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Operator splitting methods for degenerate convection–diffusion equations II: numerical examples with emphasis on reservoir simulation and sedimentation

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Abstract

We present an accurate numerical method for a large class of scalar, strongly degenerate convection–diffusion equations. Important subclasses are hyperbolic conservation laws, porous medium type equations, two-phase reservoir flow equations, and strongly degenerate equations coming from the recent theory of sedimentation–consolidation processes. The method is based on splitting the convective and the diffusive terms. The nonlinear, convective part is solved using front tracking and dimensional splitting, while the nonlinear diffusion part is solved by an implicit–explicit finite difference scheme. In addition, one version of the implemented operator splitting method has a mechanism built in for detecting and correcting unphysical entropy loss, which may occur when the time step is large. This mechanism helps us gain a large time step ability for practical computations. A detailed convergence analysis of the operator splitting method was given in Part I. Here we present numerical experiments with the method for examples modelling secondary oil recovery and sedimentation–consolidation processes. We demonstrate that the splitting method resolves sharp gradients accurately, may use large time steps, has first order convergence, exhibits small grid orientation effects, has small mass balance errors, and is rather efficient.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Norwegian University of Science and Technology, N-7491, Trondheim, Norway

    Helge Holden

  2. Department of Mathematics, University of Bergen, Johs. Brunsgt. 12, N-5008, Bergen, Norway

    Kenneth Hvistendahl Karlsen

  3. Department of Informatics, University of Oslo, P.O. Box 1080, Blindern, N-0316, Oslo, Norway

    Knut-Andreas Lie

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Holden, H., Karlsen, K.H. & Lie, KA. Operator splitting methods for degenerate convection–diffusion equations II: numerical examples with emphasis on reservoir simulation and sedimentation. Computational Geosciences 4, 287–322 (2000). https://doi.org/10.1023/A:1011582819188

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