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A monotone nonlinear finite volume method for diffusion equations and multiphase flows

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Abstract

We present a new nonlinear monotone finite volume method for diffusion equation and its application to two-phase flow model. We consider full anisotropic discontinuous diffusion or permeability tensors on conformal polyhedral meshes. The approximation of the diffusive flux uses the nonlinear two-point stencil which provides the conventional seven-point stencil for the discrete diffusion operator on cubic meshes. We show that the quality of the discrete flux in a reservoir simulator has great effect on the front behavior and the water breakthrough time. We compare two two-point flux approximations (TPFA), the proposed nonlinear TPFA and the conventional linear TPFA, and multipoint flux approximation (MPFA). The new nonlinear scheme has a number of important advantages over the traditional linear discretizations. Compared to the linear TPFA, the nonlinear TPFA demonstrates low sensitivity to grid distortions and provides appropriate approximation in case of full anisotropic permeability tensor. For nonorthogonal grids or full anisotropic permeability tensors, the conventional linear TPFA provides no approximation, while the nonlinear flux is still first-order accurate. The computational work for the new method is higher than the one for the conventional TPFA, yet it is rather competitive. Compared to MPFA, the new scheme provides sparser algebraic systems and thus is less computational expensive. Moreover, it is monotone which means that the discrete solution preserves the nonnegativity of the differential solution.

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Authors and Affiliations

  1. Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia

    Kirill Nikitin, Kirill Terekhov & Yuri Vassilevski

  2. Nuclear Safety Institute, Russian Academy of Sciences, Moscow, Russia

    Kirill Nikitin & Kirill Terekhov

  3. Moscow Institute of Physics and Technology, Dolgoprudny, Russia

    Yuri Vassilevski

Authors
  1. Kirill Nikitin
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  2. Kirill Terekhov
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  3. Yuri Vassilevski
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Correspondence to Kirill Nikitin.

Additional information

This work has been supported in part by RFBR grants 11-01-00971, 12-01-31275, 12-01-33084, Russian Presidential grant MK-7159.2013.1, Federal target programs “Scientific and scientific-pedagogical personnel of innovative Russia” and “Research and development for priority directions of science and technology complex of Russia”, ExxonMobil Upstream Research Company, and project “Breakthrough” of Rosatom

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Nikitin, K., Terekhov, K. & Vassilevski, Y. A monotone nonlinear finite volume method for diffusion equations and multiphase flows. Comput Geosci 18, 311–324 (2014). https://doi.org/10.1007/s10596-013-9387-6

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  • DOI: https://doi.org/10.1007/s10596-013-9387-6

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