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Title: A scalable variational inequality approach for flow through porous media models with pressure-dependent viscosity

Abstract

Mathematical models for flow through porous media typically enjoy the so-called maximum principles, which place bounds on the pressure field. It is highly desirable to preserve these bounds on the pressure field in predictive numerical simulations, that is, one needs to satisfy discrete maximum principles (DMP). Unfortunately, many of the existing formulations for flow through porous media models do not satisfy DMP. This paper presents a robust, scalable numerical formulation based on variational inequalities (VI), to model non-linear flows through heterogeneous, anisotropic porous media without violating DMP. VI is an optimization technique that places bounds on the numerical solutions of partial differential equations. To crystallize the ideas, a modification to Darcy equations by taking into account pressure-dependent viscosity will be discretized using the lowest-order Raviart–Thomas (RT0) and Variational Multi-scale (VMS) finite element formulations. It will be shown that these formulations violate DMP, and, in fact, these violations increase with an increase in anisotropy. It will be shown that the proposed VI-based formulation provides a viable route to enforce DMP. Moreover, it will be shown that the proposed formulation is scalable, and can work with any numerical discretization and weak form. Aseriesof numerical benchmark problems are solved to demonstrate the effectsmore » of heterogeneity, anisotropy and non-linearity on DMP violations under the two chosen formulations (RT0 and VMS), and that of non-linearity on solver convergence for the proposed VI-based formulation. Parallel scalability on modern computational platforms will be illustrated through strong-scaling studies, which will prove the efficiency of the proposed formulation in a parallel setting. Algorithmic scalability as the problem size is scaled up will be demonstrated through novel static-scaling studies. The performed static-scaling studies can serve as a guide for users to be able to select an appropriate discretization for a given problem size.« less

Authors:
 [1];  [2]; ORCiD logo [1]
  1. University of Houston, Houston, TX (United States). Department of Civil and Environmental Engineering
  2. University of Houston, Houston, TX (United States). Department of Civil and Environmental Engineering; Rice University, Houston, TX (United States). Department of Computational and Applied Mathematics
Publication Date:
Research Org.:
Lawrence Berkeley National Laboratory, Berkeley, CA (United States). National Energy Research Scientific Computing Center (NERSC).
Sponsoring Org.:
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)
OSTI Identifier:
1463653
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Computational Physics
Additional Journal Information:
Journal Volume: 359; Journal Issue: C; Journal ID: ISSN 0021-9991
Publisher:
Elsevier
Country of Publication:
United States
Language:
English
Subject:
42 ENGINEERING; Variational inequalities; Pressure-dependent viscosity; Anisotropy; Maximum principles; Flow though porous media; Parallel computing

Citation Formats

Mapakshi, N. K., Chang, J., and Nakshatrala, K. B. A scalable variational inequality approach for flow through porous media models with pressure-dependent viscosity. United States: N. p., 2018. Web. doi:10.1016/j.jcp.2018.01.022.
Mapakshi, N. K., Chang, J., & Nakshatrala, K. B. A scalable variational inequality approach for flow through porous media models with pressure-dependent viscosity. United States. doi:10.1016/j.jcp.2018.01.022.
Mapakshi, N. K., Chang, J., and Nakshatrala, K. B. Sun . "A scalable variational inequality approach for flow through porous media models with pressure-dependent viscosity". United States. doi:10.1016/j.jcp.2018.01.022. https://www.osti.gov/servlets/purl/1463653.
@article{osti_1463653,
title = {A scalable variational inequality approach for flow through porous media models with pressure-dependent viscosity},
author = {Mapakshi, N. K. and Chang, J. and Nakshatrala, K. B.},
abstractNote = {Mathematical models for flow through porous media typically enjoy the so-called maximum principles, which place bounds on the pressure field. It is highly desirable to preserve these bounds on the pressure field in predictive numerical simulations, that is, one needs to satisfy discrete maximum principles (DMP). Unfortunately, many of the existing formulations for flow through porous media models do not satisfy DMP. This paper presents a robust, scalable numerical formulation based on variational inequalities (VI), to model non-linear flows through heterogeneous, anisotropic porous media without violating DMP. VI is an optimization technique that places bounds on the numerical solutions of partial differential equations. To crystallize the ideas, a modification to Darcy equations by taking into account pressure-dependent viscosity will be discretized using the lowest-order Raviart–Thomas (RT0) and Variational Multi-scale (VMS) finite element formulations. It will be shown that these formulations violate DMP, and, in fact, these violations increase with an increase in anisotropy. It will be shown that the proposed VI-based formulation provides a viable route to enforce DMP. Moreover, it will be shown that the proposed formulation is scalable, and can work with any numerical discretization and weak form. Aseriesof numerical benchmark problems are solved to demonstrate the effects of heterogeneity, anisotropy and non-linearity on DMP violations under the two chosen formulations (RT0 and VMS), and that of non-linearity on solver convergence for the proposed VI-based formulation. Parallel scalability on modern computational platforms will be illustrated through strong-scaling studies, which will prove the efficiency of the proposed formulation in a parallel setting. Algorithmic scalability as the problem size is scaled up will be demonstrated through novel static-scaling studies. The performed static-scaling studies can serve as a guide for users to be able to select an appropriate discretization for a given problem size.},
doi = {10.1016/j.jcp.2018.01.022},
journal = {Journal of Computational Physics},
number = C,
volume = 359,
place = {United States},
year = {2018},
month = {4}
}

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