Multidimensional upstream weighting for multiphase transport in porous media

Truly multidimensional methods for hyperbolic equations use flow-based information to determine the computational stencil, as opposed to applying one-dimensional methods dimension by dimension. By doing this, the numerical errors are less correlated with the underlying computational grid. This can be important for reducing bias in flow problems that are inherently unstable at simulation scale, such as in certain porous media problems. In this work, a monotone, multi-D framework for multiphase flow and transport in porous media is developed. A local coupling of the fluxes is introduced through the use of interaction regions, resulting in a compact stencil. A relaxed volume formulation of the coupled hyperbolic–elliptic system is used that allows for nonzero residuals in the pressure equation to be handled robustly. This formulation ensures nonnegative masses and saturations (volume fractions) that sum to one (Acs et al., SPE J 25(4):543–553, 1985). Though the focus of the paper is on immiscible flow, an extension of the methods to a class of more general scalar hyperbolic equations is also presented. Several test problems demonstrate that the truly multi-D schemes reduce biasing due to the computational grid.