Accurate Cell-Centered Discretizations for Modeling Multiphase Flow in Porous Media on General Hexahedral and Simplicial Grids

Mary F. Wheeler, Guangri Xue, Ivan Yotov

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


Summary We introduce an accurate cell-centered method for modeling Darcy flow on general quadrilateral, hexahedral, and simplicial grids. We refer to these discretizations as the multipoint-flux mixed-finite-element (MFMFE) method. The MFMFE method is locally conservative with continuous fluxes and can be viewed within a variational framework as a mixed finite-element method with special approximating spaces and quadrature rules. We study two versions of the method: with a symmetric quadrature rule on smooth grids and a nonsymmetric quadrature rule on rough grids. The framework allows for handling hexahedral grids with nonplanar faces defined by trilinear mappings from the reference cube. Moreover, the MFMFE method allows for local elimination of the velocity, which leads to a cell-centered pressure system. Theoretical and numerical results demonstrate first-order convergence on rough grids. Second-order superconvergence is observed on smooth grids. We also discuss a new splitting scheme for modeling multiphase flows that can treat higher-order transport discretizations for saturations. We apply the MFMFE method to obtain physically consistent approximations to the velocity and a reference pressure on quadrilateral or hexahedral grids, and a discontinuous Galerkin method for saturations. For higher-order saturations, we propose an efficient post-processing technique that gives accurate velocities in the interior of the gridblocks. Computational results are provided for flow in highly heterogeneous reservoirs, including different capillary pressures arising from different rock types.
Original languageEnglish (US)
Pages (from-to)779-793
Number of pages15
JournalSPE Journal
Issue number3
StatePublished - 2012
Externally publishedYes

Bibliographical note

KAUST Repository Item: Exported on 2021-09-17
Acknowledged KAUST grant number(s): KUS-F1-032-04
Acknowledgements: Mary F. Wheeler is supported by the NSF-CDI under contract number DMS 0835745, DOE grant DE-FG02-04ER25617, and the Center for Frontiers of Subsurface Energy Security under Contract No. DE-SC0001114. Guangri Xue is supported by Award No. KUS-F1-032-04, made by the King Abdullah University of Science and Technology. Ivan Yotov is partially supported by the DOE grant DE-FG02-04ER25618, the NSF grant DMS 0813901, and the J. Tinsley Oden Faculty Fellowship, Institute for Computational Engineering and Sciences, University of Texas at Austin.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.


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