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

Journal Article · · Journal of Computational Physics
 [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
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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.

Research Organization:
Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States). National Energy Research Scientific Computing Center (NERSC)
Sponsoring Organization:
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)
OSTI ID:
1463653
Journal Information:
Journal of Computational Physics, Vol. 359, Issue C; ISSN 0021-9991
Publisher:
ElsevierCopyright Statement
Country of Publication:
United States
Language:
English
Citation Metrics:
Cited by: 13 works
Citation information provided by
Web of Science

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Cited By (3)

A global sensitivity analysis and reduced-order models for hydraulically fractured horizontal wells journal February 2020
On Numerical Stabilization in Modeling Double-Diffusive Viscous Fingering journal January 2020
Comparative study of finite element methods using the Time-Accuracy-Size (TAS) spectrum analysis preprint January 2018

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Related Subjects

42 ENGINEERING
Variational inequalities
Pressure-dependent viscosity
Anisotropy
Maximum principles
Flow though porous media
Parallel computing