Multidimensional upstream weighting for multiphase transport on general grids
 Eirik Keilegavlen,
 Jeremy E. Kozdon,
 Bradley T. Mallison
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The governing equations for multiphase flow in porous media have a mixed character, with both nearly elliptic and nearly hyperbolic variables. The flux for each phase can be decomposed into two parts: (1) a geometry and rockdependent term that resembles a singlephase flux; and (2) a mobility term representing fluid properties and rock–fluid interactions. The first term is commonly discretized by two or multipoint flux approximations (TPFA and MPFA, respectively). The mobility is usually treated with singlepoint upstream weighting (SPU), also known as dimensional or donor cell upstream weighting. It is well known that when simulating processes with adverse mobility ratios, SPU suffers from grid orientation effects. An important example of this, which will be considered in this work, is the displacement of a heavy oil by water. For these adverse mobility ratio flows, the governing equations are unstable at the modeling scale, rendering a challenging numerical problem. These challenges must be addressed in order to avoid systematic biasing of simulation results. In this work, we present a framework for multidimensional upstream weighting for multiphase flow with buoyancy on general twodimensional grids. The methodology is based on a dual grid, and the resulting transport methods are provably monotone. The multidimensional transport methods are coupled with MPFA methods to solve the pressure equation. Both explicit and fully implicit approaches are considered for time integration of the transport equations. The results show considerable reduction of grid orientation effects compared to SPU, and the explicit multidimensional approach allows larger time steps. For the implicit method, the total number of nonlinear iterations is also reduced when multidimensional upstream weighting is used.
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 Title
 Multidimensional upstream weighting for multiphase transport on general grids
 Journal

Computational Geosciences
Volume 16, Issue 4 , pp 10211042
 Cover Date
 20120901
 DOI
 10.1007/s1059601293017
 Print ISSN
 14200597
 Online ISSN
 15731499
 Publisher
 Springer Netherlands
 Additional Links
 Topics
 Keywords

 MultiD transport
 Porous media
 Phasebased upwinding
 Twophase
 Interaction regions
 Monotonicity
 Hyperbolic equations
 Finite volume
 General grids
 Industry Sectors
 Authors

 Eirik Keilegavlen ^{(1)}
 Jeremy E. Kozdon ^{(2)}
 Bradley T. Mallison ^{(3)}
 Author Affiliations

 1. Department of Mathematics, University of Bergen, P.O. Box 7800, 5020, Bergen, Norway
 2. Geophysics Department, 397 Panama Mall, Stanford, CA, 94305, USA
 3. Chevron ETC, 6001 Bollinger Canyon Rd, D2104, San Ramon, CA, 945832324, USA