Errors in the Upstream Mobility Scheme for Counter-Current Two-Phase Flow With Discontinuous Permeabilities

The upstream mobility scheme (UM) is widely used to solve hyperbolic conservation laws numerically. When applied to a homogeneous porous medium this scheme has been shown convergent. When heterogeneities are introduced through the permeability, the flux function attains a spatial discontinuity. In earlier works UM for some examples of countercurrent flow has been shown to perform badly. We have looked at the performance of UM for the counter-current flow of CO2 and brine in a 1D vertical column. The solutions computed from UM are compared to the physically relevant solution found by the modified Godunov flux approximation. Through four examples we show that UM may not converge to the physically correct solution. The scheme is ill-conditioned since a small perturbation in the permeability may give a large difference in the solution. Without knowledge of the physically correct solution it is impossible to rule out the solution produced by UM. Even if UM performs well in most cases, we emphasize that there exists systems where the scheme approximates a completely different solution than the physically relevant one. Since this scheme is widely used in reservoir simulation it is important to be aware of that the scheme can perform this badly.