, but most research focused on solutions for Two-Point Flux Approximation (TPFA)-based discretizations that do not readily extend to our problem formulation. Therefore, we designed a dedicated two-stage strategy, inspired by the CPR algorithm, where a block preconditioner is used for the pressure part with the aim at exploiting the inner $$2\times 2$$ 2 × 2 structure. The proposed preconditioning framework is tested by an extensive experimentation, comprising both synthetic and realistic applications in Cartesian and non-Cartesian domains.

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Linear Solvers for Reservoir Simulation Problems: An Overview and Recent Developments

Nardean

Ferronato

Abushaikha

2022

Arch Computat Methods Eng

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Linear solvers for reservoir simulation applications are the objective of this review. Specifically, we focus on techniques for Fully Implicit (FI) solution methods, in which the set of governing Partial Differential Equations (PDEs) is properly discretized in time (usually by the Backward Euler scheme), and space, and tackled by assembling and linearizing a single system of equations to solve all the model unknowns simultaneously. Due to the usually large size of these systems arising from real-world models, iterative methods, specifically Krylov subspace solvers, have become conventional choices; nonetheless, their success largely revolves around the quality of the preconditioner that is supplied to accelerate their convergence. These two intertwined elements, i.e., the solver and the preconditioner, are the focus of our analysis, especially the latter, which is still the subject of extensive research. The progressive increase in reservoir model size and complexity, along with the introduction of additional physics to the classical flow problem, display the limits of existing solvers. Intensive usage of computational and memory resources are frequent drawbacks in practice, resulting in unpleasantly slow convergence rates. Developing efficient, robust, and scalable preconditioners, often relying on physics-based assumptions, is the way to avoid potential bottlenecks in the solving phase. In this work, we proceed in reviewing principles and state-of-the-art of such linear solution tools to summarize and discuss the main advances and research directions for reservoir simulation problems. We compare the available preconditioning options, showing the connections existing among the different approaches, and try to develop a general algebraic framework.

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Accelerating multiscale simulation of complex geomodels by use of dynamically adapted basis functions

Cited by 12 publications

References 46 publications

Dynamic Coarsening and Local Reordered Nonlinear Solvers for Simulating Transport in Porous Media

Dynamic Coarsening and Local Reordered Nonlinear Solvers for Simulating Transport in Porous Media

Block constrained pressure residual preconditioning for two-phase flow in porous media by mixed hybrid finite elements

Linear Solvers for Reservoir Simulation Problems: An Overview and Recent Developments

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