In multi-phase flow simulation, the prevailing approach to discretizing flux terms treats the elliptic and the hyperbolic terms in the equations separately. With this concept, the flux is calculated analogously as for single phase flow, and later multiplied by the upstream mobility. This approach is valid when the mobility is a scalar quantity, which is the case for most traditional models. However, tensorial relative permeability (and thus tensorial mobilities) in general arise on all scales, as is seen in both laboratory and field experiments. Furthermore, upscaling methods almost invariably lead to anisotropic relative permeabilities. The saturation dependency of the fluid permeability tensor means that the upstream direction is no longer uniquely defined, which challenges common numerical schemes. In this work, we give examples of how the relative permeability has preferential flow directions on different scales. We then study how to incorporate tensorial relative permeability fields into control volume methods. In particular, we address two immediate challenges: Firstly, the non-determinacy of the upstream direction invalidates the use of upstream weighting for the saturation equation. We discuss the applicability of Godunov methods to handle flow cases where interfaces have either no or two upstream directions. Secondly, there is a marked increase in computational complexity associated with the pressure equation. We discuss the possibility of mitigating this computational complexity through mass lumping methods. The validity and computational efficiency of our approaches is discussed on theoretical terms and with the support of numerical implementations.

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