DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

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 Lab. (LBNL), Berkeley, CA (United States). National Energy Research Scientific Computing Center (NERSC)
Sponsoring Org.:
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)
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. https://doi.org/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. https://doi.org/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}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record

Citation Metrics:
Cited by: 6 works
Citation information provided by
Web of Science

Save / Share:

Works referenced in this record:

The finite element method with Lagrangian multipliers
journal, June 1973


Isothermals, isopiestics and isometrics relative to viscosity
journal, February 1893


Numerical solution of saddle point problems
journal, April 2005


Variational inequality approach to enforcing the non-negative constraint for advection–diffusion equations
journal, June 2017

  • Chang, J.; Nakshatrala, K. B.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 320
  • DOI: 10.1016/j.cma.2017.03.022

Large-Scale Optimization-Based Non-negative Computational Framework for Diffusion Equations: Parallel Implementation and Performance Studies
journal, July 2016


Modification to Darcy-Forchheimer Model due to Pressure-Dependent Viscosity: Consequences and Numerical Solutions
journal, January 2017


Maximum principle and uniform convergence for the finite element method
journal, February 1973


Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems
journal, January 2009

  • Cockburn, Bernardo; Gopalakrishnan, Jayadeep; Lazarov, Raytcho
  • SIAM Journal on Numerical Analysis, Vol. 47, Issue 2
  • DOI: 10.1137/070706616

Two classes of mixed finite element methods
journal, July 1988

  • Franca, Leopoldo P.; Hughes, Thomas J. R.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 69, Issue 1
  • DOI: 10.1016/0045-7825(88)90168-5

On steady flows of fluids with pressure– and shear–dependent viscosities
journal, March 2005

  • Franta, M.; Málek, J.; Rajagopal, K. R.
  • Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 461, Issue 2055
  • DOI: 10.1098/rspa.2004.1360

Numerical simulations and global existence of solutions of two-dimensional flows of fluids with pressure- and shear-dependent viscosities
journal, January 2003


Discrete Maximum Principle for the Weak Galerkin Method for Anisotropic Diffusion Problems
journal, July 2015


Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods
journal, November 1995


A semismooth equation approach to the solution of nonlinear complementarity problems
journal, December 1996

  • De Luca, Tecla; Facchinei, Francisco; Kanzow, Christian
  • Mathematical Programming, Vol. 75, Issue 3
  • DOI: 10.1007/BF02592192

Cross-Loop Optimization of Arithmetic Intensity for Finite Element Local Assembly
journal, January 2015

  • Luporini, Fabio; Varbanescu, Ana Lucia; Rathgeber, Florian
  • ACM Transactions on Architecture and Code Optimization, Vol. 11, Issue 4
  • DOI: 10.1145/2687415

An Algorithm for the Optimization of Finite Element Integration Loops
journal, July 2017

  • Luporini, Fabio; Ham, David A.; Kelly, Paul H. J.
  • ACM Transactions on Mathematical Software, Vol. 44, Issue 1
  • DOI: 10.1145/3054944

Global Analysis of the Flows of Fluids with Pressure-Dependent Viscosities
journal, December 2002

  • MÁLek, Josef; NeČAs, JindŘIch; Rajagopal, K. R.
  • Archive for Rational Mechanics and Analysis, Vol. 165, Issue 3
  • DOI: 10.1007/s00205-002-0219-4

On mesh restrictions to satisfy comparison principles, maximum principles, and the non-negative constraint: Recent developments and new results
journal, September 2016

  • Mudunuru, Maruti; Nakshatrala, Kalyana Babu
  • Mechanics of Advanced Materials and Structures, Vol. 24, Issue 7
  • DOI: 10.1080/15502287.2016.1166160

The Semismooth Algorithm for Large Scale Complementarity Problems
journal, November 2001

  • Munson, Todd S.; Facchinei, Francisco; Ferris, Michael C.
  • INFORMS Journal on Computing, Vol. 13, Issue 4
  • DOI: 10.1287/ijoc.13.4.294.9734

A Note on Preconditioning for Indefinite Linear Systems
journal, January 2000

  • Murphy, Malcolm F.; Golub, Gene H.; Wathen, Andrew J.
  • SIAM Journal on Scientific Computing, Vol. 21, Issue 6
  • DOI: 10.1137/S1064827599355153

Enforcing the non-negativity constraint and maximum principles for diffusion with decay on general computational grids
journal, August 2010

  • Nagarajan, H.; Nakshatrala, K. B.
  • International Journal for Numerical Methods in Fluids, Vol. 67, Issue 7
  • DOI: 10.1002/fld.2389

A numerical study of fluids with pressure-dependent viscosity flowing through a rigid porous medium
journal, May 2010

  • Nakshatrala, K. B.; Rajagopal, K. R.
  • International Journal for Numerical Methods in Fluids, Vol. 67, Issue 3
  • DOI: 10.1002/fld.2358

A Mixed Formulation for a Modification to Darcy Equation Based on Picard Linearization and Numerical Solutions to Large-Scale Realistic Problems
journal, September 2013

  • Nakshatrala, K. B.; Turner, D. Z.
  • International Journal for Computational Methods in Engineering Science and Mechanics, Vol. 14, Issue 6
  • DOI: 10.1080/15502287.2013.822942

Non-negative mixed finite element formulations for a tensorial diffusion equation
journal, October 2009


A numerical framework for diffusion-controlled bimolecular-reactive systems to enforce maximum principles and the non-negative constraint
journal, November 2013

  • Nakshatrala, K. B.; Mudunuru, M. K.; Valocchi, A. J.
  • Journal of Computational Physics, Vol. 253
  • DOI: 10.1016/j.jcp.2013.07.010

A Numerical Methodology for Enforcing Maximum Principles and the Non-Negative Constraint for Transient Diffusion Equations
journal, January 2016

  • Nakshatrala, K. B.; Nagarajan, H.; Shabouei, M.
  • Communications in Computational Physics, Vol. 19, Issue 1
  • DOI: 10.4208/cicp.180615.280815a

Adaptive spacetime discontinuous Galerkin method for hyperbolic advection-diffusion with a non-negativity constraint: HYPERBOLIC ADVECTION-DIFFUSION WITH NON-NEGATIVITY CONSTRAINT
journal, September 2015

  • Pal, Raj Kumar; Abedi, Reza; Madhukar, Amit
  • International Journal for Numerical Methods in Engineering, Vol. 105, Issue 13
  • DOI: 10.1002/nme.4999

On a Discrete Maximum Principle
journal, June 1966

  • Varga, Richard S.
  • SIAM Journal on Numerical Analysis, Vol. 3, Issue 2
  • DOI: 10.1137/0703029

Works referencing / citing this record:

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