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This work presents an advanced numerical framework for simulating two-phase and three-phase flows in high-contrast porous media by integrating semi-discrete Lagrangian-Eulerian (SDLE) schemes with Generalized Multiscale Finite Elements (GMsFEM), which is based on the work [1,2]; see also [3,4,5,6]. Novel and key highlights of the proposed approach include: 1) A novel class of SDLE schemes is combined with enhanced GMsFEM, specifically designed to handle high-contrast multiscale porous media. 2) Hyperbolic-Transport Subproblem: The approach utilizes a non-splitting semi-discrete Lagrangian-Eulerian method (i.e., no dimensional splitting technique is employed); Numerical experiments and potential MPI parallel computing results are used to validate the Lagrangian-Eulerian method's performance . 3) Elliptic Pressure-Velocity-Flow Subproblem: A new design and proof-of-concept GMsFEM approach is applied. 4) Stability, Accuracy, and Theoretical Connections: The method is subject to a new weak CFL stability condition and satisfies a weak version of the positivity principle proposed by P. Lax and X.-D. Liu for multidimensional hyperbolic systems. A connection is established between the numerical results and the work of A. Bressan regarding local existence and continuous dependence for discontinuous ODEs, interpreting no-flow curves as a forward vector field with locally bounded variation. We also simulated the SPE10 oil exploration benchmark on quadrilateral grids. In conclusion, we will discuss how integrating a novel class of semi-discrete Lagrangian-Eulerian Schemes subject to a new weak CFL stability condition with enhanced generalized multiscale finite elements for two-phase flow and three-phase flow simulations in high-contrast multiscale porous media.
[1] E. Abreu, P. Ferraz, J. R. François, J. Galvis (Submitted to Journal of Computational and Applied Mathematics, under review R1 - Positive report with minor issues), Integrating Semi-Discrete
Lagrangian-Eulerian Schemes with Generalized Multiscale Finite Elements for Enhanced Two- and Three-Phase Flow Simulations.
[2] E. Abreu, P. Ferraz, J. R. François, J. Galvis, presented at (SIAM-GS25) MS18 (Recent Advances in Multiscale Model Reduction) SIAM Conference on Mathematical & Computational Issues in the Geosciences (GS25), A Semi-Discrete Lagrangian-Eulerian Approach with Enhanced Generalized Multiscale Finite Elements for 3-Phase Flows in High-Contrast Multiscale Porous Media. https://meetings.siam.org/sess/dsp_talk.cfm?p=149952 | https://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=85535
[3] E. Abreu, V. Matos, J. Perez and P. Rodriguez-Bermudez. Riemann problem solutions for a balance law under Dirac-Delta source with a discontinuous flux, Journal of Hyperbolic Differential Equations, v. 21(1) (2024) p. 1-32. LINK: https://doi.org/10.1142/S0219891624500012
[4] E. Abreu, C. Díaz, J. Galvis, J. Pérez, On the Conservation Properties in Multiple Scale Coupling and Simulation for Darcy Flow with Hyperbolic-Transport in Complex Flows. MULTISCALE MODELING & SIMULATION, v.18 (2020) p.1375-1408. LINK: https://epubs.siam.org/doi/10.1137/20M1320250
[5] E. Abreu, P. Ferraz, W. Lambert. A study of non-equilibrium wave groups in two-phase flow in high-contrast porous media with relative permeability hysteresis, Communications in Nonlinear Science and Numerical Simulation v. 127 (2023) 107552. LINK: https://doi.org/10.1016/j.cnsns.2023.107552
[6] E. Abreu, L. Hernandez, D. Pardo, E. Abreu, J. Muñoz-Matute, and Ciro Díaz, Juan Galvis. An exponential integration generalized multiscale finite element method for parabolic problems. Journal of Computational Physics v. 479, 15 April 2023, 112014 LINK: https://doi.org/10.1016/j.jcp.2023.112014
| References | [1] E. Abreu, P. Ferraz, J. R. François, J. Galvis (Submitted to Journal of Computational and Applied Mathematics, under review R1 - Positive report with minor issues), Integrating Semi-Discrete Lagrangian-Eulerian Schemes with Generalized Multiscale Finite Elements for Enhanced Two- and Three-Phase Flow Simulations. [2] E. Abreu, P. Ferraz, J. R. François, J. Galvis, presented at (SIAM-GS25) MS18 (Recent Advances in Multiscale Model Reduction) SIAM Conference on Mathematical & Computational Issues in the Geosciences (GS25), A Semi-Discrete Lagrangian-Eulerian Approach with Enhanced Generalized Multiscale Finite Elements for 3-Phase Flows in High-Contrast Multiscale Porous Media. [3] E. Abreu, V. Matos, J. Perez and P. Rodriguez-Bermudez. Riemann problem solutions for a balance law under Dirac-Delta source with a discontinuous flux, Journal of Hyperbolic Differential Equations, v. 21(1) (2024) p. 1-32. LINK: https://doi.org/10.1142/S0219891624500012 [4] E. Abreu, C. Díaz, J. Galvis, J. Pérez, On the Conservation Properties in Multiple Scale Coupling and Simulation for Darcy Flow with Hyperbolic-Transport in Complex Flows. MULTISCALE MODELING & SIMULATION, v.18 (2020) p.1375-1408. LINK: https://epubs.siam.org/doi/10.1137/20M1320250 [5] E. Abreu, P. Ferraz, W. Lambert. A study of non-equilibrium wave groups in two-phase flow in high-contrast porous media with relative permeability hysteresis, Communications in Nonlinear Science and Numerical Simulation v. 127 (2023) 107552. LINK: https://doi.org/10.1016/j.cnsns.2023.107552 [6] E. Abreu, L. Hernandez, D. Pardo, E. Abreu, J. Muñoz-Matute, and Ciro Díaz, Juan Galvis. An exponential integration generalized multiscale finite element method for parabolic problems. Journal of Computational Physics v. 479, 15 April 2023, 112014 LINK: https://doi.org/10.1016/j.jcp.2023.112014 |
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| Country | Brazil |
| Green Housing & Porous Media Focused Abstracts | This abstract is related to Green Housing |
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