Commun. Comput. Phys., 11 (2012), pp. 19-47.

Mixed Multiscale Finite Volume Methods for Elliptic Problems in Two-Phase Flow Simulations

Lijian Jiang 1*, Ilya D. Mishev 2

1 Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN 55455, USA.
2 ExxonMobil Upstream Research Company, Houston, TX 77252, USA.

Received 17 September 2010; Accepted (in revised version) 18 March 2011
Available online 5 September 2011


We develop a framework for constructing mixed multiscale finite volume methods for elliptic equations with multiple scales arising from flows in porous media. Some of the methods developed using the framework are already known [20]; others are new. New insight is gained for the known methods and extra flexibility is provided by the new methods. We give as an example a mixed MsFV on uniform mesh in 2-D. This method uses novel multiscale velocity basis functions that are suited for using global information, which is often needed to improve the accuracy of the multiscale simulations in the case of continuum scales with strong non-local features. The method efficiently captures the small effects on a coarse grid. We analyze the new mixed MsFV and apply it to solve two-phase flow equations in heterogeneous porous media. Numerical examples demonstrate the accuracy and efficiency of the proposed method for modeling the flows in porous media with non-separable and separable scales.

AMS subject classifications: 65N30, 65N20, 34E13

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Key words: Mixed multiscale finite volume methods, elliptic equations, two-phase flows, heterogeneous porous media.

*Corresponding author.
Email: (L. Jiang), (I. D. Mishev)

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