Computational Geosciences

, Volume 22, Issue 2, pp 565–586 | Cite as

Monotone nonlinear finite-volume method for challenging grids

  • M. Schneider
  • B. Flemisch
  • R. Helmig
  • K. Terekhov
  • H. Tchelepi
Original Paper


This article presents a new positivity-preserving finite-volume scheme with a nonlinear two-point flux approximation, which uses optimization techniques for the face stencil calculation. The gradient is reconstructed using harmonic averaging points with the constraint that the sum of the coefficients included in the face stencils must be positive. We compare the proposed scheme to a nonlinear two-point scheme available in literature and a few linear schemes. Using two test cases, taken from the FVCA6 benchmarks, the accuracy of the scheme is investigated. Furthermore, it is shown that the scheme is linearity-preserving on highly complex corner-point grids. Moreover, a two-phase flow problem on the Norne formation, a geological formation in the Norwegian Sea, is simulated. It is demonstrated that the proposed scheme is consistent in contrast to the linear Two-Point Flux Approximation scheme, which is industry standard for simulating subsurface flow on corner-point grids.


Finite-volume method Monotone discretization Corner-point grid Challenging grids 


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The authors Bernd Flemisch, Rainer Helmig, and Martin Schneider would like to thank the German Research Foundation (DFG) for financial support of the project within the Cluster of Excellence in Simulation Technology (EXC 310/2) at the University of Stuttgart.


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute for Modelling Hydraulic and Environmental Systems (IWS)University StuttgartStuttgartGermany
  2. 2.Energy Resources EngineeringStanford UniversityStanfordUSA

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